Пусть в прямоугольной системе координат Oxy алгебраическая линия второго по-рядка задана уравнением. 4. I.1. Кривые второго порядка и их свойства.


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ÌèíèñòåðñòâîîáðàçîâàíèÿèíàóêèÐîññèéñêîéÔåäåðàöèè
Ôåäåðàëüíîåãîñóäàðñòâåííîåàâòîíîìíîåîáðàçîâàòåëüíîåó÷ðåæäåíèå
âûñøåãîïðîôåññèîíàëüíîãîîáðàçîâàíèÿ
¾Êàçàíñêèé(Ïðèâîëæñêèé)ôåäåðàëüíûéóíèâåðñèòåò¿
ÈÍÑÒÈÒÓÒÌÀÒÅÌÀÒÈÊÈÈÌÅÕÀÍÈÊÈÈÌ.ËÎÁÀ×ÅÂÑÊÎÃÎ
ÊÀÔÅÄÐÀÂÛÑØÅÉÌÀÒÅÌÀÒÈÊÈÈÌÀÒÅÌÀÒÈ×ÅÑÊÎÃÎÌÎÄÅËÈÐÎÂÀÍÈß
Íàïðàâëåíèå:050100.68:Ïåäàãîãè÷åñêîåîáðàçîâàíèå
Ïðîôèëü:Èíôîðìàöèîííûåòåõíîëîãèèâôèçèêî-ìàòåìàòè÷åñêîìîáðàçîâàíèè
ÂÛÏÓÑÊÍÀßÊÂÀËÈÔÈÊÀÖÈÎÍÍÀßÐÀÁÎÒÀ
ÌÀÃÈÑÒÅÐÑÊÀßÄÈÑÑÅÐÒÀÖÈß
¾Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåäâèæåíèÿíåáåñíûõòåë
ñïîìîùüþïàêåòàïðîãðàììMaple¿
Ðàáîòàçàâåðøåíà:
¾
¿
2014ã.
(ÀñëàíÎñìàí.)
Ðàáîòàäîïóùåíàêçàùèòå:
Íàó÷íûéðóêîâîäèòåëü,
äîêòîðôèç.-ìàò.íàóê
¾
¿
2014ã.
(ÑóøêîâÑ.Â.)
Ðåöåíçåíò:
Ïðîô.êàôåäðûòåîðèèîòíîñèòåëüíîñòè
èãðàâèòàöèè
¾
¿
2014ã.
(ÕóñíóòäèíîâÍ.Ð.)
Äàòàçàùèòû:
¾
¿
2014ã.Îöåíêà
Çàâåäóþùèéêàôåäðîé
Äîêòîðôèç.-ìàò.íàóê,ïðîôåññîð
¾
¿
2014ã.
(ÈãíàòüåâÞ.Ã.)
Êàçàíü2014ãîä
Îãëàâëåíèå
Ââåäåíèå
2
IÊðèâûåâòîðîãîïîðÿäêà
3
I.1Êðèâûåâòîðîãîïîðÿäêàèèõñâîéñòâà
...................
3
I.2Ýëëèïñ
.....................................
12
I.3Ãèïåðáîëà
....................................
17
I.4Ïàðàáîëà
....................................
23
I.5Íåêîòîðûåîáùèåñâîéñòâàýëëèïñà,ãèïåðáîëû,ïàðàáîëû
........
27
IIÌàòåìàòè÷åñêèåîñíîâûäâèæåíèÿâïîëåòÿæåñòè
31
II.1Äâèæåíèåìàòåðèàëüíîéòî÷êèïîääåéñòâèåìñèëûòÿãîòåíèÿ(çàäà÷à
äâóõòåë)
....................................
31
II.2ÄâèæåíèåèñêóññòâåííûõñïóòíèêîâÇåìëè
.................
36
IIIÌàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
41
III.1Êðèâûåâòîðîãîïîðÿäêà
...........................
41
III.2Äâèæåíèåíåáåñíûõòåë
............................
51
III.3ÄâèæåíèåèñêóññòâåííûõñïóòíèêîâÇåìëè
.................
65
Çàêëþ÷åíèå
70
Ëèòåðàòóðà
72
Ââåäåíèå
Çàäà÷èîäâèæåíèèòåëàâîâíåøíåìïîëåòÿãîòåíèÿèîäâèæåíèèäâóõòåëïîä
äåéñòâèåìâçàèìíîéñèëûòÿãîòåíèÿïðåäñòàâëÿþòîäíóèçöåíòðàëüíûõïðîáëåìêëàñ-
ñè÷åñêîéôèçèêè.Âàæíåéøèìïðèëîæåíèåìýòîéïðîáëåìûÿâëÿåòñÿîïèñàíèåäâèæå-
íèÿïëàíåòâÑîëíå÷íîéñèñòåìå,àòàêæåîïèñàíèåäâèæåíèÿèñêóññòâåííûõñïóòíèêîâ
Çåìëè.Ôåíîìåíîëîãè÷åñêèåçàêîíûäâèæåíèÿïëàíåòáûëèïîëó÷åíûÊåïëåðîìâ17
âåêå.ÏîçæåÍüþòîíïîëó÷èëìàòåìàòè÷åñêîåîáîñíîâàíèåçàêîíîâÊåïëåðàíàîñíîâå
òåîðèèïîñòðîåííîéèìòåîðèèòÿãîòåíèÿ.
Âñâîåéäèññåðòàöèîííîéðàáîòåÿñòàâëþöåëüþèçó÷åíèåäâèæåíèÿíåáåñíûõòåë
ñïîìîùüþêîìïüþòåðíîãîìîäåëèðîâàíèÿ.Äëÿäîñòèæåíèÿýòîéöåëèñòàâÿòñÿñëåäó-
þùèåçàäà÷è:

Èçó÷èòüîñíîâíûåçàêîíûäâèæåíèÿíåáåñíûõòåë(çàêîíòÿãîòåíèÿ,çàêîíûÊåïëå-
ðà)

Èçó÷èòüâîçìîæíîñòèïàêåòàMapleäëÿêîìïüþòåðíîãîìîäåëèðîâàíèÿ

ÑîçäàòüïðîãðàììóïîñòðîåíèÿêðèâûõâòîðîãîïîðÿäêàâïàêåòåMaple

Ðàçðàáîòàòüêîìïüþòåðíûåìîäåëèäâèæåíèÿíåáåñíûõòåë,èëëþñòðèðóþùèåðàç-
ëè÷íûåçàêîíûÊåïëåðà

ÏîñòðîèòüêîìïüþòåðíûåìîäåëèäâèæåíèÿèñêóññòâåííûõñïóòíèêîâÇåìëè
2
ÃëàâàI
Êðèâûåâòîðîãîïîðÿäêà
I.1Êðèâûåâòîðîãîïîðÿäêàèèõñâîéñòâà
I.1.1Êàíîíè÷åñêèåóðàâíåíèÿëèíèéâòîðîãîïîðÿäêà
Àëãåáðàè÷åñêîéëèíèåéâòîðîãîïîðÿäêàíàçûâàåòñÿãåîìåòðè÷åñêîåìåñòîòî÷åê
ïëîñêîñòè,êîòîðîåâêàêîé-ëèáîàôôèííîéñèñòåìåêîîðäèíàò
Ox
1
x
2
ìîæåòáûòüçà-
äàíîóðàâíåíèåìâèäà
p
(
x
1
;x
2
)=0
;
ãäå
p
(
x
1
;x
2
)
ìíîãî÷ëåíâòîðîéñòåïåíèäâóõïåðåìåííûõ
x
1
,
x
2
:
Òðåáóåòñÿíàéòèïðÿìîóãîëüíóþñèñòåìóêîîðäèíàò,âêîòîðîéóðàâíåíèåëèíèè
ïðèíÿëîáûíàèáîëååïðîñòîéâèä.
Ðåçóëüòàòîìðåøåíèÿïîñòàâëåííîéçàäà÷èÿâëÿåòñÿñëåäóþùàÿîñíîâíàÿòåîðåìà.
Òåîðåìà.1:(Êëàññèôèêàöèÿàëãåáðàè÷åñêèõëèíèéâòîðîãîïîðÿäêà).
Äëÿ
ëþáîéàëãåáðàè÷åñêîéëèíèèâòîðîãîïîðÿäêàñóùåñòâóåòïðÿìîóãîëüíàÿñèñòåìàêî-
îðäèíàò
Oxy;
âêîòîðîéóðàâíåíèåýòîéëèíèèïðèíèìàåòîäèíèçñëåäóþùèõäåâÿòè
êàíîíè÷åñêèõâèäîâ:
Òàáëèöà.1
:Êàíîíè÷åñêèåóðàâíåíèÿëèíèéâòîðîãîïîðÿäêà
1.
x
2
a
2
+
y
2
b
2
=1
óðàâíåíèåýëëèïñà;
2.
x
2
a
2
+
y
2
b
2
=

1
óðàâíåíèåìíèìîãîýëëèïñà;
3.
x
2
a
2
+
y
2
b
2
=0
óðàâíåíèåïàðûìíèìûõïåðå-
ñåêàþùèõñÿïðÿìûõ;
3
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
4.
x
2
a
2

y
2
b
2
=1
óðàâíåíèåãèïåðáîëû;
5.
x
2
a
2

y
2
b
2
=0
óðàâíåíèåïàðûïåðåñåêàþùèõ-
ñÿïðÿìûõ;
6.
y
2
=2

p

x
óðàâíåíèåïàðàáîëû;
7.
y
2

b
2
=0
óðàâíåíèåïàðûïàðàëëåëüíûõ
ïðÿìûõ;
8.
y
2
+
b
2
=0
óðàâíåíèåïàðûìíèìûõïàðàë-
ëåëüíûõïðÿìûõ;
9.
y
2
=0
óðàâíåíèåïàðûìíèìûõñîâïà-
äàþùèõïðÿìûõ.
Òåîðåìà1.äà¼ò
àíàëèòè÷åñêèåîïðåäåëåíèÿ
ëèíèéâòîðîãîïîðÿäêà.Ëèíèè(1),
(4),(5),(6),(7),(9)
íàçûâàþòñÿâåùåñòâåííûìè(äåéñòâèòåëüíûìè)
,àëèíèè
(2),(3),(8)
ìíèìûìè
.
Ïðèâåäåìäîêàçàòåëüñòâîòåîðåìû,ïîñêîëüêóîíîôàêòè÷åñêèñîäåðæèòàëãîðèòì
ðåøåíèÿïîñòàâëåííîéçàäà÷è.
Áåçîãðàíè÷åíèÿîáùíîñòèìîæíîïðåäïîëàãàòü,÷òîóðàâíåíèåëèíèèâòîðîãîïî-
ðÿäêàçàäàíîâïðÿìîóãîëüíîéñèñòåìåêîîðäèíàò
Oxy:
Âïðîòèâíîìñëó÷àåìîæíîïå-
ðåéòèîòíåïðÿìîóãîëüíîéñèñòåìûêîîðäèíàò
Ox
1
x
2
êïðÿìîóãîëüíîé
Oxy;
ïðèýòîì
óðàâíåíèåëèíèèáóäåòèìåòüòîòæåâèäèòóæåñòåïåíüñîãëàñíîòåîðåìåîáèíâàðè-
àíòíîñòèïîðÿäêààëãåáðàè÷åñêîéëèíèè.
Ïóñòüâïðÿìîóãîëüíîéñèñòåìåêîîðäèíàò
Oxy
àëãåáðàè÷åñêàÿëèíèÿâòîðîãîïî-
ðÿäêàçàäàíàóðàâíåíèåì
4
I.1.Êðèâûåâòîðîãîïîðÿäêàèèõñâîéñòâà
a
11
x
2
+2
a
12
xy
+
a
22
y
2
+2
a
1
x
+2
a
2
y
+
a
0
=0
;
(I.1)
âêîòîðîìõîòÿáûîäèíèçñòàðøèõêîýôôèöèåíòîâ
a
11
;a
12
;a
22
îòëè÷åíîòíóëÿ,ò.å.
ëåâàÿ÷àñòü(
I.1
)ìíîãî÷ëåíäâóõïåðåìåííûõ
x
,
y
âòîðîéñòåïåíè.Êîýôôèöèåíòûïðè
ïåðâûõñòåïåíÿõïåðåìåííûõ
x
è
y
,àòàêæåïðèèõïðîèçâåäåíèè
x

y
âçÿòûóäâîåííûìè
ïðîñòîäëÿóäîáñòâàäàëüíåéøèõïðåîáðàçîâàíèé.
Äëÿïðèâåäåíèÿóðàâíåíèÿ(
I.1
)êêàíîíè÷åñêîìóâèäóèñïîëüçóþòñÿñëåäóþùèå
ïðåîáðàçîâàíèÿïðÿìîóãîëüíûõêîîðäèíàò:
ïîâîðîòíàóãîë
'
(
x
=
x
0

cos
'

y
0

sin
';
y
=
x
0

sin
'
+
y
0

cos
'
;
(I.2)
ïàðàëëåëüíûéïåðåíîñ
(
x
=
x
0
+
x
0
;
y
=
y
0
+
y
0
;
(I.3)
èçìåíåíèåíàïðàâëåíèéêîîðäèíàòíûõîñåé(îòðàæåíèÿâêîîðäèíàòíûõîñÿõ):
îñèîðäèíàò
(
x
=
x
0
;
y
=

y
0
;
îñèàáñöèññ
(
x
=

x
0
;
y
=
y
0
;
îáåèõîñåé
(
x
=

x
0
;
y
=

y
0
;
(I.4)
ïåðåèìåíîâàíèåêîîðäèíàòíûõîñåé(îòðàæåíèåâïðÿìîé
y
=
x
)
(
x
=
y
0
;
y
=
x
0
;
(I.5)
ãäå
x;y
è
x
0
;y
0
êîîðäèíàòûïðîèçâîëüíîéòî÷êèâñòàðîé
(
Oxy
)
èíîâîé
(
O
0
x
0
y
0
)
ñèñòåìàõêîîðäèíàòñîîòâåòñòâåííî.
Êðîìåïðåîáðàçîâàíèÿêîîðäèíàòîáå÷àñòèóðàâíåíèÿìîæíîóìíîæàòüíàîòëè÷íîå
îòíóëÿ÷èñëî.
Ðàññìîòðèìñíà÷àëà÷àñòíûåñëó÷àè,êîãäàóðàâíåíèå(
I.1
)èìååòâèä:
(
I
)
:

2

y
2
+
a
0
=0
;
2
6
=0;
(
II
)
:

2

y
2
+2

a
1

x
=0
;
2
6
=0
;a
1
6
=0;
(
III
)
:

2

y
2
+

1

x
2
+
a
0
=0
;
1
6
=0
;
2
6
=0
:
Ýòèóðàâíåíèÿ(òàêæåìíîãî÷ëåíûâëåâûõ÷àñòÿõ)íàçûâàþòñÿ
ïðèâåä¼ííûìè
.
Ïîêàæåì,÷òîïðèâåä¼ííûåóðàâíåíèÿ(I),(II),(III)ñâîäÿòñÿêêàíîíè÷åñêèì(1)(9).
Óðàâíåíèå(I)
.Åñëèâóðàâíåíèè(I)ñâîáîäíûé÷ëåíðàâåííóëþ
(
a
0
=0)
;
òî,
ðàçäåëèâîáå÷àñòèóðàâíåíèÿ

2

y
2
=0
íàñòàðøèéêîýôôèöèåíò
(

0
6
=0)
;
ïîëó÷èì
y
2
=0

óðàâíåíèåäâóõñîâïàäàþùèõïðÿìûõ
(9),ñîäåðæàùèõîñüàáñöèññ
(
y
=0)
:
Åñëèæåñâîáîäíûé÷ëåíîòëè÷åíîòíóëÿ
(
a
0
6
=0)
;
òîðàçäåëèìîáå÷àñòèóðàâíåíèÿ
(I)íàñòàðøèéêîýôôèöèåíò
(

2
6
=0)
:
y
2
+
a
0

2
=0
:
Åñëèâåëè÷èíà
a
0

2
îòðèöàòåëüíàÿ,
òî,îáîçíà÷èâå¼÷åðåç

b
2
;
ãäå
b
=
q

a
0

2
;
ïîëó÷àåì
y
2

b
2
=0

óðàâíåíèåïàðû
5
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
ïàðàëëåëüíûõïðÿìûõ
(7):
y
=
b
èëè
y
=

b:
Åñëèæåâåëè÷èíà
a
0

2
ïîëîæèòåëüíàÿ,òî,
îáîçíà÷èâå¼÷åðåç
b
2
;
ãäå
b
=
q
a
0

2
;
ïîëó÷àåì
y
2
+
b
2
=0

óðàâíåíèåïàðûìíèìûõ
ïàðàëëåëüíûõïðÿìûõ
(8).Ýòîóðàâíåíèåíåèìååòäåéñòâèòåëüíûõðåøåíèé,ïîýòîìó
íàêîîðäèíàòíîéïëîñêîñòèíåòòî÷åê,îòâå÷àþùèõýòîìóóðàâíåíèþ.Îäíàêîâîáëàñòè
êîìïëåêñíûõ÷èñåëóðàâíåíèå
y
2
+
b
2
=0
èìååòäâàñîïðÿæ¼ííûõðåøåíèÿ
y
=

ib;
êîòîðûåèëëþñòðèðóþòñÿøòðèõîâûìèëèíèÿìè(ñì.ïóíêò8òåîðåìû1.).
Óðàâíåíèå(II).
Ðàçäåëèìóðàâíåíèåíàñòàðøèéêîýôôèöèåíò
(

2
6
=0)
èïåðåíå-
ñåìëèíåéíûé÷ëåíâïðàâóþ÷àñòü:
y
2
=

2
a
1

2
x:
Åñëèâåëè÷èíà
a
1

2
îòðèöàòåëüíàÿ,òî,
îáîçíà÷àÿ
p
=

a
1

2

0
;
ïîëó÷àåì
y
2
=2

p

x

óðàâíåíèåïàðàáîëû
(6).Åñëèâåëè÷èíà
a
1

2
ïîëîæèòåëüíàÿ,òî,èçìåíÿÿíàïðàâëåíèåîñèàáñöèññ,ò.å.âûïîëíÿÿâòîðîåïðåîá-
ðàçîâàíèåâ
I.4
,ïîëó÷àåìóðàâíåíèå
(
y
0
)
2
=
2
a
1

2
x
0
èëè
(
y
0
)
2
=2

p

x
0
;
ãäå
p
=
a
1

2

0
:
Ýòîóðàâíåíèåïàðàáîëûâíîâîéñèñòåìåêîîðäèíàò
O
0
x
0
y
0
:
Óðàâíåíèå(III).
Âîçìîæíûäâàñëó÷àÿ:ëèáîñòàðøèåêîýôôèöèåíòûîäíîãîçíàêà
(
ýëëèïòè÷åñêèéñëó÷àé
),ëèáîïðîòèâîïîëîæíûõçíàêîâ(
ãèïåðáîëè÷åñêèéñëó-
÷àé
).
Â
ýëëèïòè÷åñêîìñëó÷àå
(

1


2

0)
ïðè
a
0
6
=0
ïåðåíîñèìñâîáîäíûé÷ëåíâ
ïðàâóþ÷àñòüèäåëèìîáå÷àñòèíà

a
0
6
=0
:
(
III
)
,

1

x
2
+

2

y
2
=

a
0
,

1

a
0

x
2
+

2

a
0

y
2
=1
:
Åñëèçíàêñòàðøèõêîýôôèöèåíòîâ

1
;
2
ïðîòèâîïîëîæåíçíàêó
a
0
;
òî,îáîçíà÷àÿ
ïîëîæèòåëüíûåâåëè÷èíû

a
0

1
è

a
0

2
÷åðåç
a
2
è
b
2
;
ïîëó÷àåì
x
2
a
2
+
y
2
b
2
=1

óðàâíåíèå
ýëëèïñà
(1).
Åñëèçíàêñòàðøèõêîýôôèöèåíòîâ

1
;
2
ñîâïàäàåòñîçíàêîì
a
0
;
òî,îáîçíà÷àÿïî-
ëîæèòåëüíûåâåëè÷èíû
a
0

1
è
a
0

2
÷åðåç
a
2
è
b
2
;
ïîëó÷àåì-
x
2
a
2

y
2
b
2
=1
,
x
2
a
2
+
y
2
b
2
=

1

óðàâíåíèåìíèìîãîýëëèïñà
(2).Ýòîóðàâíåíèåíåèìååòäåéñòâèòåëüíûõðåøåíèé.
Îäíàêîîíîèìååòðåøåíèÿâîáëàñòèêîìïëåêñíûõ÷èñåë,êîòîðûåèëëþñòðèðóþòñÿ
øòðèõîâîéëèíèåé(ñì.ïóíêò2òåîðåìû1.).
Ìîæíîñ÷èòàòü,÷òîâóðàâíåíèÿõýëëèïñà(äåéñòâèòåëüíîãîèëèìíèìîãî)êîýôôè-
öèåíòûóäîâëåòâîðÿþòíåðàâåíñòâó
a

b;
âïðîòèâíîìñëó÷àåýòîãîìîæíîäîáèòüñÿ,
ïåðåèìåíîâûâàÿêîîðäèíàòíûåîñè,ò.å.äåëàÿïðåîáðàçîâàíèå
I.5
ñèñòåìûêîîðäèíàò.
Åñëèñâîáîäíûé÷ëåíóðàâíåíèÿ(III)ðàâåííóëþ
(
a
0
=0)
;
òî,îáîçíà÷àÿïîëîæè-
òåëüíûåâåëè÷èíû
1
j

1
j
è
1
j

2
j
÷åðåç
a
2
è
b
2
;
ïîëó÷àåì
x
2
a
2
+
y
2
b
2
=0

óðàâíåíèåïàðû
ìíèìûõïåðåñåêàþùèõñÿïðÿìûõ
(3).Ýòîìóóðàâíåíèþóäîâëåòâîðÿåòòîëüêîòî÷êà
ñêîîðäèíàòàìè
x
=0
è
y
=0
;
ò.å.òî÷êà
O
íà÷àëîêîîðäèíàò.Îäíàêîâîáëàñòè
êîìïëåêñíûõ÷èñåëëåâóþ÷àñòüóðàâíåíèÿìîæíîðàçëîæèòüíàìíîæèòåëè
x
2
a
2
+
y
2
b
2
=

y
b
+
i
x
a

y
b

i
x
a

;
6
I.1.Êðèâûåâòîðîãîïîðÿäêàèèõñâîéñòâà
ïîýòîìóóðàâíåíèåèìååòñîïðÿæåííûåðåøåíèÿ
y
=

i
b
a

x;
êîòîðûåèëëþñòðèðóþòñÿ
øòðèõîâûìèëèíèÿìè,ïåðåñåêàþùèìèñÿâíà÷àëåêîîðäèíàò(ñì.ïóíêò3òåîðåìû1.).
Â
ãèïåðáîëè÷åñêîìñëó÷àå
(

1


2

0)
ïðè
a
0
6
=0
ïåðåíîñèìñâîáîäíûé÷ëåíâ
ïðàâóþ÷àñòüèäåëèìîáå÷àñòèíà

a
0
6
=0
:
(
III
)
,

1

x
2
+

2

y
2
=

a
0
,

1

a
0

x
2
+

2

a
0

y
2
=1
:
Âåëè÷èíû

a
0

1
è

a
0

2
èìåþòïðîòèâîïîëîæíûåçíàêè.Áåçîãðàíè÷åíèÿîáùíîñòè
ñ÷èòàåì,÷òîçíàê

2
ñîâïàäàåòñîçíàêîìñâîáîäíîãî÷ëåíà
a
0
;
ò.å.
a
0

2

0
:
Âïðîòèâíîì
ñëó÷àåíóæíîïåðåèìåíîâàòüêîîðäèíàòíûåîñè,ò.å.ñäåëàòüïðåîáðàçîâàíèå
I.5
ñèñòå-
ìûêîîðäèíàò.Îáîçíà÷àÿïîëîæèòåëüíûåâåëè÷èíû

a
0

1
è
a
0

2
÷åðåç
a
2
è
b
2
;
ïîëó÷àåì
x
2
a
2

y
2
b
2
=1

óðàâíåíèåãèïåðáîëû
(4).
Ïóñòüâóðàâíåíèè(III)ñâîáîäíûé÷ëåíðàâåííóëþ
(
a
0
=0)
:
Òîãäàìîæíîñ÷èòàòü,
÷òî

1

0
;
2

0
(âïðîòèâíîìñëó÷àåîáå÷àñòèóðàâíåíèÿóìíîæèìíà1).Îáî-
çíà÷àÿïîëîæèòåëüíûåâåëè÷èíû
1

1
è

1

2
÷åðåç
a
2
è
b
2
;
ïîëó÷àåì
x
2
a
2

y
2
b
2
=0

óðàâíåíèåïàðûïåðåñåêàþùèõñÿïðÿìûõ
(5).Óðàâíåíèÿïðÿìûõíàõîäÿòñÿâðåçóëüòà-
òåðàçëîæåíèÿíàìíîæèòåëèëåâîé÷àñòèóðàâíåíèÿ
x
2
a
2

y
2
b
2
=

x
a

y
b

x
a
+
y
b

=0
;
ò.å.
y
=

b
a

x:
Òàêèìîáðàçîì,ïðèâåä¼ííûåóðàâíåíèÿ(I),(II),(III)àëãåáðàè÷åñêîéëèíèèâòîðîãî
ïîðÿäêàñâîäÿòñÿêîäíîìóèçêàíîíè÷åñêèõâèäîâ(1)(9),ïåðå÷èñëåííûõâ(òåîðåìå
1.)
Îñòàëîñüïîêàçàòü,÷òîîáùååóðàâíåíèå
I.1
ìîæíîñâåñòèêïðèâåä¼ííûìïðèïî-
ìîùèïðåîáðàçîâàíèéïðÿìîóãîëüíîéñèñòåìûêîîðäèíàò.
Óïðîùåíèåîáùåãîóðàâíåíèÿ
I.1
ïðîèçâîäèòñÿâäâàýòàïà.Íàïåðâîìýòàïåïðè
ïîìîùèïîâîðîòàñèñòåìûêîîðäèíàò¾óíè÷òîæàåòñÿ¿÷ëåíñïðîèçâåäåíèåìíåèçâåñò-
íûõ.Åñëèïðîèçâåäåíèÿíåèçâåñòíûõíåò
(
a
12
=0)
;
òîïîâîðîòäåëàòüíåíàäî(âýòîì
ñëó÷àåïåðåõîäèìñðàçóêîâòîðîìóýòàïó).Íàâòîðîìýòàïåïðèïîìîùèïàðàëëåëüíîãî
ïåðåíîñà"óíè÷òîæàþòñÿ"îäèíèëèîáà÷ëåíàïåðâîéñòåïåíè.Âðåçóëüòàòåïîëó÷àþòñÿ
ïðèâåä¼ííûå
óðàâíåíèÿ(I),(II),(III).
Ïåðâûéýòàï:
ïðåîáðàçîâàíèåóðàâíåíèÿëèíèèâòîðîãîïîðÿäêàïðèïîâîðîòåïðÿ-
ìîóãîëüíîéñèñòåìûêîîðäèíàò.
Åñëèêîýôôèöèåíò
a
12
6
=0
;
âûïîëíèìïîâîðîòñèñòåìûêîîðäèíàòíàóãîë
':
Ïîä-
ñòàâëÿÿâûðàæåíèÿ
I.2
âóðàâíåíèå
I.1
,ïîëó÷àåì:
a
11

(
x
0
cos
'

y
0
sin
'
)
2
+2

a
12

(
x
0
cos
'

y
0
sin
'
)

(
x
0
sin
'
+
y
0
cos
'
)+
+
a
22

(
x
0
sin
'
+
y
0
cos
'
)
2
+2

a
1
(
x
0
cos
'

y
0
sin
'
)+
+2

a
2

(
x
0
sin
'
+
y
0
cos
'
)+
a
0
=0
:
7
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
Ïðèâîäÿïîäîáíûå÷ëåíû,ïðèõîäèìêóðàâíåíèþâèäà
I.1
:
a
0
11

(
x
0
)
2
+2

a
0
12

x
0

y
0
+
a
0
22

(
y
0
)
2
+2

a
0
1

x
0
+2

a
0
2

y
0
+
a
0
0
=0
;
(I.6)
ãäå
a
0
11
=
a
11

cos
2
'
+2

a
12
cos
'

sin
'
+
a
22

sin
2
'
;
a
0
12
=

a
11

cos
'

sin
'
+
a
12

(cos
2
'

sin
2
'
)+
a
22

cos
'

sin
'
;
a
0
22
=
a
11

sin
2
'

2

a
12

cos
'

sin
'
+
a
22

cos
2
'
;
a
0
1
=
a
1

cos
'
+
a
2

sin
'
;
a
0
2
=

a
1

sin
'
+
a
2

cos
'
;
a
0
0
=
a
0
:
Îïðåäåëèìóãîë
'
òàê,÷òîáû
a
0
12
=0
:
Ïðåîáðàçóåìâûðàæåíèåäëÿ
a
0
12
;
ïåðåõîäÿê
äâîéíîìóóãëó:
a
0
12
=

1
2

a
11

sin2
'
+
a
12

cos2
'
+
1
2

a
22

sin2
'
=
a
22

a
11
2

sin2
'
+
a
12

cos2
':
Óãîë
'
äîëæåíóäîâëåòâîðÿòüîäíîðîäíîìóòðèãîíîìåòðè÷åñêîìóóðàâíåíèþ
a
22

a
11
2

sin2
'
+
a
12

cos2
'
=0
;
êîòîðîåðàâíîñèëüíîóðàâíåíèþ
ctg2
'
=
a
11

a
22
2

a
12
;
(I.7)
ïîñêîëüêó
a
12
6
=0
:
Ýòîóðàâíåíèåèìååòáåñêîíå÷íîåêîëè÷åñòâîêîðíåé
'
=
1
2
arcctg
a
11

a
22
2

a
12
+

2
n;n
2
Z
:
Âûáåðåìëþáîéèçíèõ,íàïðèìåð,óãîë
'
èçèíòåðâàëà
0
'

2
:
Òîãäàâóðàâíåíèè
I.6
èñ÷åçíåò÷ëåí
2

a
0
12

x
0

y
0
;
ïîñêîëüêó
a
0
12
=0
:
Îáîçíà÷èâîñòàâøèåñÿñòàðøèå
êîýôôèöèåíòû÷åðåç

1
=
a
0
11
è

2
=
a
0
22
;
ïîëó÷èìóðàâíåíèå

1

(
x
0
)
2
+

2

(
y
0
)
2
+2

a
0
1

x
0
+2

a
0
2

y
0
+
a
0
0
=0
:
(I.8)
Óðàâíåíèå
I.8
ÿâëÿåòñÿóðàâíåíèåìâòîðîéñòåïåíè(ïðèïðåîáðàçîâàíèè
I.2
ïîðÿäîê
ëèíèèñîõðàíÿåòñÿ),ò.å.õîòÿáûîäèíèçñòàðøèõêîýôôèöèåíòîâ

1
èëè

2
îòëè÷åíîò
íóëÿ.Äàëååáóäåìñ÷èòàòü,÷òîèìåííîêîýôôèöèåíòïðè
(
y
0
)
2
íåðàâåííóëþ
(

2
6
=0)
:
Âïðîòèâíîìñëó÷àå(ïðè

2
=0
è

1
6
=0
)ñëåäóåòñäåëàòüïîâîðîòñèñòåìûêîîðäèíàò
íàóãîë
'
+

2
;
êîòîðûéòàêæåóäîâëåòâîðÿåòóñëîâèþ
I.7
.Òîãäàâìåñòîêîîðäèíàò
x
0
,
y
0
â
I.8
ïîëó÷èì
y
0
,

x
0
ñîîòâåòñòâåííî,ò.å.îòëè÷íûéîòíóëÿêîýôôèöèåíò

1
áóäåò
ïðè
(
y
0
)
2
:
Âòîðîéýòàï
:ïðåîáðàçîâàíèåóðàâíåíèÿëèíèèâòîðîãîïîðÿäêàïðèïàðàëëåëüíîì
ïåðåíîñåïðÿìîóãîëüíîéñèñòåìûêîîðäèíàò.
Óðàâíåíèå
I.8
ìîæíîóïðîñòèòü,âûäåëÿÿïîëíûåêâàäðàòû.Íóæíîðàññìîòðåòü
äâàñëó÷àÿ:

1
6
=0
èëè

1
=0
(ñîãëàñíîïðåäïîëîæåíèþ

2
6
=0)
;
êîòîðûåíàçûâàþòñÿ
öåíòðàëüíûé
(âêëþ÷àþùèéýëëèïòè÷åñêèéèãèïåðáîëè÷åñêèéñëó÷àè)èëè
ïàðàáî-
ëè÷åñêèé
ñîîòâåòñòâåííî.Ãåîìåòðè÷åñêèéñìûñëýòèõíàçâàíèéðàñêðûâàåòñÿâäàëü-
íåéøåì.
8
I.1.Êðèâûåâòîðîãîïîðÿäêàèèõñâîéñòâà
Öåíòðàëüíûéñëó÷àé:

1
6
=0
è

2
6
=0
:
Âûäåëÿÿïîëíûåêâàäðàòûïîïåðåìåííûì
x
0
,
y
0
;
ïîëó÷àåì

1
"
(
x
0
)
2
+2
a
0
1

1
x
0
+

a
0
1

1

2
#
+

2
"
(
y
0
)
2
+2
a
0
2

2
y
0
+

a
0
2

2

2
#



1

a
0
1

1

2


2

a
0
2

2

2
+
a
0
0
=0
,
,

1

x
0
+
a
0
1

1

2
+

2

y
0
+
a
0
2

2

2


1

a
0
1

1

2


2

a
0
2

2

2
+
a
0
0
=0
:
Ïîñëåçàìåíûïåðåìåííûõ
8







:
x
00
=
x
0
+
a
0
1

1
;
y
00
=
y
0
+
a
0
2

2
;
(I.9)
ïîëó÷àåìóðàâíåíèå

1
(
x
00
)
2
+

2
(
y
00
)
2
+
a
00
0
=0
;
(I.10)
ãäå
a
00
0
=


1

a
0
1

1

2


2

a
0
2

2

2
+
a
0
0
:
Ïàðàáîëè÷åñêèéñëó÷àé:

1
=0
è

2
6
=0
:
Âûäåëÿÿïîëíûéêâàäðàòïîïåðåìåííîé
y
0
,ïîëó÷àåì

2

"
(
y
0
)
2
+2

a
0
2

2

y
0
+

a
0
2

2

2
#
+2

a
0
1

x
0


2

a
0
2

2

2
+
a
0
0
=0
,
,

2

y
0
+
a
0
2

2

2
+2

a
0
1

x
0


2

a
0
2

2

2
+
a
0
0
=0
:
(I.11)
Åñëè
a
0
1
6
=0
,òîïîñëåäíååóðàâíåíèåïðèâîäèòñÿêâèäó

2

y
0
+
a
0
2

2

2
+2

a
0
1

"
x
0
+
a
0
0
2

a
0
1


2
2

a
0
1


a
0
2

2

2
#
=0
:
Ñäåëàâçàìåíóïåðåìåííûõ
8







:
x
00
=
x
0
+
a
0
0
2

a
0
1


2
2

a
0
1


a
0
2

2

2
;
y
00
=
y
0
+
a
0
2

2
;
(I.12)
ïîëó÷èì,

2

(
y
00
)
2
+2

a
00
1

x
00
=0
;
(I.13)
9
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
ãäå
a
00
1
=
a
0
1
.
Åñëè
a
0
1
=0
,òîóðàâíåíèå
I.11
ïðèâîäèòñÿêâèäó,

2

(
y
00
)
2
+
a
00
0
=0
;
(I.14)
ãäå
a
00
0
=


2

a
0
2

2

2
+
a
0
0
=0
;
8



:
x
00
=
x
0
;
y
00
=
y
0
+
a
0
2

2
:
(I.15)
Çàìåíûïåðåìåííûõ
I.9
,
I.12
,
I.15
ñîîòâåòñòâóþòïàðàëëåëüíîìóïåðåíîñóñèñòåìûêî-
îðäèíàò
Ox
0
y
0
(ñì.ïóíêò1"a"çàìå÷àíèé2.3).
Òàêèìîáðàçîì,ïðèïîìîùèïàðàëëåëüíîãîïåðåíîñàñèñòåìûêîîðäèíàò
O
0
x
0
y
0
ïî-
ëó÷àåìíîâóþñèñòåìóêîîðäèíàò
O
00
x
00
y
00
,âêîòîðîéóðàâíåíèåëèíèèâòîðîãîïîðÿäêà
ïðèíèìàåòâèä
I.9
,èëè
I.13
,èëè
I.14
.Ýòèóðàâíåíèÿÿâëÿþòñÿïðèâåä¼ííûìè(âèäà
(III),(II)èëè(I)ñîîòâåòñòâåííî).
Îñíîâíàÿòåîðåìà1.îïðèâåäåíèèóðàâíåíèÿàëãåáðàè÷åñêîéëèíèèâòîðîãîïîðÿä-
êàêêàíîíè÷åñêîìóâèäóäîêàçàíà.
Çàìå÷àíèÿ1.
1.Ñèñòåìàêîîðäèíàò,âêîòîðîéóðàâíåíèåàëãåáðàè÷åñêîéëèíèèâòîðîãîïîðÿäêà
èìååòêàíîíè÷åñêèéâèä,íàçûâàåòñÿ
êàíîíè÷åñêîé
.Êàíîíè÷åñêàÿñèñòåìàêîîðäèíàò
îïðåäåëÿåòñÿíåîäíîçíà÷íî.Íàïðèìåð,èçìåíÿÿíàïðàâëåíèåîñèîðäèíàòíàïðîòèâîïî-
ëîæíîå,ñíîâàïîëó÷àåìêàíîíè÷åñêóþñèñòåìóêîîðäèíàò,òàêêàêçàìåíàïåðåìåííîé
y
íà
(

y
)
íåèçìåíÿåòóðàâíåíèé(1)(9).Ïîýòîìóîðèåíòàöèÿêàíîíè÷åñêîéñèñòåìûêî-
îðäèíàòíåèìååòïðèíöèïèàëüíîãîçíà÷åíèÿ,å¼âñåãäàìîæíîñäåëàòüïðàâîé,èçìåíèâ
ïðèíåîáõîäèìîñòèíàïðàâëåíèåîñèîðäèíàò.
2.Ðàíååïîêàçàíî,÷òîïðåîáðàçîâàíèÿïðÿìîóãîëüíûõñèñòåìêîîðäèíàòíàïëîñêî-
ñòèñâîäÿòñÿêîäíîìóèçïðåîáðàçîâàíèé:
(
x
=
x
0
+
x
0

cos
'

y
0

sin
';
y
=
y
0
+
x
0

sin
'
+
y
0

cos
';
èëè
(
x
=
x
0
+
x
0

cos
'
+
y
0

sin
';
y
=
y
0
+
x
0

sin
'

y
0

cos
':
Ïîýòîìóçàäà÷àïðèâåäåíèÿóðàâíåíèÿëèíèèâòîðîãîïîðÿäêàêêàíîíè÷åñêîìóâè-
äóñâîäèòñÿêíàõîæäåíèþíà÷àëà
O
0
(
x
0
;y
0
)
êàíîíè÷åñêîéñèñòåìûêîîðäèíàò
O
0
x
0
y
0
è
óãëà
'
íàêëîíàå¼îñèàáñöèññ
O
0
x
0
êîñèàáñöèññ
Ox
èñõîäíîéñèñòåìûêîîðäèíàò
Oxy
.
3.Âñëó÷àÿõ(3),(5),(7),(8),(9)ëèíèèíàçûâàþòñÿ
ðàñïàäàþùèìèñÿ
,ïîñêîëüêó
ñîîòâåòñòâóþùèåèììíîãî÷ëåíûâòîðîéñòåïåíèðàçëàãàþòñÿâïðîèçâåäåíèåìíîãî-
÷ëåíîâïåðâîéñòåïåíè.
I.1.2Ñõåìàïðèâåäåíèÿóðàâíåíèÿëèíèèâòîðîãîïîðÿäêàêêà-
íîíè÷åñêîìóâèäó
Ïóñòüâïðÿìîóãîëüíîéñèñòåìåêîîðäèíàò
Oxy
àëãåáðàè÷åñêàÿëèíèÿâòîðîãîïî-
ðÿäêàçàäàíàóðàâíåíèåì
I.1
:
a
11

x
2
+2

a
12

x

y
+
a
22

y
2
+2

a
1

x
+2

a
2

y
+
a
0
=0
:
10
I.1.Êðèâûåâòîðîãîïîðÿäêàèèõñâîéñòâà
×òîáûïðèâåñòèóðàâíåíèåêêàíîíè÷åñêîìóâèäó,íóæíîâûïîëíèòüñëåäóþùèå
äåéñòâèÿ.
1.Åñëèâóðàâíåíèèèìååòñÿ÷ëåíñïðîèçâåäåíèåìíåèçâåñòíûõ
(
a
12
6
=0)
,òîäåëàåì
ïîâîðîòñèñòåìûêîîðäèíàò:
(
x
=
x
0

cos
'

y
0

sin
';
y
=
x
0

sin
'
+
y
0

cos
'
íàóãîë
'

0
'

2

,óäîâëåòâîðÿþùèéðàâåíñòâó
ctg2
'
=
a
11

a
22
2
a
12
.Ïðèýòîìïîëó-
÷èì"ïî÷òè"ïðèâåä¼ííîåóðàâíåíèåëèíèèâòîðîãîïîðÿäêà:

1

(
x
0
)
2
+

2

(
y
0
)
2
+2

a
0
1

x
0
+2

a
0
2

y
0
+
a
0
=0
:
Åñëè
a
12
=0
,ïåðåõîäèìêïóíêòó2,ïîâîðîòñèñòåìûêîîðäèíàòäåëàòüíåíóæíî,
òàêêàêèñõîäíîåóðàâíåíèåèìååò"ïî÷òè"ïðèâåä¼ííûéâèä.
2.Âûïîëíÿåìïàðàëëåëüíûéïåðåíîññèñòåìûêîîðäèíàò:
à)åñëèâóðàâíåíèèíåòëèíåéíûõ÷ëåíîâ,òîïåðåõîäèìêïóíêòó3;
á)åñëèâóðàâíåíèèèìååòñÿëèíåéíûé÷ëåíñêàêîé-ëèáîíåèçâåñòíîéèêâàäðàòè÷-
íûé÷ëåíñýòîéæåíåèçâåñòíîé,òî,äîïîëíÿÿýòè÷ëåíûäîïîëíîãîêâàäðàòà,äåëàåì
çàìåíó,÷òîáûâóðàâíåíèèíåñòàëîëèíåéíîãî÷ëåíàñýòîéíåèçâåñòíîé.Íàïðèìåð,
åñëèâóðàâíåíèè

1
6
=0
è
a
0
1
6
=0
,òîâûïîëíÿåìïðåîáðàçîâàíèÿ:

1
(
x
0
)
2
+2

a
0
1

x
0
=

1
"
(
x
0
)
2
+2
a
0
1

1
x
0
+

a
0
1

1

2
#


1


a
0
1

1

2
=
=

1


x
0
+
a
0
1

1

2


1


a
0
1

1

2
;
àçàòåìçàìåíóíåèçâåñòíûõ
x
00
=
x
0
+
a
0

1
;y
00
=
y
0
,ïîñëåêîòîðîéâóðàâíåíèèíåáóäåò
ëèíåéíîãî÷ëåíàñíåèçâåñòíîé
x
00
;
â)åñëèâóðàâíåíèèèìååòñÿòîëüêîîäèíëèíåéíûé÷ëåíñêàêîé-ëèáîíåèçâåñòíîé,
àêâàäðàòýòîéíåèçâåñòíîéîòñóòñòâóåò,òîïðèïîìîùèçàìåíûýòîéïåðåìåííîéíàäî
ñäåëàòüðàâíûìíóëþñâîáîäíûé÷ëåíóðàâíåíèÿ.Íàïðèìåð,åñëèóðàâíåíèåèìååòâèä

1

(
x
0
)
2
+2

a
0
2

y
0
+
a
0
=0
;
òî,âûïîëíÿÿçàìåíóíåèçâåñòíûõ
x
00
=
x
0
;y
00
=
y
0
+
a
0
2
a
0
2
,ïîëó÷àåìóðàâíåíèåáåç
ñâîáîäíîãî÷ëåíà:

1

(
x
00
)
2
+2

a
0
2

y
00
=0
:
3.Ïîëó÷åííîåâðåçóëüòàòåóïðîùåíèé(ïóíêò2)óðàâíåíèåèìååò¾ïî÷òè¿êàíîíè÷å-
ñêèéâèä[9].Äëÿîêîí÷àòåëüíîãîóïðîùåíèÿ¾ïî÷òè¿êàíîíè÷åñêîãîóðàâíåíèÿïðè
íåîáõîäèìîñòèïðèìåíÿþòñÿñëåäóþùèåïðåîáðàçîâàíèÿ:
à)ïåðåèìåíîâàíèåêîîðäèíàòíûõîñåé:
x
0
=
y
00
;y
0
=
x
00
;
á)èçìåíåíèåíàïðàâëåíèÿêîîðäèíàòíîéîñè,íàïðèìåðîñèàáñöèññ:
x
0
=

x
00
;
y
0
=
y
00
;
â)óìíîæåíèåîáåèõ÷àñòåéóðàâíåíèÿíàîòëè÷íûéîòíóëÿìíîæèòåëü;
11
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
ã)ïåðåíîñ÷ëåíîâèçîäíîé÷àñòèóðàâíåíèÿâäðóãóþ.
Âðåçóëüòàòåýòèõïðåîáðàçîâàíèéóðàâíåíèåïðèâîäèòñÿêêàíîíè÷åñêîìóâèäó.Çà-
ìåíóíåèçâåñòíûõ,ïðèâîäÿùóþóðàâíåíèåïîâåðõíîñòèêêàíîíè÷åñêîìóâèäó,îïðåäå-
ëÿåìêàêêîìïîçèöèþâñåõçàìåí,ïðèìåíÿåìûõâõîäåðåøåíèÿ.
I.2Ýëëèïñ
I.2.1Ôîêàëüíîåñâîéñòâîýëëèïñà
Ýëëèïñîì
íàçûâàåòñÿãåîìåòðè÷åñêîåìåñòîòî÷åêïëîñêîñòè,ñóììàðàññòîÿíèéîò
êàæäîéèçêîòîðûõäîäâóõçàäàííûõòî÷åê
F
1

F
2
åñòüâåëè÷èíàïîñòîÿííàÿ
(2
a
)
,
áîëüøàÿðàññòîÿíèÿ
(2
c
)
ìåæäóýòèìèçàäàííûìèòî÷êàìè(
Ðèñ.1
,à).Ýòîãåîìåòðè-
÷åñêîåîïðåäåëåíèåâûðàæàåò
ôîêàëüíîåñâîéñòâîýëëèïñà
.
Ðèñ.1
:Ôîêàëüíîåñâîéñòâîýëëèïñà
Òî÷êè
F
1
è
F
2
íàçûâàþòñÿ
ôîêóñàìè
ýëëèïñà,ðàññòîÿíèåìåæäóôîêóñàìè
2
c
=
F
1
F
2
-
ôîêóñíûìðàññòîÿíèåì
,ñåðåäèíà
O
îòðåçêà
F
1
F
2
-
öåíòðîì
ýëëèïñà,÷èñëî
(2
a
)
-
äëèíîéáîëüøîéîñè
ýëëèïñà(ñîîòâåòñòâåííî,÷èñëî
a
-
áîëüøîéïîëóîñüþ
ýëëèïñà).Îòðåçêè
F
1
M
è
F
2
M
,ñîåäèíÿþùèåïðîèçâîëüíóþòî÷êó
M
ýëëèïñàñåãî
ôîêóñàìè,íàçûâàþòñÿ
ôîêàëüíûìèðàäèóñàìè
òî÷êè
M
.Îòðåçîê,ñîåäèíÿþùèé
äâåòî÷êèýëëèïñà,íàçûâàåòñÿ
õîðäîéýëëèïñà
.
Îòíîøåíèå
e
=
c
a
íàçûâàåòñÿ
ýêñöåíòðèñèòåòîì
ýëëèïñà.Èçîïðåäåëåíèÿ
(2
a�
2
c
)
ñëåäóåò,÷òî
0
6
e
1
:
Ïðè
e
=0
,ò.å.ïðè
c
=0
,ôîêóñû
F
1
è
F
2
,àòàêæåöåíòð
O
ñîâïàäàþò,èýëëèïñÿâëÿåòñÿ
îêðóæíîñòüþðàäèóñà
a
(
Ðèñ.1
,á).
Ãåîìåòðè÷åñêîåîïðåäåëåíèåýëëèïñà,âûðàæàþùåååãîôîêàëüíîåñâîéñòâî,ýêâèâà-
ëåíòíîåãîàíàëèòè÷åñêîìóîïðåäåëåíèþëèíèè,çàäàâàåìîéêàíîíè÷åñêèìóðàâíå-
íèåìýëëèïñà:
12
I.2.Ýëëèïñ
x
2
a
2
+
y
2
b
2
=1
:
(I.16)
Äåéñòâèòåëüíî,ââåäåìïðÿìîóãîëüíóþñèñòåìóêîîðäèíàò(
Ðèñ.1
,
a
).Öåíòð
O
ýë-
ëèïñàïðèìåìçàíà÷àëîñèñòåìûêîîðäèíàò;ïðÿìóþ,ïðîõîäÿùóþ÷åðåçôîêóñû
(
ôîêàëüíóþîñü
èëè
ïåðâóþîñü
ýëëèïñà),ïðèìåìçàîñüàáñöèññ(ïîëîæèòåëüíîå
íàïðàâëåíèåíàíåéîòòî÷êè
F
1
êòî÷êå
F
2
);ïðÿìóþ,ïåðïåíäèêóëÿðíóþôîêàëüíîé
îñèèïðîõîäÿùóþ÷åðåçöåíòðýëëèïñà(
âòîðóþîñü
ýëëèïñà),ïðèìåìçàîñüîðäèíàò
(íàïðàâëåíèåíàîñèîðäèíàòâûáèðàåòñÿòàê,÷òîáûïðÿìîóãîëüíàÿñèñòåìàêîîðäèíàò
Oxy
îêàçàëàñüïðàâîé).
Ñîñòàâèìóðàâíåíèåýëëèïñà,ïîëüçóÿñüåãîãåîìåòðè÷åñêèìîïðåäåëåíèåì,âûðà-
æàþùèìôîêàëüíîåñâîéñòâî.Ââûáðàííîéñèñòåìåêîîðäèíàòîïðåäåëÿåìêîîðäèíàòû
ôîêóñîâ
F
1
(

c;
0)
,
F
2
(
c;
0)
.Äëÿïðîèçâîëüíîéòî÷êè
M
(
x;y
)
,ïðèíàäëåæàùåéýëëèïñó,
èìååì:
j
F
1
M
j
+
j
F
2
M
j
=2
a:
Çàïèñûâàÿýòîðàâåíñòâîâêîîðäèíàòíîéôîðìå,ïîëó÷àåì:
p
(
x
+
c
)
2
+
y
2
+
p
(
x

c
)
2
+
y
2
=2
a:
Ïåðåíîñèìâòîðîéðàäèêàëâïðàâóþ÷àñòü,âîçâîäèìîáå÷àñòèóðàâíåíèÿâêâàäðàòè
ïðèâîäèìïîäîáíûå÷ëåíû:
(
x
+
c
)
2
+
y
2
=4
a
2

4
a
p
(
x

c
)
2
+
y
2
+(
x

c
)
2
+
y
2
,
4
a
p
(
x

c
)
2
+
y
2
=4
a
2

4
cx:
Ðàçäåëèâíà4,âîçâîäèìîáå÷àñòèóðàâíåíèÿâêâàäðàò:
a
2
(
x

c
)
2
+
a
2

y
2
=
a
4

2
a
2

c

x
+
c
2

x
2
,
(
a
2

c
2
)

x
2
+
a
2

y
2
=
a
2

(
a
2

c
2
)
:
Îáîçíà÷èâ
b
=
p
a
2

c
2

0
;
ïîëó÷àåì
b
2

x
2
+
a
2

y
2
=
a
2

b
2
:
Ðàçäåëèâîáå÷àñòèíà
a
2

b
2
6
=0
;
ïðèõîäèìêêàíîíè÷åñêîìóóðàâíåíèþýëëèïñà:
x
2
a
2
+
y
2
b
2
=1
:
Ñëåäîâàòåëüíî,âûáðàííàÿñèñòåìàêîîðäèíàòÿâëÿåòñÿêàíîíè÷åñêîé.
Åñëèôîêóñûýëëèïñàñîâïàäàþò,òîýëëèïñïðåäñòàâëÿåòñîáîéîêðóæíîñòü(
Ðèñ.
1
),ïîñêîëüêó
a
=
b
.Âýòîìñëó÷àåêàíîíè÷åñêîéáóäåòëþáàÿïðÿìîóãîëüíàÿñèñòåìà
êîîðäèíàòñíà÷àëîìâòî÷êå
O

F
1

F
2
,aóðàâíåíèå
x
2
+
y
2
=
a
2
ÿâëÿåòñÿ
óðàâíåíèåìîêðóæíîñòè
ñöåíòðîìâòî÷êå
O
èðàäèóñîì,ðàâíûì
a
.
Ïðîâîäÿðàññóæäåíèÿâîáðàòíîìïîðÿäêå,ìîæíîïîêàçàòü,÷òîâñåòî÷êè,êîîðäè-
íàòûêîòîðûõóäîâëåòâîðÿþòóðàâíåíèþ(
I.16
),èòîëüêîîíè,ïðèíàäëåæàòãåîìåòðè÷å-
ñêîìóìåñòóòî÷åê,íàçûâàåìîìóýëëèïñîì.Äðóãèìèñëîâàìè,àíàëèòè÷åñêîåîïðåäåëå-
íèåýëëèïñàýêâèâàëåíòíîåãîãåîìåòðè÷åñêîìóîïðåäåëåíèþ,âûðàæàþùåìóôîêàëüíîå
ñâîéñòâîýëëèïñà.
13
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
I.2.2Äèðåêòîðèàëüíîåñâîéñòâîýëëèïñà
Äèðåêòðèñàìèýëëèïñà
íàçûâàþòñÿäâåïðÿìûå,ïðîõîäÿùèåïàðàëëåëüíîîñè
îðäèíàòêàíîíè÷åñêîéñèñòåìûêîîðäèíàòíàîäèíàêîâîìðàññòîÿíèè
a
2
c
îòíå¼.Ïðè
c
=
0
,êîãäàýëëèïñÿâëÿåòñÿîêðóæíîñòüþ,äèðåêòðèñíåò(ìîæíîñ÷èòàòü,÷òîäèðåêòðèñû
áåñêîíå÷íîóäàëåíû).
Ýëëèïññýêñöåíòðèñèòåòîì
0
e
1
ìîæíîîïðåäåëèòü,êàê
ãåîìåòðè÷åñêîåìå-
ñòîòî÷åêïëîñêîñòè,äëÿêàæäîéèçêîòîðûõîòíîøåíèåðàññòîÿíèÿäîçàäàííîé
òî÷êèF(ôîêóñà)êðàññòîÿíèþäîçàäàííîéïðÿìîéd(äèðåêòðèñû),íåïðîõîäÿùåé
÷åðåççàäàííóþòî÷êó,ïîñòîÿííîèðàâíîýêñöåíòðèñèòåòóe
(
äèðåêòîðèàëüíîå
ñâîéñòâîýëëèïñà
).ÇäåñüFèdîäèíèçôîêóñîâýëëèïñàèîäíàèçåãîäèðåêòðèñ,
ðàñïîëîæåííûåïîîäíóñòîðîíóîòîñèîðäèíàòêàíîíè÷åñêîéñèñòåìûêîîðäèíàò,ò.å.
F
1
;d
1
èëè
F
2
;d
2
:
Ðèñ.2
:Äèðåêòîðèàëüíîåñâîéñòâîýëëèïñà
Âñàìîìäåëå,íàïðèìåð,äëÿôîêóñà
F
2
èäèðåêòðèñû
d
2
(
Ðèñ.2
)óñëîâèå
r
2

2
=
e
ìîæíîçàïèñàòüâêîîðäèíàòíîéôîðìå:
p
(
x

c
)
2
+
y
2
=
e


a
2
c

x
!
:
Èçáàâëÿÿñüîòèððàöèîíàëüíîñòèèçàìåíÿÿ
e
=
c
a
,
a
2

c
2
=
b
2
,ïðèõîäèìêêàíîíè÷å-
ñêîìóóðàâíåíèþýëëèïñà(
I.16
)Àíàëîãè÷íûåðàññóæäåíèÿìîæíîïðîâåñòèäëÿôîêóñà
F
1
èäèðåêòðèñû
d
1
:
r
1

1
=
e:
I.2.3Óðàâíåíèåýëëèïñàâïîëÿðíîéñèñòåìåêîîðäèíàò
Óðàâíåíèåýëëèïñàâïîëÿðíîéñèñòåìåêîîðäèíàò
F
1
r'
èìååòâèä
r
=
p
1

e

cos
'
;
(I.17)
ãäå
p
=
b
2
a
-
ôîêàëüíûéïàðàìåòðýëëèïñà
.
Âñàìîìäåëå,âûáåðåìâêà÷åñòâåïîëþñàïîëÿðíîéñèñòåìûêîîðäèíàòëåâûéôîêóñ
F
1
ýëëèïñà,àâêà÷åñòâåïîëÿðíîéîñèëó÷
F
1
F
2
(
Ðèñ.2
,â).Òîãäàäëÿïðîèçâîëüíîé
14
I.2.Ýëëèïñ
òî÷êè
M
(
r;'
)
,ñîãëàñíîãåîìåòðè÷åñêîìóîïðåäåëåíèþ(ôîêàëüíîìóñâîéñòâó)ýëëèïñà,
èìååì
r
+
MF
2
=2
a
.Âûðàæàåìðàññòîÿíèåìåæäóòî÷êàìè
M
(
r;'
)
è
F
2
(2
c;
0)
F
2
M
=
p
(2
c
)
2
+
r
2

2

(2
c
)

r

cos(
'

0)=
p
r
2

4

c

r

cos
'
+4

c
2
:
Ñëåäîâàòåëüíî,âêîîðäèíàòíîéôîðìåóðàâíåíèåýëëèïñà
F
1
M
+
F
2
M
=2
a
èìååòâèä
r
+
p
r
2

4

c

r

cos
'
+4

c
2
=2

a:
Óåäèíÿåìðàäèêàë,âîçâîäèìîáå÷àñòèóðàâíåíèÿâêâàäðàò,äåëèìíà4èïðèâîäèì
ïîäîáíûå÷ëåíû:
r
2

4

c

r

cos
'
+4

c
2
=4

a
2

4

a

r
+
r
2
,
a


1

c
a

cos
'


r
=
a
2

c
2
:
Âûðàæàåìïîëÿðíûéðàäèóñ
r
èäåëàåìçàìåíó
e
=
c
a
;b
2
=
a
2

c
2
;p
=
b
2
a
:
r
=
a
2

c
2
a

(1

e

cos
'
)
,
r
=
b
2
a

(1

e

cos
'
)
,
r
=
p
1

e

cos
'
;
÷òîèòðåáîâàëîñüäîêàçàòü.
I.2.4Ãåîìåòðè÷åñêèéñìûñëêîýôôèöèåíòîââóðàâíåíèèýëëèï-
ñà
Íàéäåìòî÷êèïåðåñå÷åíèÿýëëèïñà(
Ðèñ.2
,à)ñêîîðäèíàòíûìèîñÿìè(
âåðøèíû
ýëëèïñà
).Ïîäñòàâëÿÿâóðàâíåíèå
y
=0
,íàõîäèìòî÷êèïåðåñå÷åíèÿýëëèïñàñîñüþ
àáñöèññ(ñôîêàëüíîéîñüþ):
x
=

a
.Ñëåäîâàòåëüíî,äëèíàîòðåçêàôîêàëüíîéîñè,
çàêëþ÷åííîãîâíóòðèýëëèïñà,ðàâíà
2
a
.Ýòîòîòðåçîê,êàêîòìå÷åíîâûøå,íàçûâàåòñÿ
áîëüøîéîñüþýëëèïñà
,à÷èñëî
a

áîëüøîéïîëóîñüþýëëèïñà
.Ïîäñòàâëÿÿ
x
=
0
,ïîëó÷àåì
y
=

b
.Ñëåäîâàòåëüíî,äëèíàîòðåçêàâòîðîéîñèýëëèïñà,çàêëþ÷åííîãî
âíóòðèýëëèïñà,ðàâíà
2
b
.Ýòîòîòðåçîêíàçûâàåòñÿ
ìàëîéîñüþýëëèïñà
,à÷èñëî
b

ìàëîéïîëóîñüþýëëèïñà
.
Äåéñòâèòåëüíî,
b
=
p
a
2

c
2
6
p
a
2
=
a
,ïðè÷¼ìðàâåíñòâî
b
=
a
ïîëó÷àåòñÿòîëüêî
âñëó÷àå
c
=0
,êîãäàýëëèïñÿâëÿåòñÿîêðóæíîñòüþ.Îòíîøåíèå
k
=
b
a
6
1
íàçûâàåòñÿ
êîýôôèöèåíòîìñæàòèÿýëëèïñà
.
Çàìå÷àíèå1.
1.
Ïðÿìûå
x
=

a;y
=

b
îãðàíè÷èâàþòíàêîîðäèíàòíîéïëîñêîñòè
îñíîâíîé
ïðÿìîóãîëüíèê
,âíóòðèêîòîðîãîíàõîäèòñÿýëëèïñ(
Ðèñ.2
,
a
).
2.
Ýëëèïñìîæíîîïðåäåëèòü,
êàêãåîìåòðè÷åñêîåìåñòîòî÷åê,ïîëó÷àåìîåâðå-
çóëüòàòåñæàòèÿîêðóæíîñòèêå¼äèàìåòðó
.
Äåéñòâèòåëüíî,ïóñòüâïðÿìîóãîëüíîéñèñòåìåêîîðäèíàò
Oxy
óðàâíåíèåîêðóæ-
íîñòèèìååòâèä
x
2
+
y
2
=
a
2
.Ïðèñæàòèèêîñèàáñöèñññêîýôôèöèåíòîì
0
k
6
1
êîîðäèíàòûïðîèçâîëüíîéòî÷êè
M
(
x;y
)
,ïðèíàäëåæàùåéîêðóæíîñòè,èçìåíÿþòñÿïî
çàêîíó
(
x
0
=
x;
y
0
=
k

y:
15
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
Ïîäñòàâëÿÿâóðàâíåíèåîêðóæíîñòè
x
=
x
0
è
y
=
1
k

y
0
,ïîëó÷àåìóðàâíåíèåäëÿ
êîîðäèíàòîáðàçà
M
0
(
x
0
;y
0
)
òî÷êè
M
(
x;y
)
:
(
x
0
)
2
+(
1
k

y
0
)
2
=
a
2
,
(
x
0
)
2
a
2
+
(
y
0
)
2
k
2

a
2
=1
,
(
x
0
)
2
a
2
+
(
y
0
)
2
b
2
=1
;
ïîñêîëüêó
b
=
k

a
.Ýòîêàíîíè÷åñêîåóðàâíåíèåýëëèïñà.
3.
Êîîðäèíàòíûåîñè(êàíîíè÷åñêîéñèñòåìûêîîðäèíàò)ÿâëÿþòñÿ
îñÿìèñèì-
ìåòðèè
ýëëèïñà(íàçûâàþòñÿ
ãëàâíûìèîñÿìèýëëèïñà
),àåãîöåíòð
öåíòðîì
ñèììåòðèè
.
Äåéñòâèòåëüíî,åñëèòî÷êà
M
(
x;y
)
ïðèíàäëåæèòýëëèïñó
x
2
a
2
+
y
2
b
2
=1
,òîèòî÷êè
M
0
(
x;

y
)
è
M
00
(

x;y
)
,ñèììåòðè÷íûåòî÷êå
M
îòíîñèòåëüíîêîîðäèíàòíûõîñåé,òàêæå
ïðèíàäëåæàòòîìóæåýëëèïñó.
4.
Èçóðàâíåíèÿýëëèïñàâïîëÿðíîéñèñòåìåêîîðäèíàò
r
=
p
1

e
cos
'
(
Ðèñ.2
,â),
âûÿñíÿåòñÿ
ãåîìåòðè÷åñêèéñìûñëôîêàëüíîãîïàðàìåòðà
ýòîïîëîâèíàäëèíûõîðäû
ýëëèïñà,ïðîõîäÿùåé÷åðåçåãîôîêóñïåðïåíäèêóëÿðíîôîêàëüíîéîñè(
r
=
p
ïðè
'
=

2
).
Ðèñ.3
5.
Ýêñöåíòðèñèòåò
e
õàðàêòåðèçóåòôîðìóýëëèïñà,àèìåííîîòëè÷èåýëëèïñàîò
îêðóæíîñòè.×åìáîëüøå
e
,òåìýëëèïñáîëååâûòÿíóò,à÷åìáëèæå
e
êíóëþ,òåì
áëèæåýëëèïñêîêðóæíîñòè(
Ðèñ.3
,
a
).
Äåéñòâèòåëüíî,ó÷èòûâàÿ,÷òî
e
=
c
a
è
c
2
=
a
2

b
2
,ïîëó÷àåì
e
2
=
c
2
a
2
=
a
2

b
2
a
2
=1


b
a

2
=1

k
2
;
ãäå
k
êîýôôèöèåíòñæàòèÿýëëèïñà,
0
k
6
1
.Ñëåäîâàòåëüíî,
e
=
p
1

k
2
.×åì
áîëüøåñæàòýëëèïñïîñðàâíåíèþñîêðóæíîñòüþ,òåììåíüøåêîýôôèöèåíòñæàòèÿ
k
èáîëüøåýêñöåíòðèñèòåò.Äëÿîêðóæíîñòè
k
=1
è
e
=0
.
6.
Óðàâíåíèå
x
2
a
2
+
y
2
b
2
=1
ïðè
ab
îïðåäåëÿåòýëëèïñ,ôîêóñûêîòîðîãîðàñïî-
ëîæåíûíàîñè
Oy
(
Ðèñ.3
,á).Ýòîóðàâíåíèåñâîäèòñÿêêàíîíè÷åñêîìóïðèïîìîùè
ïåðåèìåíîâàíèÿêîîðäèíàòíûõîñåé.
16
I.3.Ãèïåðáîëà
7.
Óðàâíåíèå
(
x

x
0
)
2
a
2
+
(
y

y
0
)
2
b
2
=1
,
a

b
,îïðåäåëÿåòýëëèïññöåíòðîìâòî÷-
êå
O
0
(
x
0
;y
0
)
,îñèêîòîðîãîïàðàëëåëüíûêîîðäèíàòíûìîñÿì(
Ðèñ.3
,â).Ýòîóðàâíåíèå
ñâîäèòñÿêêàíîíè÷åñêîìóïðèïîìîùèïàðàëëåëüíîãîïåðåíîñà.
Ïðè
a
=
b
=
R
óðàâíåíèå
(
x

x
0
)
2
+(
y

y
0
)
2
=
R
2
îïèñûâàåòîêðóæíîñòüðàäèóñà
R
ñöåíòðîìâòî÷êå
O
0
(
x
0
;y
0
)
.
8.
Ïàðàìåòðè÷åñêîåóðàâíåíèåýëëèïñà
âêàíîíè÷åñêîéñèñòåìåêîîðäèíàòèìå-
åòâèä
(
x
=
a

cos
t;
y
=
b

sin
t;
0
6
t
2
:
Äåéñòâèòåëüíî,ïîäñòàâëÿÿýòèâûðàæåíèÿâóðàâíåíèå(
I.16
),ïðèõîäèìêîñíîâíî-
ìóòðèãîíîìåòðè÷åñêîìóòîæäåñòâó
cos
2
t
+sin
2
t
=1
:
I.3Ãèïåðáîëà
Ãèïåðáîëîé
íàçûâàåòñÿãåîìåòðè÷åñêîåìåñòîòî÷åêïëîñêîñòè,ìîäóëüðàçíîñòè
ðàññòîÿíèéîòêàæäîéèçêîòîðûõäîäâóõçàäàííûõòî÷åê
F
1
è
F
2
åñòüâåëè÷èíàïî-
ñòîÿííàÿ
(2
a
)
,ìåíüøàÿðàññòîÿíèÿ
(2
c
)
ìåæäóýòèìèçàäàííûìèòî÷êàìè(
Ðèñ.4
,
a
).
Ýòîãåîìåòðè÷åñêîåîïðåäåëåíèåâûðàæàåò
ôîêàëüíîåñâîéñòâîãèïåðáîëû
.
I.3.1Ôîêàëüíîåñâîéñòâîãèïåðáîëû
Òî÷êè
F
1
è
F
2
íàçûâàþòñÿ
ôîêóñàìè
ãèïåðáîëû,ðàññòîÿíèå
2
c
=
F
1
F
2
ìåæäóíèìè

ôîêóñíûìðàññòîÿíèåì
,ñåðåäèíà
O
îòðåçêà
F
1
F
2

öåíòðîì
ãèïåðáîëû,÷èñëî
2
a

äëèíîéäåéñòâèòåëüíîéîñè
ãèïåðáîëû(ñîîòâåòñòâåííî,
a

äåéñòâèòåëü-
íîéïîëóîñüþ
ãèïåðáîëû).Îòðåçêè
F
1
M
è
F
2
M
,ñîåäèíÿþùèåïðîèçâîëüíóþòî÷êó
M
ãèïåðáîëûñå¼ôîêóñàìè,íàçûâàþòñÿ
ôîêàëüíûìèðàäèóñàìè
òî÷êè
M
.Îòðåçîê,
ñîåäèíÿþùèéäâåòî÷êèãèïåðáîëû,íàçûâàåòñÿ
õîðäîéãèïåðáîëû
.
Ðèñ.4
:Ôîêàëüíîåñâîéñòâîãèïåðáîëû
17
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
Îòíîøåíèå
e
=
c
a
,ãäå
c
=
p
a
2
+
b
2
,íàçûâàåòñÿ
ýêñöåíòðèñèòåòîì
ãèïåðáîëû.
Èçîïðåäåëåíèÿ
(2
a
2
c
)
ñëåäóåò,÷òî
e�
1
.
Ãåîìåòðè÷åñêîåîïðåäåëåíèåãèïåðáîëû,âûðàæàþùååå¼ôîêàëüíîåñâîéñòâî,ýêâè-
âàëåíòíîå¼àíàëèòè÷åñêîìóîïðåäåëåíèþëèíèè,çàäàâàåìîéêàíîíè÷åñêèìóðàâíå-
íèåìãèïåðáîëû
:
x
2
a
2

y
2
b
2
=1
:
(I.18)
Äåéñòâèòåëüíî,ââåäåìïðÿìîóãîëüíóþñèñòåìóêîîðäèíàò(
Ðèñ.4
,á).Öåíòð
O
ãè-
ïåðáîëûïðèìåìçàíà÷àëîñèñòåìûêîîðäèíàò;ïðÿìóþ,ïðîõîäÿùóþ÷åðåçôîêóñû
(ôîêàëüíóþîñü
),ïðèìåìçàîñüàáñöèññ(ïîëîæèòåëüíîåíàïðàâëåíèåíàíåéîòòî÷-
êè
F
1
êòî÷êå
F
2
);ïðÿìóþ,ïåðïåíäèêóëÿðíóþîñèàáñöèññèïðîõîäÿùóþ÷åðåçöåíòð
ãèïåðáîëû,ïðèìåìçàîñüîðäèíàò(íàïðàâëåíèåíàîñèîðäèíàòâûáèðàåòñÿòàê,÷òîáû
ïðÿìîóãîëüíàÿñèñòåìàêîîðäèíàò
Oxy
îêàçàëàñüïðàâîé).
Ñîñòàâèìóðàâíåíèåãèïåðáîëû,èñïîëüçóÿãåîìåòðè÷åñêîåîïðåäåëåíèå,âûðàæàþ-
ùååôîêàëüíîåñâîéñòâî.Ââûáðàííîéñèñòåìåêîîðäèíàòîïðåäåëÿåìêîîðäèíàòûôî-
êóñîâ
F
1
(

c;
0)
è
F
2
(
c;
0)
.Äëÿïðîèçâîëüíîéòî÷êè
M
(
x;y
)
,ïðèíàäëåæàùåéãèïåðáîëå,
èìååì:



j
F
1
M
j�j
F
2
M
j



=2
a:
Çàïèñûâàÿýòîóðàâíåíèåâêîîðäèíàòíîéôîðìå,ïîëó÷àåì:
p
(
x
+
c
)
2
+
y
2

p
(
x

c
)
2
+
y
2
=

2
a:
Âûïîëíÿÿïðåîáðàçîâàíèÿ,àíàëîãè÷íûåïðåîáðàçîâàíèÿì,èñïîëüçóåìûìïðèâûâîäå
óðàâíåíèÿýëëèïñà(ò.å.èçáàâëÿÿñüîòèððàöèîíàëüíîñòè),ïðèõîäèìêêàíîíè÷åñêîìó
óðàâíåíèþãèïåðáîëû:
x
2
a
2

y
2
b
2
=1
;
ãäå
b
=
p
c
2

a
2
,ò.å.âûáðàííàÿñèñòåìàêîîðäèíàòÿâëÿåòñÿêàíîíè÷åñêîé.
Ïðîâîäÿðàññóæäåíèÿâîáðàòíîìïîðÿäêå,ìîæíîïîêàçàòü,÷òîâñåòî÷êè,êîîðäè-
íàòûêîòîðûõóäîâëåòâîðÿþòóðàâíåíèþ(
I.18
),èòîëüêîîíè,ïðèíàäëåæàòãåîìåòðè-
÷åñêîìóìåñòóòî÷åê,íàçûâàåìîìóãèïåðáîëîé.Òàêèìîáðàçîì,àíàëèòè÷åñêîåîïðåäå-
ëåíèåãèïåðáîëûýêâèâàëåíòíîåãîãåîìåòðè÷åñêîìóîïðåäåëåíèþ.
I.3.2Äèðåêòîðèàëüíîåñâîéñòâîãèïåðáîëû
Äèðåêòðèñàìèãèïåðáîëû
íàçûâàþòñÿäâåïðÿìûå,ïðîõîäÿùèåïàðàëëåëüíîîñè
îðäèíàòêàíîíè÷åñêîéñèñòåìûêîîðäèíàòíàîäèíàêîâîìðàññòîÿíèè
a
2
c
îòíå¼(
Ðèñ.
5
,
a
).Ïðè
a
=0
,êîãäàãèïåðáîëàâûðîæäàåòñÿâïàðóïåðåñåêàþùèõñÿïðÿìûõ,äèðåê-
òðèñûñîâïàäàþò.
18
I.3.Ãèïåðáîëà
Ðèñ.5
:Äèðåêòîðèàëüíîåñâîéñòâîãèïåðáîëû
Ãèïåðáîëóñýêñöåíòðèñèòåòîì
e�
1
ìîæíîîïðåäåëèòü,
êàêãåîìåòðè÷åñêîåìåñòî
òî÷åêïëîñêîñòè,äëÿêàæäîéèçêîòîðûõîòíîøåíèåðàññòîÿíèÿäîçàäàííîéòî÷êè
F
(ôîêóñà)êðàññòîÿíèþäîçàäàííîéïðÿìîé
d
(äèðåêòðèñû),íåïðîõîäÿùåé÷åðåç
çàäàííóþòî÷êó,ïîñòîÿííîèðàâíîýêñöåíòðèñèòåòóe(
äèðåêòîðèàëüíîåñâîé-
ñòâîãèïåðáîëû)
.Çäåñü
F
è
d
îäèíèçôîêóñîâãèïåðáîëûèîäíàèçå¼äèðåêòðèñ,
ðàñïîëîæåííûåïîîäíóñòîðîíóîòîñèîðäèíàòêàíîíè÷åñêîéñèñòåìûêîîðäèíàò.
Âñàìîìäåëå,íàïðèìåð,äëÿôîêóñà
F
2
èäèðåêòðèñû
d
2
(
Ðèñ.5
,
a
)óñëîâèå
r
2

2
=
e
ìîæíîçàïèñàòüâêîîðäèíàòíîéôîðìå:
p
(
x

c
)
2
+
y
2
=
e

x

a
2
c
!
:
Èçáàâëÿÿñüîòèððàöèîíàëüíîñòèèçàìåíÿÿ
e
=
c
a
,
c
2

a
2
=
b
2
,ïðèõîäèìêêàíîíè-
÷åñêîìóóðàâíåíèþãèïåðáîëû(
I.18
).Àíàëîãè÷íûåðàññóæäåíèÿìîæíîïðîâåñòèäëÿ
ôîêóñà
F
1
èäèðåêòðèñû
d
1
:
r
1

1
=
e
,
p
(
x
+
c
)
2
+
y
2
=
e

x
+
a
2
c
!
:
I.3.3Óðàâíåíèåãèïåðáîëûâïîëÿðíîéñèñòåìåêîîðäèíàò
Óðàâíåíèåïðàâîéâåòâèãèïåðáîëûâïîëÿðíîéñèñòåìåêîîðäèíàò
F
2
r'
(
Ðèñ.5
,á)
èìååòâèä
r
=
p
1

e

cos
'
;
ãäå
p
=
p
2
a

ôîêàëüíûéïàðàìåòðãèïåðáîëû
.
Âñàìîìäåëå,âûáåðåìâêà÷åñòâåïîëþñàïîëÿðíîéñèñòåìûêîîðäèíàòïðàâûéôî-
êóñ
F
2
ãèïåðáîëû,àâêà÷åñòâåïîëÿðíîéîñèëó÷ñíà÷àëîìâòî÷êå
F
2
,ïðèíàäëåæà-
ùèéïðÿìîé
F
1
F
2
,íîíåñîäåðæàùèéòî÷êè
F
1
(
Ðèñ.5
,á).Òîãäàäëÿïðîèçâîëüíîéòî÷êè
19
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
M
(
r;'
)
,ïðèíàäëåæàùåéïðàâîéâåòâèãèïåðáîëû,ñîãëàñíîãåîìåòðè÷åñêîìóîïðåäåëå-
íèþ(ôîêàëüíîìóñâîéñòâó)ãèïåðáîëû,èìååì
F
1
M

r
=2
a
.Âûðàæàåìðàññòîÿíèå
ìåæäóòî÷êàìè
M
(
r;'
)
è
F
1
(2
c;
)
F
1
M
=
p
(2
c
)
2
+
r
2

2

(2
c
)

r

cos(
'


)=
p
r
2
+4

c

r

cos
'
+4

c
2
:
Ñëåäîâàòåëüíî,âêîîðäèíàòíîéôîðìåóðàâíåíèåãèïåðáîëûèìååòâèä
p
r
2
+4

c

r

cos
'
+4

c
2

r
=2
a:
Óåäèíÿåìðàäèêàë,âîçâîäèìîáå÷àñòèóðàâíåíèÿâêâàäðàò,äåëèìíà4èïðèâîäèì
ïîäîáíûå÷ëåíû:
r
2
+4

c

r

cos
'
+4

c
2
=4

a
2
+4

a

r
+
r
2
,
a

1

c
a
cos
'

r
=
c
2

a
2
:
Âûðàæàåìïîëÿðíûéðàäèóñ
r
èäåëàåìçàìåíû
e
=
c
a
,
b
2
=
c
2

a
2
,
p
=
b
2
a
:
r
=
c
2

a
2
a
(1

e
cos
'
)
,
r
=
b
2
a
(1

e
cos
'
)
,
r
=
p
1

e
cos
'
;
÷òîèòðåáîâàëîñüäîêàçàòü.Çàìåòèì,÷òîâïîëÿðíûõêîîðäèíàòàõóðàâíåíèÿãèïåðáî-
ëûèýëëèïñàñîâïàäàþò,íîîïèñûâàþòðàçíûåëèíèè,ïîñêîëüêóîòëè÷àþòñÿýêñöåí-
òðèñèòåòàìè(
e�
1
äëÿãèïåðáîëû,
0
6
e
1
äëÿýëëèïñà).
I.3.4Ãåîìåòðè÷åñêèéñìûñëêîýôôèöèåíòîââóðàâíåíèèãèïåð-
áîëû
Íàéäåìòî÷êèïåðåñå÷åíèÿãèïåðáîëû(
Ðèñ.6
,a)ñîñüþàáñöèññ(
âåðøèíûãèïåðáî-
ëû
).Ïîäñòàâëÿÿâóðàâíåíèå
y
=0
,íàõîäèìàáñöèññûòî÷åêïåðåñå÷åíèÿ:
x
=

a
.Ñëå-
äîâàòåëüíî,âåðøèíûèìåþòêîîðäèíàòû
(

a;
0)
;
(
a;
0)
.Äëèíàîòðåçêà,ñîåäèíÿþùåãî
âåðøèíû,ðàâíà
2
a
.Ýòîòîòðåçîêíàçûâàåòñÿ
äåéñòâèòåëüíîéîñüþãèïåðáîëû
,
à÷èñëî
a

äåéñòâèòåëüíîéïîëóîñüþãèïåðáîëû
.Ïîäñòàâëÿÿ
x
=0
,ïîëó÷à-
åì
y
=

ib
.Äëèíàîòðåçêàîñèîðäèíàò,ñîåäèíÿþùåãîòî÷êè
(0
;

b
)
;
(0
;b
)
,ðàâíà
2
b
.
Ýòîòîòðåçîêíàçûâàåòñÿ
ìíèìîéîñüþãèïåðáîëû
,à÷èñëî
b

ìíèìîéïîëó-
îñüþãèïåðáîëû
.Ãèïåðáîëàïåðåñåêàåòïðÿìóþ,ñîäåðæàùóþäåéñòâèòåëüíóþîñü,è
íåïåðåñåêàåòïðÿìóþ,ñîäåðæàùóþìíèìóþîñü.
Çàìå÷àíèå2.
1.
Ïðÿìûå
x
=

a;y
=

b
îãðàíè÷èâàþòíàêîîðäèíàòíîéïëîñêîñòè
îñíîâíîé
ïðÿìîóãîëüíèê
,âíåêîòîðîãîíàõîäèòñÿãèïåðáîëà(
Ðèñ.6
,a).
2.
Ïðÿìûå
y
=

b
a
x
,ñîäåðæàùèåäèàãîíàëèîñíîâíîãîïðÿìîóãîëüíèêà,íàçûâàþòñÿ
àñèìïòîòàìèãèïåðáîëû
(
Ðèñ.6
,a).
Äëÿðàâíîñòîðîííåéãèïåðáîëû,îïèñûâàåìîéóðàâíåíèåì
x
2
a
2

y
2
a
2
=1
(ò.å.ïðè
a
=
b
),îñíîâíîéïðÿìîóãîëüíèêÿâëÿåòñÿêâàäðàòîì,äèàãîíàëèêîòîðîãîïåðïåíäèêóëÿð-
íû.Ïîýòîìóàñèìïòîòûðàâíîñòîðîííåéãèïåðáîëûòàêæåïåðïåíäèêóëÿðíû,èèõìîæ-
íîâçÿòüâêà÷åñòâåêîîðäèíàòíûõîñåéïðÿìîóãîëüíîéñèñòåìûêîîðäèíàò
Ox
0
y
0
(
Ðèñ.
20
I.3.Ãèïåðáîëà
6
,á).Âýòîéñèñòåìåêîîðäèíàòóðàâíåíèåãèïåðáîëûèìååòâèä
y
0
=
a
2
2
x
0
(ãèïåðáîëàñîâ-
ïàäàåòñãðàôèêîìýëåìåíòàðíîéôóíêöèè,âûðàæàþùåéîáðàòíî-ïðîïîðöèîíàëüíóþ
çàâèñèìîñòü).
Ðèñ.6
Âñàìîìäåëå,ïîâåðí¼ìêàíîíè÷åñêóþñèñòåìóêîîðäèíàòíàóãîë
'
=


4
(
Ðèñ.6
,á).
Ïðèýòîìêîîðäèíàòûòî÷êèâñòàðîéèíîâîéñèñòåìàõêîîðäèíàòñâÿçàíûðàâåíñòâàìè
8







:
x
=
p
2
2

x
0
+
p
2
2

y
0
;
y
=

p
2
2

x
0
+
p
2
2

y
0
;
,
8







:
x
=
p
2
2

(
x
0
+
y
0
)
;
y
=
p
2
2

(
y
0

x
0
)
:
(I.19)
Ïîäñòàâëÿÿýòèâûðàæåíèÿâóðàâíåíèå
x
2
a
2

y
2
a
2
=1
ðàâíîñòîðîííåéãèïåðáîëûè
ïðèâîäÿïîäîáíûå÷ëåíû,ïîëó÷àåì
1
2
(
x
0
+
y
0
)
2
a
2

1
2
(
y
0

x
0
)
2
a
2
=1
,
2

x
0

y
0
=
a
2
,
y
0
=
a
2
2

x
0
:
3.
Êîîðäèíàòíûåîñè(êàíîíè÷åñêîéñèñòåìûêîîðäèíàò)ÿâëÿþòñÿ
îñÿìèñèì-
ìåòðèè
ãèïåðáîëû(íàçûâàþòñÿ
ãëàâíûìèîñÿìèãèïåðáîëû
),àå¼öåíòð
öåí-
òðîìñèììåòðèè
.
Äåéñòâèòåëüíî,åñëèòî÷êà
M
(
x;y
)
ïðèíàäëåæèòãèïåðáîëå
x
2
a
2

y
2
b
2
=1
;
òîèòî÷êè
M
0
(
x;

y
)
è
M
00
(

x;y
)
,ñèììåòðè÷íûåòî÷êå
M
îòíîñèòåëüíîêîîðäèíàòíûõîñåé,òàêæå
ïðèíàäëåæàòòîéæåãèïåðáîëå.
Îñüñèììåòðèè,íàêîòîðîéðàñïîëàãàþòñÿôîêóñûãèïåðáîëû,ÿâëÿåòñÿôîêàëüíîé
îñüþ.
4.
Èçóðàâíåíèÿãèïåðáîëûâïîëÿðíûõêîîðäèíàòàõ
r
=
p
1

e

cos
'
(
Ðèñ.5
,á)âû-
ÿñíÿåòñÿ
ãåîìåòðè÷åñêèéñìûñëôîêàëüíîãîïàðàìåòðà
ýòîïîëîâèíàäëèíû
õîðäûãèïåðáîëû,ïðîõîäÿùåé÷åðåçå¼ôîêóñïåðïåíäèêóëÿðíîôîêàëüíîéîñè(
r
=
p
ïðè
'
=

2
).
21
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
5.
Ýêñöåíòðèñèòåò
e
õàðàêòåðèçóåòôîðìóãèïåðáîëû.×åìáîëüøå
e
,òåìøèðåâåòâè
ãèïåðáîëû,à÷åìáëèæå
e
êåäèíèöå,òåìâåòâèãèïåðáîëûóæå(
Ðèñ.7
,a).
Äåéñòâèòåëüíî,âåëè÷èíà

óãëàìåæäóàñèìïòîòàìèãèïåðáîëû,ñîäåðæàùåãîå¼
âåòâü,îïðåäåëÿåòñÿîòíîøåíèåìñòîðîíîñíîâíîãîïðÿìîóãîëüíèêà:
tg

2
=
b
a
.Ó÷èòû-
âàÿ,÷òî
e
=
c
a
è
c
2
=
a
2
+
b
2
,ïîëó÷àåì
e
2
=
c
2
a
2
=
a
2
+
b
2
a
2
=1+

b
a

2
=1+tg
2

2
:
×åìáîëüøå
e
,òåìáîëüøåóãîë

.Äëÿðàâíîñòîðîííåéãèïåðáîëû(
a
=
b
)èìååì
e
=
p
2
è

=

2
.Äëÿ
e�
p
2
óãîë

òóïîé,àäëÿ
1
e
p
2
óãîë

îñòðûé(
Ðèñ.7
,a).
Ðèñ.7
6.
Äâåãèïåðáîëû,îïðåäåëÿåìûåâîäíîéèòîéæåñèñòåìåêîîðäèíàòóðàâíåíèÿìè
x
2
a
2

y
2
b
2
=1
è

x
2
a
2
+
y
2
b
2
=1
íàçûâàþòñÿñîïðÿæ¼ííûìèäðóãñäðóãîì.Ñîïðÿæ¼í-
íûåãèïåðáîëûèìåþòîäíèèòåæåàñèìïòîòû(
Ðèñ.7
,á).Óðàâíåíèå
ñîïðÿæ¼ííîé
ãèïåðáîëû

x
2
a
2
+
y
2
b
2
=1
ïðèâîäèòñÿêêàíîíè÷åñêîìóïðèïîìîùèïåðåèìåíîâàíèÿ
êîîðäèíàòíûõîñåé.
7.
Óðàâíåíèå
(
x

x
0
)
2
a
2

(
y

y
0
)
2
b
2
=1
îïðåäåëÿåòãèïåðáîëóñöåíòðîìâòî÷êå
O
0
(
x
0
;y
0
)
,îñèêîòîðîéïàðàëëåëüíûêîîðäèíàòíûìîñÿì(
Ðèñ.7
,â).Ýòîóðàâíåíèåñâî-
äèòñÿêêàíîíè÷åñêîìóïðèïîìîùèïàðàëëåëüíîãîïåðåíîñà.Óðàâíåíèå

(
x

x
0
)
2
a
2
+
(
y

y
0
)
2
b
2
=1
îïðåäåëÿåòñîïðÿæ¼ííóþãèïåðáîëóñöåíòðîìâòî÷êå
O
0
(
x
0
;y
0
)
.
8.
Ïàðàìåòðè÷åñêîåóðàâíåíèåãèïåðáîëû
âêàíîíè÷åñêîéñèñòåìåêîîðäèíàò
èìååòâèä
(
x
=
a

ch
t;
y
=
b

sh
t;
t
2
R
;
(I.20)
22
I.4.Ïàðàáîëà
ãäå
ch
t
=
e
t
+
e

t
2

ãèïåðáîëè÷åñêèéêîñèíóñ
,a
sh
t
=
e
t

e

t
2
ãèïåðáîëè÷åñêèéñèíóñ
.
Äåéñòâèòåëüíî,ïîäñòàâëÿÿâûðàæåíèÿêîîðäèíàòâóðàâíåíèå(
I.18
),ïðèõîäèìê
îñíîâíîìóãèïåðáîëè÷åñêîìóòîæäåñòâó
ch
2
t

sh
2
t
=1
.
I.4Ïàðàáîëà
Ïàðàáîëîé
íàçûâàåòñÿãåîìåòðè÷åñêîåìåñòîòî÷åêïëîñêîñòè,ðàâíîóäàë¼ííûõîò
çàäàííîéòî÷êè
F
èçàäàííîéïðÿìîé
d
,íåïðîõîäÿùåé÷åðåççàäàííóþòî÷êó.Ýòî
ãåîìåòðè÷åñêîåîïðåäåëåíèåâûðàæàåòäèðåêòîðèàëüíîåñâîéñòâîïàðàáîëû.
I.4.1Äèðåêòîðèàëüíîåñâîéñòâîïàðàáîëû
Òî÷êà
F
íàçûâàåòñÿ
ôîêóñîìïàðàáîëû
,ïðÿìàÿ
d

äèðåêòðèñîéïàðàáîëû
,
ñåðåäèíà
O
ïåðïåíäèêóëÿðà,îïóùåííîãîèçôîêóñàíàäèðåêòðèñó,
âåðøèíîéïàðà-
áîëû
,ðàññòîÿíèå
p
îòôîêóñàäîäèðåêòðèñû
ïàðàìåòðîìïàðàáîëû
,àðàññòîÿíèå
p
2
îòâåðøèíûïàðàáîëûäîå¼ôîêóñà
ôîêóñíûìðàññòîÿíèåì
(
Ðèñ.8
,a).Ïðÿìàÿ,
ïåðïåíäèêóëÿðíàÿäèðåêòðèñåèïðîõîäÿùàÿ÷åðåçôîêóñ,íàçûâàåòñÿ
îñüþïàðàáîëû
(ôîêàëüíîéîñüþïàðàáîëû)
.Îòðåçîê
FM
,ñîåäèíÿþùèéïðîèçâîëüíóþòî÷êó
M
ïàðàáîëûñå¼ôîêóñîì,íàçûâàåòñÿ
ôîêàëüíûìðàäèóñîì
òî÷êè
M
.Îòðåçîê,ñîåäè-
íÿþùèéäâåòî÷êèïàðàáîëû,íàçûâàåòñÿ
õîðäîéïàðàáîëû
.
Ðèñ.8
Äëÿïðîèçâîëüíîéòî÷êèïàðàáîëûîòíîøåíèåðàññòîÿíèÿäîôîêóñàêðàññòîÿíèþ
äîäèðåêòðèñûðàâíîåäèíèöå.Ñðàâíèâàÿäèðåêòîðèàëüíûåñâîéñòâàýëëèïñà,ãèïåðáî-
ëûèïàðàáîëû,çàêëþ÷àåì,÷òî
ýêñöåíòðèñèòåòïàðàáîëû
ïîîïðåäåëåíèþðàâåí
åäèíèöå
(
e
=1)
.
Ãåîìåòðè÷åñêîåîïðåäåëåíèåïàðàáîëû,âûðàæàþùååå¼äèðåêòîðèàëüíîåñâîéñòâî,
ýêâèâàëåíòíîå¼àíàëèòè÷åñêîìóîïðåäåëåíèþëèíèè,çàäàâàåìîéêàíîíè÷åñêèìóðàâ-
íåíèåìïàðàáîëû
:
y
2
=2

p

x:
(I.21)
23
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
Äåéñòâèòåëüíî,ââåäåìïðÿìîóãîëüíóþñèñòåìóêîîðäèíàò(
Ðèñ.8
,á).Âåðøèíó
O
ïàðàáîëûïðèìåìçàíà÷àëîñèñòåìûêîîðäèíàò;ïðÿìóþ,ïðîõîäÿùóþ÷åðåçôîêóñ
ïåðïåíäèêóëÿðíîäèðåêòðèñå,ïðèìåìçàîñüàáñöèññ(ïîëîæèòåëüíîåíàïðàâëåíèåíà
íåéîòòî÷êè
O
êòî÷êå
F
);ïðÿìóþ,ïåðïåíäèêóëÿðíóþîñèàáñöèññèïðîõîäÿùóþ÷åðåç
âåðøèíóïàðàáîëû,ïðèìåìçàîñüîðäèíàò(íàïðàâëåíèåíàîñèîðäèíàòâûáèðàåòñÿ
òàê,÷òîáûïðÿìîóãîëüíàÿñèñòåìàêîîðäèíàò
Oxy
îêàçàëàñüïðàâîé).
Ñîñòàâèìóðàâíåíèåïàðàáîëû,èñïîëüçóÿå¼ãåîìåòðè÷åñêîåîïðåäåëåíèå,âûðàæàþ-
ùååäèðåêòîðèàëüíîåñâîéñòâîïàðàáîëû.Ââûáðàííîéñèñòåìåêîîðäèíàòîïðåäåëÿåì
êîîðäèíàòûôîêóñà
F

p
2
;0

èóðàâíåíèåäèðåêòðèñû
x
=

p
2
.Äëÿïðîèçâîëüíîéòî÷êè
M
(
x;y
)
,ïðèíàäëåæàùåéïàðàáîëå,èìååì:
FM
=
MM
d
;
ãäå
M
d


p
2
;y

îðòîãîíàëüíàÿïðîåêöèÿòî÷êè
M
(
x;y
)
íàäèðåêòðèñó.Çàïèñûâàåì
ýòîóðàâíåíèåâêîîðäèíàòíîéôîðìå:
s

x

p
2

2
+
y
2
=
x
+
p
2
:
Âîçâîäèìîáå÷àñòèóðàâíåíèÿâêâàäðàò:

x

p
2

2
+
y
2
=
x
2
+
p

x
+
p
2
4
:
Ïðèâîäÿïîäîáíûå
÷ëåíû,ïîëó÷àåìêàíîíè÷åñêîåóðàâíåíèåïàðàáîëû
y
2
=2

p

x;
ò.å.âûáðàííàÿñèñòåìàêîîðäèíàòÿâëÿåòñÿêàíîíè÷åñêîé.
Ïðîâîäÿðàññóæäåíèÿâîáðàòíîìïîðÿäêå,ìîæíîïîêàçàòü,÷òîâñåòî÷êè,êîîð-
äèíàòûêîòîðûõóäîâëåòâîðÿþòóðàâíåíèþ(
I.21
),èòîëüêîîíè,ïðèíàäëåæàòãåîìåò-
ðè÷åñêîìóìåñòóòî÷åê,íàçûâàåìîìóïàðàáîëîé.Òàêèìîáðàçîì,àíàëèòè÷åñêîåîïðå-
äåëåíèåïàðàáîëûýêâèâàëåíòíîåãîãåîìåòðè÷åñêîìóîïðåäåëåíèþ,âûðàæàþùåìóäè-
ðåêòîðèàëüíîåñâîéñòâîïàðàáîëû.
I.4.2Óðàâíåíèåïàðàáîëûâïîëÿðíîéñèñòåìåêîîðäèíàò
Óðàâíåíèåïàðàáîëûâïîëÿðíîéñèñòåìåêîîðäèíàò
Fr'
(
Ðèñ.8
,â)èìååòâèä
r
=
p
1

e

cos
'
;
ãäå
p
ïàðàìåòðïàðàáîëû,à
e
=1
å¼ýêñöåíòðèñèòåò.
Âñàìîìäåëå,âêà÷åñòâåïîëþñàïîëÿðíîéñèñòåìûêîîðäèíàòâûáåðåìôîêóñ
F
ïàðàáîëû,àâêà÷åñòâåïîëÿðíîéîñèëó÷ñíà÷àëîìâòî÷êå
F
,ïåðïåíäèêóëÿðíûé
äèðåêòðèñåèíåïåðåñåêàþùèéå¼(
Ðèñ.8
,â).Òîãäàäëÿïðîèçâîëüíîéòî÷êè
M
(
r;'
)
,
ïðèíàäëåæàùåéïàðàáîëå,ñîãëàñíîãåîìåòðè÷åñêîìóîïðåäåëåíèþ(äèðåêòîðèàëüíîìó
ñâîéñòâó)ïàðàáîëû,èìååì
MM
d
=
r
.Ïîñêîëüêó
MM
d
=
p
+
r

cos
'
,ïîëó÷àåì
óðàâíåíèåïàðàáîëûâêîîðäèíàòíîéôîðìå:
p
+
r

cos
'
,
r
=
p
1

cos
'
;
÷òîèòðåáîâàëîñüäîêàçàòü.Çàìåòèì,÷òîâïîëÿðíûõêîîðäèíàòàõóðàâíåíèÿýëëèïñà,
ãèïåðáîëûèïàðàáîëûñîâïàäàþò,íîîïèñûâàþòðàçíûåëèíèè,ïîñêîëüêóîòëè÷àþòñÿ
ýêñöåíòðèñèòåòàìè(
0
6
e
1
äëÿýëëèïñà,
e
=1
äëÿïàðàáîëû,
e�
1
äëÿãèïåðáîëû).
24
I.4.Ïàðàáîëà
I.4.3Ãåîìåòðè÷åñêèéñìûñëïàðàìåòðàâóðàâíåíèèïàðàáîëû
Ïîÿñíèì
ãåîìåòðè÷åñêèéñìûñëïàðàìåòðà
p
âêàíîíè÷åñêîìóðàâíåíèèïàðàáîëû.
Ïîäñòàâëÿÿâóðàâíåíèå(
I.21
)
x
=
p
2
,ïîëó÷àåì
y
2
=
p
2
,ò.å.
y
=

p
.Ñëåäîâàòåëü-
íî,ïàðàìåòð
p
ýòîïîëîâèíàäëèíûõîðäûïàðàáîëû,ïðîõîäÿùåé÷åðåçå¼ôîêóñ
ïåðïåíäèêóëÿðíîîñèïàðàáîëû.
Ôîêàëüíûìïàðàìåòðîìïàðàáîëû
,òàêæåêàêäëÿýëëèïñàèäëÿãèïåðáîëû,
íàçûâàåòñÿïîëîâèíàäëèíûõîðäû,ïðîõîäÿùåé÷åðåçå¼ôîêóñïåðïåíäèêóëÿðíîôî-
êàëüíîéîñè(
Ðèñ.8
,â).Èçóðàâíåíèÿïàðàáîëûâïîëÿðíûõêîîðäèíàòàõïðè
'
=

2
ïîëó÷àåì
r
=
p
,ò.å.ïàðàìåòðïàðàáîëûñîâïàäàåòñå¼ôîêàëüíûìïàðàìåòðîì.
Çàìå÷àíèå3.
1.
Ïàðàìåòð
p
ïàðàáîëûõàðàêòåðèçóåòå¼ôîðìó.×åìáîëüøå
p
,òåìøèðåâåòâè
ïàðàáîëû,÷åìáëèæå
p
êíóëþ,òåìâåòâèïàðàáîëûóæå(
Ðèñ.9
).
Ðèñ.9
2.
Óðàâíåíèå
y
2
=

2

p

x
(ïðè
p�
0
)îïðåäåëÿåòïàðàáîëó,êîòîðàÿðàñïîëîæåíà
ñëåâàîòîñèîðäèíàò(
Ðèñ.10
,a).Ýòîóðàâíåíèåñâîäèòñÿêêàíîíè÷åñêîìóïðèïîìîùè
èçìåíåíèÿíàïðàâëåíèÿîñèàáñöèññ.Íàðèñ.(
Ðèñ.10
,a)èçîáðàæåíûçàäàííàÿñèñòåìà
êîîðäèíàò
Oxy
èêàíîíè÷åñêàÿ
Ox
0
y
0
.
3.
Óðàâíåíèå
(
y

y
0
)
2
=2

p

(
x

x
0
)
;p�
0
,îïðåäåëÿåòïàðàáîëóñâåðøèíîé
O
0
(
x
0
;y
0
)
,îñüêîòîðîéïàðàëëåëüíàîñèàáñöèññ(
Ðèñ.10
,á).Ýòîóðàâíåíèåñâîäèòñÿê
êàíîíè÷åñêîìóïðèïîìîùèïàðàëëåëüíîãîïåðåíîñà.
Óðàâíåíèå
(
x

x
0
)
2
=2

p

(
y

y
0
)
;p�
0
,òàêæåîïðåäåëÿåòïàðàáîëóñâåðøèíîé
O
0
(
x
0
;y
0
)
,îñüêîòîðîéïàðàëëåëüíàîñèîðäèíàò(
Ðèñ.10
,â).Ýòîóðàâíåíèåñâîäèòñÿê
êàíîíè÷åñêîìóïðèïîìîùèïàðàëëåëüíîãîïåðåíîñàèïåðåèìåíîâàíèÿêîîðäèíàòíûõ
îñåé.Íàðèñ.(
Ðèñ.10
,á,â)èçîáðàæåíûçàäàííûåñèñòåìûêîîðäèíàò
Oxy
èêàíîíè÷å-
ñêèåñèñòåìûêîîðäèíàò
O
0
x
0
y
0
.
4.
Ãðàôèêêâàäðàòíîãîòðåõ÷ëåíà
y
=
ax
2
+
bx
+
c
,
a
6
=0
;
ÿâëÿåòñÿïàðàáîëîéñâåðøèíîéâòî÷êå
O
0


b
2
a
;

b
2

4
ac
4
a

;
îñüêîòîðîéïàðàëëåëüíàîñè
îðäèíàò,âåòâèïàðàáîëûíàïðàâëåíûââåðõ(ïðè
a�
0
)èëèâíèç(ïðè
a
0
:
)Äåéñòâè-
25
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
Ðèñ.10
òåëüíî,âûäåëÿÿïîëíûéêâàäðàò,ïîëó÷àåìóðàâíåíèå
y
=
a

x
+
b
2
a

2

b
2
4
a
+
c
,

x
+
b
2
a

2
=
1
a

y
+
b
2

4
ac
4
a
!
;
êîòîðîåïðèâîäèòñÿêêàíîíè÷åñêîìóâèäó
(
y
0
)
2
=2

p

x
0
,ãäå
p
=




1
2
a




,ïðèïîìîùè
çàìåíû
y
0
=
x
+
b
2
a
è
x
0
=


y
+
b
2

4
ac
4
a
!
.Çíàêâûáèðàåòñÿñîâïàäàþùèìñîçíàêîì
ñòàðøåãîêîýôôèöèåíòà
a
.Ýòàçàìåíàñîîòâåòñòâóåòêîìïîçèöèè:ïàðàëëåëüíîãîïåðå-
íîñàñ
x
0
=

b
2
a
è
y
0
=

b
2

4
ac
4
a
,ïåðåèìåíîâàíèÿêîîðäèíàòíûõîñåé,àâñëó÷àå
a
0
åù¼èèçìåíåíèÿíàïðàâëåíèÿêîîðäèíàòíîéîñè.(
Ðèñ.11
,a,â)èçîáðàæåíûçàäàííûå
ñèñòåìûêîîðäèíàò
Oxy
èêàíîíè÷åñêèåñèñòåìûêîîðäèíàò
O
0
x
0
y
0
äëÿñëó÷àåâ
a�
0
è
a
0
ñîîòâåòñòâåííî.
Ðèñ.11
26
I.5.Íåêîòîðûåîáùèåñâîéñòâàýëëèïñà,ãèïåðáîëû,ïàðàáîëû
5.
Îñüàáñöèññêàíîíè÷åñêîéñèñòåìûêîîðäèíàòÿâëÿåòñÿ
îñüþñèììåòðèèïà-
ðàáîëû
,ïîñêîëüêóçàìåíàïåðåìåííîé
y
íà

y
íåèçìåíÿåòóðàâíåíèÿ(
I.21
).Äðóãè-
ìèñëîâàìè,êîîðäèíàòûòî÷êè
M
(
x;y
)
,ïðèíàäëåæàùåéïàðàáîëå,èêîîðäèíàòûòî÷êè
M
0
(
x;

y
)
,ñèììåòðè÷íîéòî÷êå
M
îòíîñèòåëüíîîñèàáñöèññ,óäîâëåòâîðÿþòóðàâíåíèþ
(
I.21
).Îñèêàíîíè÷åñêîéñèñòåìûêîîðäèíàòíàçûâàþòñÿ
ãëàâíûìèîñÿìèïàðàáî-
ëû
.
I.5Íåêîòîðûåîáùèåñâîéñòâàýëëèïñà,ãèïåðáîëû,ïà-
ðàáîëû
1.
Äèðåêòîðèàëüíîåñâîéñòâîìîæåòáûòüèñïîëüçîâàíîêàêåäèíîåîïðåäåëåíèåýë-
ëèïñà,ãèïåðáîëû,ïàðàáîëû(
Ðèñ.12
):
ãåîìåòðè÷åñêîåìåñòîòî÷åêïëîñêîñòè,äëÿ
êàæäîéèçêîòîðûõîòíîøåíèåðàññòîÿíèÿäîçàäàííîéòî÷êè
F
(ôîêóñà)êðàññòî-
ÿíèþäîçàäàííîéïðÿìîé
d
(äèðåêòðèñû),íåïðîõîäÿùåé÷åðåççàäàííóþòî÷êó,ïî-
ñòîÿííîèðàâíîýêñöåíòðèñèòåòó
e
,íàçûâàåòñÿ:
a)
ýëëèïñîì,
åñëè
0
6
e
1
;
b)
ãèïåðáîëîé,
åñëè
e�
1
;
c)
ïàðàáîëîé,
åñëè
e
=1
.
Ðèñ.12
2.
Ýëëèïñ,ãèïåðáîëà,ïàðàáîëàïîëó÷àþòñÿâñå÷åíèÿõêðóãîâîãîêîíóñàïëîñêîñòÿìè
èïîýòîìóíàçûâàþòñÿ
êîíè÷åñêèìèñå÷åíèÿìè
.Ýòîñâîéñòâîòàêæåìîæåòñëóæèòü
ãåîìåòðè÷åñêèìîïðåäåëåíèåìýëëèïñà,ãèïåðáîëû,ïàðàáîëû.
27
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
3.
Ê÷èñëóîáùèõñâîéñòâýëëèïñà,ãèïåðáîëûèïàðàáîëûìîæíîîòíåñòè
áèññåê-
òîðèàëüíîåñâîéñòâî
èõêàñàòåëüíûõ.Ïîä
êàñàòåëüíîé
êëèíèèâíåêîòîðîéå¼
òî÷êå
K
ïîíèìàåòñÿïðåäåëüíîåïîëîæåíèåñåêóùåé
KM
,êîãäàòî÷êà
M
,îñòàâàÿñüíà
ðàññìàòðèâàåìîéëèíèè,ñòðåìèòñÿêòî÷êå
K
.Ïðÿìàÿ,ïåðïåíäèêóëÿðíàÿêàñàòåëüíîé
êëèíèèèïðîõîäÿùàÿ÷åðåçòî÷êóêàñàíèÿ,íàçûâàåòñÿ
íîðìàëüþ
êýòîéëèíèè.
Áèññåêòîðèàëüíîåñâîéñòâîêàñàòåëüíûõ(èíîðìàëåé)êýëëèïñó,ãèïåðáîëåèïà-
ðàáîëåôîðìóëèðóåòñÿñëåäóþùèìîáðàçîì:
êàñàòåëüíàÿ(íîðìàëü)êýëëèïñóèëèê
ãèïåðáîëåîáðàçóåòðàâíûåóãëûñôîêàëüíûìèðàäèóñàìèòî÷êèêàñàíèÿ
(
Ðèñ.13
,à,á);
êàñàòåëüíàÿ(íîðìàëü)êïàðàáîëåîáðàçóåòðàâíûåóãëûñôîêàëüíûìðàäèóñîìòî÷êè
êàñàíèÿèïåðïåíäèêóëÿðîì,îïóùåííûìèçíå¼íàäèðåêòðèñó
(
Ðèñ.13
,â).Äðóãèìè
ñëîâàìè,êàñàòåëüíàÿêýëëèïñóâòî÷êå
K
ÿâëÿåòñÿáèññåêòðèñîéâíåøíåãîóãëàòðå-
óãîëüíèêà
F
1
KF
2
(àíîðìàëüáèññåêòðèñîéâíóòðåííåãîóãëà
F
1
KF
2
òðåóãîëüíèêà);
êàñàòåëüíàÿêãèïåðáîëåÿâëÿåòñÿáèññåêòðèñîéâíóòðåííåãîóãëàòðåóãîëüíèêà
F
1
KF
2
(àíîðìàëüáèññåêòðèñîéâíåøíåãîóãëà);êàñàòåëüíàÿêïàðàáîëåÿâëÿåòñÿáèññåê-
òðèñîéâíóòðåííåãîóãëàòðåóãîëüíèêà
FKK
d
(àíîðìàëüáèññåêòðèñîéâíåøíåãî
óãëà).Áèññåêòîðèàëüíîåñâîéñòâîêàñàòåëüíîéêïàðàáîëåìîæíîñôîðìóëèðîâàòüòàê
æå,êàêäëÿýëëèïñàèãèïåðáîëû,åñëèñ÷èòàòü,÷òîóïàðàáîëûèìååòñÿâòîðîéôîêóñ
âáåñêîíå÷íîóäàë¼ííîéòî÷êå.
Ðèñ.13
4.
Èçáèññåêòîðèàëüíûõñâîéñòâñëåäóþò
îïòè÷åñêèåñâîéñòâà
ýëëèïñà,ãèïåð-
áîëûèïàðàáîëû,ïîÿñíÿþùèåôèçè÷åñêèéñìûñëòåðìèíà"ôîêóñ".Ïðåäñòàâèìñåáå
ïîâåðõíîñòè,îáðàçîâàííûåâðàùåíèåìýëëèïñà,ãèïåðáîëûèëèïàðàáîëûâîêðóãôî-
êàëüíîéîñè.Åñëèíàýòèïîâåðõíîñòèíàíåñòèîòðàæàþùååïîêðûòèå,òîïîëó÷àþòñÿ
ýëëèïòè÷åñêîå,ãèïåðáîëè÷åñêîåèïàðàáîëè÷åñêîåçåðêàëà.Ñîãëàñíîçàêîíóîïòèêè,
óãîëïàäåíèÿëó÷àñâåòàíàçåðêàëîðàâåíóãëóîòðàæåíèÿ,ò.å.ïàäàþùèéèîòðàæåí-
íûéëó÷èîáðàçóþòðàâíûåóãëûñíîðìàëüþêïîâåðõíîñòè,ïðè÷¼ìîáàëó÷àèîñü
âðàùåíèÿíàõîäÿòñÿâîäíîéïëîñêîñòè.Îòñþäàïîëó÷àåìñëåäóþùèåñâîéñòâà:
åñëèèñòî÷íèêñâåòàíàõîäèòñÿâîäíîìèçôîêóñîâýëëèïòè÷åñêîãîçåðêàëà,òî
ëó÷èñâåòà,îòðàçèâøèñüîòçåðêàëà,ñîáèðàþòñÿâäðóãîìôîêóñå(
Ðèñ.14
,à);
åñëèèñòî÷íèêñâåòàíàõîäèòñÿâîäíîìèçôîêóñîâãèïåðáîëè÷åñêîãîçåðêàëà,
òîëó÷èñâåòà,îòðàçèâøèñüîòçåðêàëà,ðàñõîäÿòñÿòàê,êàêåñëèáûîíèèñõîäèëèèç
äðóãîãîôîêóñà(
Ðèñ.14
,á);
åñëèèñòî÷íèêñâåòàíàõîäèòñÿâôîêóñåïàðàáîëè÷åñêîãîçåðêàëà,òîëó÷èñâåòà,
îòðàçèâøèñüîòçåðêàëà,èäóòïàðàëëåëüíîôîêàëüíîéîñè(
Ðèñ.14
,â).
28
I.5.Íåêîòîðûåîáùèåñâîéñòâàýëëèïñà,ãèïåðáîëû,ïàðàáîëû
Ðèñ.14
5.
Äèàìåòðàëüíîåñâîéñòâî
ýëëèïñà,ãèïåðáîëûèïàðàáîëûìîæíîñôîðìóëè-
ðîâàòüñëåäóþùèìîáðàçîì:

ñåðåäèíûïàðàëëåëüíûõõîðäýëëèïñà(ãèïåðáîëû)ëåæàòíàîäíîéïðÿìîé,ïðîõî-
äÿùåé÷åðåçöåíòðýëëèïñà(ãèïåðáîëû)
;

ñåðåäèíûïàðàëëåëüíûõõîðäïàðàáîëûëåæàòíàïðÿìîé,êîëëèíåàðíîéîñèñèì-
ìåòðèèïàðàáîëû
.
Ãåîìåòðè÷åñêîåìåñòîñåðåäèíâñåõïàðàëëåëüíûõõîðäýëëèïñà(ãèïåðáîëû,ïàðàáî-
ëû)íàçûâàþò
äèàìåòðîìýëëèïñà(ãèïåðáîëû,ïàðàáîëû)
,ñîïðÿæ¼ííûìêýòèì
õîðäàì.
Ýòîîïðåäåëåíèåäèàìåòðàâóçêîìñìûñëå.Ðàíååáûëîäàíîîïðåäåëåíèåäèàìåò-
ðàâøèðîêîìñìûñëå,ãäåäèàìåòðîìýëëèïñà,ãèïåðáîëû,ïàðàáîëû,àòàêæåäðóãèõ
ëèíèéâòîðîãîïîðÿäêàíàçûâàåòñÿïðÿìàÿ,ñîäåðæàùàÿñåðåäèíûâñåõïàðàëëåëüíûõ
õîðä.Âóçêîìñìûñëåäèàìåòðîìýëëèïñàÿâëÿåòñÿëþáàÿõîðäà,ïðîõîäÿùàÿ÷åðåç
åãîöåíòð(
Ðèñ.15
,a);äèàìåòðîìãèïåðáîëûÿâëÿåòñÿëþáàÿïðÿìàÿ,ïðîõîäÿùàÿ÷å-
ðåçöåíòðãèïåðáîëû(çàèñêëþ÷åíèåìàñèìïòîò),ëèáî÷àñòüòàêîéïðÿìîé(
Ðèñ.15
,á);
äèàìåòðîìïàðàáîëûÿâëÿåòñÿëþáîéëó÷,èñõîäÿùèéèçíåêîòîðîéòî÷êèïàðàáîëûè
êîëëèíåàðíûéîñèñèììåòðèè(
Ðèñ.15
,â).
Äâàäèàìåòðà,êàæäûéèõêîòîðûõäåëèòïîïîëàìâñåõîðäû,ïàðàëëåëüíûåäðó-
ãîìóäèàìåòðó,íàçûâàþòñÿ
ñîïðÿæ¼ííûìè
.Íà(
Ðèñ.14
)ïîëóæèðíûìèëèíèÿìè
èçîáðàæåíûñîïðÿæ¼ííûåäèàìåòðûýëëèïñà,ãèïåðáîëû,ïàðàáîëû.
Êàñàòåëüíóþêýëëèïñó(ãèïåðáîëå,ïàðàáîëå)âòî÷êå
K
ìîæíîîïðåäåëèòüêàê
ïðåäåëüíîåïîëîæåíèåïàðàëëåëüíûõñåêóùèõ
M
1
M
2
,êîãäàòî÷êè
M
1
è
M
2
,îñòàâàÿñü
íàðàññìàòðèâàåìîéëèíèè,ñòðåìÿòñÿêòî÷êå
K
.Èçýòîãîîïðåäåëåíèÿñëåäóåò,÷òî
êàñàòåëüíàÿ,ïàðàëëåëüíàÿõîðäàì,ïðîõîäèò÷åðåçêîíåöäèàìåòðà,ñîïðÿæ¼ííîãîê
ýòèìõîðäàì.
29
ÃëàâàI.Êðèâûåâòîðîãîïîðÿäêà
Ðèñ.15
6.
Ýëëèïñ,ãèïåðáîëàèïàðàáîëàèìåþò,êðîìåïðèâåä¼ííûõâûøå,ìíîãî÷èñëåí-
íûåãåîìåòðè÷åñêèåñâîéñòâàèôèçè÷åñêèåïðèëîæåíèÿ.Íàïðèìåð,(
Ðèñ.12
)ìîæåò
ñëóæèòüèëëþñòðàöèåéòðàåêòîðèéäâèæåíèÿêîñìè÷åñêèõîáúåêòîâ,íàõîäÿùèõñÿâ
îêðåñòíîñòèöåíòðà
F
ïðèòÿæåíèÿ.
30
ÃëàâàII
Ìàòåìàòè÷åñêèåîñíîâûäâèæåíèÿâ
ïîëåòÿæåñòè
II.1Äâèæåíèåìàòåðèàëüíîéòî÷êèïîääåéñòâèåìñè-
ëûòÿãîòåíèÿ(çàäà÷àäâóõòåë)
Êåïëåðïðîÿâèëíåîáû÷àéíóþèçîáðåòàòåëüíîñòüèçàòðàòèëîãðîìíîåêîëè÷åñòâî
òðóäà,÷òîáûíàîñíîâàíèèíàáëþäåíèé,ò.å.ýìïèðè÷åñêè,ïîëó÷èòüñâîèòðèçàêîíà
äâèæåíèÿïëàíåò.Íüþòîí,çàâåðøèâíà÷èíàíèåÃàëèëåÿ,íàø¼ëáîëååîáùèåïðèíöèïû,
èçêîòîðûõ,â÷àñòíîñòè,ñëåäóþòçàêîíûÊåïëåðà,ïðè÷åìâáîëååîáùåìâèäå,÷åìèõ
íàø¼ëÊåïëåð.ÏîýòîìóâûâîäçàêîíîâÊåïëåðàèççàêîíîâÍüþòîíàèìååòïðèíöèïè-
àëüíîåçíà÷åíèå,èìûïîñòàðàåìñÿâýòîìïàðàãðàôåäàòüèõñòðîãèåäîêàçàòåëüñòâà.
Áóäåìðåøàòüçàäà÷óäâóõòåë,èñïîëüçóÿñâîéñòâàâåêòîðîâ.Ïîìèìîèçâåñòíûõ
äåéñòâèéñëîæåíèÿèðàçëîæåíèÿ,ñâåêòîðàìèìîæíîäåëàòüèäðóãèåìàòåìàòè÷åñêèå
îïåðàöèè(óìíîæåíèå,äèôôåðåíöèðîâàíèåèò.ï.).
Ïðîèçâåäåíèåâåêòîðîâìîæåòáûòüäâóõòèïîâ:âåêòîðíîåèñêàëÿðíîå.
Ñêàëÿðíûì
ïðîèçâåäåíèåìäâóõâåêòîðîâ
a
è
b
íàçûâàåòñÿñêàëÿð
c
,îïðåäåëÿåìûéâûðàæåíèåì
c
=
a

b
=
a

b

cos
;
(II.1)
Âåêòîðíûìïðîèçâåäåíèåìâåêòîðîâ
a
è
b
íàçûâàåòñÿâåêòîð
c
,ïåðïåíäèêóëÿð-
íûéïëîñêîñòèïàðàëëåëîãðàììà,ïîñòðîåííîãîíàâåêòîðàõ
a
è
b
êàêíàñòîðîíàõèñ
äëèíîé,ðàâíîéåãîïëîùàäè,ò.å.
j
c
j
=
j
a

b
j
=
a

b

sin
:
(II.2)
Íàïðàâëåíèåâåêòîðà
c
âûáèðàåòñÿïîïðàâèëóâèíòà:åñëèñìîòðåòüñêîíöàâåêòîðà
c
,òîêðàò÷àéøèéïîâîðîò
a
ê
b
äîëæåíáûòüïðîòèâ÷àñîâîéñòðåëêè(
Ðèñ.16
).Âåêòîð-
íîåïðîèçâåäåíèåáóäåìîáîçíà÷àòüêâàäðàòíûìèñêîáêàìè,àñàìèâåêòîðûæèðíûì
øðèôòîì:
[ab]
.
31
ÃëàâàII.Ìàòåìàòè÷åñêèåîñíîâûäâèæåíèÿâïîëåòÿæåñòè
Ðèñ.16
Çàìåòèì,÷òîèçîïðåäåëåíèÿâåêòîðíîãîïðîèçâåäåíèÿñëåäóåòåãîíåêîììóòàòèâ-
íîñòü
[
ab
]=

[
ba
]
:
Êðîìåòîãî,î÷åâèäíî,÷òî
[
aa
]=0
:
Òåïåðüïåðåõîäèìêóðàâíåíèþäâèæåíèÿâçàäà÷åäâóõòåë.Ðàññìîòðèìèíåðöèàëüíóþ
ñèñòåìóîòñ÷¼òà
x
0
,
y
0
,
z
0
(
Ðèñ.17
).Ïóñòüâåêòîðû
r
1
è
r
2
èçîáðàæàþòïîëîæåíèåäâóõ
òåëñìàññàìè
m
1
è
m
2
.
Ðèñ.17
Äëÿîïðåäåë¼ííîñòèáóäåìèõíàçûâàòüÑîëíöåèïëàíåòà.Äâèæåíèåîáîèõòåëïðî-
èñõîäèòïîääåéñòâèåìäâóõñèë,
F
12
è
F
21
,îäèíàêîâûõïîâåëè÷èíå,íîïðîòèâîïî-
ëîæíîíàïðàâëåííûõ.Âåëè÷èíàýòèõñèëîïðåäåëÿåòñÿìàññàìèòåëèäëèíîéâåêòî-
ðà
r
=
r
2

r
1
:
Ïîíàïðàâëåíèþñèëà
F
21
,äåéñòâóþùàÿñîñòîðîíûòåëà
m
2
íàòåëî
32
II.1.Äâèæåíèåìàòåðèàëüíîéòî÷êèïîääåéñòâèåìñèëûòÿãîòåíèÿ(çàäà÷àäâóõòåë)
m
1
,ïàðàëëåëüíàâåêòîðó
r
,àñèëà
F
12
àíòèïàðàëëåëüíàåìó,òàêêàêèìåííîîíàâûçû-
âàåòäâèæåíèåïëàíåòûâîêðóãÑîëíöà.Ïîýòîìó
j
F
21
j
=
j
F
12
j
=
Gm
1
m
2
r
2
;
èëè,ó÷èòûâàÿ,÷òî
r
r
åäèíè÷íûéâåêòîðâíàïðàâëåíèèîò
m
1
ê
m
2
,
F
21
=
Gm
1
m
2
r
3
r
;
è
F
12
=

Gm
1
m
2
r
3
r
:
(II.3)
ÇàïèøåìâòîðîéçàêîíÍüþòîíàäëÿäâèæåíèÿïëàíåòûèÑîëíöà:
m
1
d
v
1
dt
=
Gm
1
m
2
r
3
r
;m
2
d
v
2
dt
=

Gm
1
m
2
r
3
r
:
(II.4)
Âû÷èòàÿïî÷ëåííîóðàâíåíèÿ(
II.4
)îäíîèçäðóãîãî,ïîëó÷èìóðàâíåíèåäâèæåíèÿïëà-
íåòûîòíîñèòåëüíîÑîëíöà
m
2
d
(
v
2

v
1
)
dt
=

m
2
G
(
m
1
+
m
2
)
r
3
r
:
(II.5)
Áóäåìòåïåðüðàññìàòðèâàòüâåêòîð
r
êàêðàäèóñ-âåêòîðâíåêîòîðîéïîëÿðíîéñèñòåìå
îòñ÷¼òà
(
r;'
)
ñïîëþñîìâòî÷êå
m
1
(
Ðèñ.17
,á).Ýòàñèñòåìàâîáùåìñëó÷àåíåèíåð-
öèàëüíà:âäàííîìñëó÷àåîíàÿâëÿåòñÿãåëèîöåíòðè÷åñêîé.Îáîçíà÷àÿ
v
=
v
2

v
1
è
M
=
m
1
+
m
2
;
(II.6)
ïîëó÷èì
d
v
dt
=
GM
r
3
r
:
(II.7)
Òàêèìîáðàçîì,äâèæåíèåòåëà
m
2
îòíîñèòåëüíî
m
1
ïðîèñõîäèòòàêæå,êàêèâ
èíåðöèàëüíîéñèñòåìå,íîïðèóñëîâèè,÷òîâïîëþñåñîñðåäîòî÷åíàñóììàìàññîáîèõ
òåë.Ïîêàæåì,÷òîèçóðàâíåíèÿ(
II.7
)ñëåäóþòçàêîíûÊåïëåðà.Íà÷í¼ìñîâòîðîãîèç
íèõ,òàêêàêîíïîòðåáóåòñÿäëÿâûâîäàïåðâîãî.
ÂòîðîéçàêîíÊåïëåðà.
Ïîëüçóÿñüîïðåäåëåíèåìïðîèçâîäíîé,ìîæíîïîêàçàòü,÷òîïðîèçâåäåíèÿâåêòîðîâ
äèôôåðåíöèðóþòñÿïîòåìæåïðàâèëàì,÷òîèïðîèçâåäåíèÿñêàëÿðíûõâåëè÷èí.Ðàñ-
ñìîòðèìïðîèçâîäíóþïîâðåìåíèîòìîìåíòàñêîðîñòè
d
dt
[
rv
]=[
r
d
v
dt
]+[
d
r
dt
v
]
:
Î÷åâèäíî,÷òîîáàñëàãàåìûõâïðàâîé÷àñòèýòîãîâûðàæåíèÿðàâíûíóëþ,òàêêàêâ
íèõâåêòîðíîïåðåìíîæàþòñÿêîëëèíåàðíûåâåêòîðû:óñêîðåíèå
d
v
dt
íàïðàâëåíîâäîëü
äåéñòâèÿöåíòðàëüíîéñèëûïàðàëëåëüíîâåêòîðó
r

d
r
dt
=
v
ïîîïðåäåëåíèþñêîðîñòè.
Ñëåäîâàòåëüíî,
d
dt
[
rv
]=0
è
[
rv
]=[
r
d
r
dt
]=
C
;
(II.8)
ò.å.ìîìåíòâåêòîðàñêîðîñòèåñòü
ïîñòîÿííûéâåêòîð
Ñ
.Î÷åâèäíî,÷òîîíîïðåäåëÿåò
íåèçìåííóþïëîñêîñòüîðáèòûîòíîñèòåëüíîãîäâèæåíèÿïëàíåòû,ïðîõîäÿùóþ÷åðåç
öåíòðàëüíîåòåëî.
33
ÃëàâàII.Ìàòåìàòè÷åñêèåîñíîâûäâèæåíèÿâïîëåòÿæåñòè
Ïîîïðåäåëåíèþâåêòîðíîãîïðîèçâåäåíèÿ(
II.2
)äëèíàâåêòîðà
[
r
d
r
]
ðàâíàóäâîåííîé
ïëîùàäèòðåóãîëüíèêà
dS
,ïîñòðîåííîãîíàâåêòîðàõ
r
è
d
r
(
Ðèñ.17
,á).Ïîýòîìó
C
=2
dS
dt
=2

1
2
r
2
d'
dt
èëè
d'
dt
=
C
r
2
:
(II.9)
Êîíñòàíòà
C
íàçûâàåòñÿñåêòîðíîéñêîðîñòüþèÿâëÿåòñÿìîäóëåìâåêòîðà
C
.Îíà
îçíà÷àåòïîñòîÿíñòâîïëîùàäè,çàìåòàåìîéðàäèóñ-âåêòîðîìçàåäèíèöóâðåìåíè.Òà-
êèìîáðàçîì,ìûïðèõîäèìêñëåäóþùåéóòî÷í¼ííîéôîðìóëèðîâêåâòîðîãîçàêîíà
Êåïëåðà(èëèçàêîíàïëîùàäåé):
ðàäèóñ-âåêòîð,õàðàêòåðèçóþùèéïîëîæåíèåäâèæóùåãîñÿòåëàîòíîñèòåëüíîíåïî-
äâèæíîãîöåíòðàëüíîãîòåëàâçàäà÷åäâóõòåë,âñåãäàëåæèòâíåèçìåííîéïëîñêî-
ñòèîðáèòûèçàðàâíûåïðîìåæóòêèâðåìåíèîïèñûâàåòïëîùàäèðàâíîéâåëè÷èíû.
ÏåðâûéçàêîíÊåïëåðà.
Çàïèøåìóðàâíåíèÿ(
II.4
)äëÿäâèæåíèÿòåëà
m
1
âïîëÿðíîéñèñòåìåêîîðäèíàò
(
r;'
)
âïðîåêöèèíàîñè
x
è
y
,êàêýòîèçîáðàæåíîíà(
Ðèñ.17
,á):
x
=
r
cos
';
y
=
r
sin
';
d
2
x
dt
2
=
dv
x
dt
=

GM
r
2
cos
'
;
d
2
y
dt
2
=
dv
y
dt
=

GM
r
2
sin
':
(II.10)
Ïîäåëèìýòèóðàâíåíèÿïî÷ëåííîíàóðàâíåíèÿ(
II.9
).Òîãäà
dv
x
d'
=

GM
C
cos
';
dv
y
d'
=

GM
C
sin
';
dv
x
=

GM
C
d
(sin
'
);
dv
y
=+
GM
C
d
(cos
'
)
:
(II.11)
Ôóíêöèè,äèôôåðåíöèàëûêîòîðûõðàâíû,ìîãóòðàçëè÷àòüñÿëèøüíàïîñòîÿííûåâå-
ëè÷èíû,êîòîðûåìûîáîçíà÷èìñîîòâåòñòâåííî÷åðåç
A
è
B
,
v
x
=

GM
C
sin
'
+
A;
v
y
=
GM
C
cos
'
+
B:
(II.12)
Âûáåðåìíàïðàâëåíèåîñè
x
òàê,÷òîáûïðè
'
=0
ñêîðîñòüáûëàíàïðàâëåíàòîëüêî
âäîëüîñè
y;
ò.å.
v
x
=0
:
Òîãäà
A
=0
:
Ïðîäèôôåðåíöèðóåìâûðàæåíèÿäëÿ
x
è
y
èç
ôîðìóë(
II.10
)ñöåëüþïîëó÷èòüâûðàæåíèÿäëÿëåâûõ÷àñòåéóðàâíåíèé(
II.12
).Òîãäà
dx
dt
=
dr
dt
cos
'

r
sin
'
d'
dt
=

GM
C
sin
'
+
A;
dy
dt
=
dr
dt
sin
'
+
r
cos
'
d'
dt
=
GM
C
cos
'
+
B:
34
II.1.Äâèæåíèåìàòåðèàëüíîéòî÷êèïîääåéñòâèåìñèëûòÿãîòåíèÿ(çàäà÷àäâóõòåë)
Èñêëþ÷àÿèçýòèõóðàâíåíèé
dr
dt
óìíîæåíèåìïåðâîãîíà

sin
'
,àâòîðîãîíà

cos
'
è
ïî÷ëåííûìñëîæåíèåì,ïîëó÷èìñó÷åòîì,÷òî
A
=0
r
d'
dt
=
GM
C
+
B
cos
':
Èñêëþ÷èì
d'
dt
ñïîìîùüþçàêîíàïëîùàäåé(
II.9
)
1
r
=
GM
C
2
+
B
C
cos
':
Ââåäÿîáîçíà÷åíèÿ
C
2
GM
=
p
è
BC
GM
=
e;
(II.13)
ïîëó÷èìîáùååóðàâíåíèåêîíè÷åñêèõñå÷åíèéâïîëÿðíûõêîîðäèíàòàõñïîëþñîìâ
ôîêóñåêðèâîé
r
=
p
(1+
e

cos
'
)
:
(II.14)
Âåëè÷èíà
p
íàçûâàåòñÿ
ïàðàìåòðîì

e

ýêñöåíòðèñèòåòîì
êðèâîéâòîðîãîïîðÿäêà.
Èç÷åòíîñòèôóíêöèèêîñèíóñàñëåäóåò,÷òîêðèâàÿ,èçîáðàæàåìàÿóðàâíåíèåì(
II.14
),
ñèììåòðè÷íàîòíîñèòåëüíîïðÿìûõ
'
=0
è
'
=

.Ïîñêîëüêóïðè
e
1
çàìåíà
e
íà

e
è
'
íà
(


'
)
íåìåíÿåòâèäàêðèâîé,òîâýòîìñëó÷àåèìååòñÿåùåîäíàîñü
ñèììåòðèè,ïàðàëëåëüíàÿîñè
y
,è,ñëåäîâàòåëüíî,âòîðîéôîêóñ.Ëèíèÿçàìêíóòàè
îáðàçóåòýëëèïñ.Åãîóðàâíåíèåîòíîñèòåëüíîâòîðîãîôîêóñà,î÷åâèäíî,èìååòâèä
r
=
p
(1

e

cos
'
)
:
(II.15)
Åñëè
e
=1
,âåòâèêðèâîéóõîäÿòâáåñêîíå÷íîñòüïðè
'
=

(èëè
'
=0
äëÿóðàâíåíèÿ
âèäà(
II.15
)).Ïðè
e�
1
ñóùåñòâóþòäâåàñèìïòîòûïðè
cos
'
=

1
e
(ãèïåðáîëà).Ïðè
e
=0
îðáèòàîêðóæíîñòü.
Òàêèìîáðàçîì,ïåðâûéçàêîíÊåïëåðàâóòî÷íåííîìâèäåìîæíîñôîðìóëèðîâàòü
òàê:
ïîääåéñòâèåìñèëûòÿãîòåíèÿîäíîíåáåñíîåòåëîäâèæåòñÿîòíîñèòåëüíîäðó-
ãîãîïîîäíîìóèçêîíè÷åñêèõñå÷åíèé:ãèïåðáîëå,ïàðàáîëåèëèýëëèïñó(âïðåäåëüíîì
ñëó÷àåïîïðÿìîéèëèîêðóæíîñòè).
ÂýòîéôîðìóëèðîâêåïåðâûéçàêîíÊåïëåðàñïðàâåäëèâíåòîëüêîäëÿîäíîéïëà-
íåòû,îáðàùàþùåéñÿâîêðóãÑîëíöà,íîèäëÿëþáîéêîìåòû,îðáèòàêîòîðîéìîæåò
áûòüêàêýëëèïòè÷åñêîé,òàêèïàðàáîëè÷åñêîéèëèãèïåðáîëè÷åñêîé.
ÒðåòèéçàêîíÊåïëåðà.
Êîíñòàíòàçàêîíàïëîùàäåé(
II.9
)ðàâíàóäâîåííîéïëîùàäèýëëèïñà
S
,ïîäåëåííîé
íàïåðèîäîáðàùåíèÿ
T
:
C
=
2
S
T
=
2
a
2
T
p
1

e
2
:
Ó÷èòûâàÿñîîòíîøåíèÿ(
II.13
),èñâîéñòâàýëëèïñà,ïîëó÷àåì
4

2
a
4
(1

e
2
)
T
=
a
(1

e
2
)
CM;
35
ÃëàâàII.Ìàòåìàòè÷åñêèåîñíîâûäâèæåíèÿâïîëåòÿæåñòè
îòêóäà,ó÷èòûâàÿ,÷òî
M
=
m
1
+
m
2
,íàõîäèìñòðîãóþìàòåìàòè÷åñêóþôîðìóëèðîâêó
òðåòüåãîçàêîíàÊåïëåðà
T
2
a
3
(
m
1
+
m
2
)=
4

2
G
:
(II.16)
èëèâñëîâåñíîéôîðìå:
îòíîøåíèåêâàäðàòàñèäåðè÷åñêîãîïåðèîäàîáðàùåíèÿäâóõòåëêêóáóñðåäíåãî
ðàññòîÿíèÿìåæäóíèìè,óìíîæåííîåíàñóììóìàññîáîèõòåë,åñòüóíèâåðñàëüíàÿ
ïîñòîÿííàÿ.
Óòî÷í¼ííîéòðåòèéçàêîíÊåïëåðàèãðàåòîñîáåííîâàæíóþðîëüâàñòðîíîìèè,òàê
êàêïîçâîëÿåòîïðåäåëèòüëèáîñóììóìàññîáðàùàþùèõñÿòåë(êàê,íàïðèìåð,âñëó-
÷àåäâîéíûõçâ¼çä),ëèáîìàññóöåíòðàëüíîãîòåëà,êàêâñëó÷àåòåëÑîëíå÷íîéñèñòå-
ìû,åñëèìàññîéñïóòíèêàìîæíîïðåíåáðå÷üèëèåãîîòíîñèòåëüíàÿìàññàèçâåñòíàèç
êàêèõ-ëèáîäîïîëíèòåëüíûõñîîáðàæåíèé.
II.2ÄâèæåíèåèñêóññòâåííûõñïóòíèêîâÇåìëè
Õîòÿäâèæåíèåèñêóññòâåííûõñïóòíèêîâíåáåñíûõòåëïîä÷èíÿåòñÿòåìæåçàêîíàì,
÷òîèäâèæåíèååñòåñòâåííûõ,íåêîòîðûåîñîáåííîñòèèõîðáèòèóñëîâèÿ,îïðåäåëÿþ-
ùèåõàðàêòåðèõäâèæåíèÿ,çàñëóæèâàþòîòäåëüíîãîðàññìîòðåíèÿ.
ÈñêóññòâåííûåñïóòíèêèÇåìëè(ÈÑÇ)âûâîäÿòíàîðáèòóñïîìîùüþäâóõèëèìíî-
ãîñòóïåí÷àòûõðàêåò.Ïîñëåäíÿÿñòóïåíüðàêåòûñîîáùàåòñïóòíèêóîïðåäåë¼ííóþñêî-
ðîñòüíàçàäàííîéâûñîòå.Òåëî,çàïóùåííîåãîðèçîíòàëüíîíàâûñîòå
h
îòïîâåðõíîñòè
Çåìëè,ñòàíåòÈÑÇ,åñëèåãîñêîðîñòüâýòîòìîìåíòáóäåòóäîâëåòâîðÿòüíåêîòîðûì
óñëîâèÿì.
Åñëèñêîðîñòüçàïóñêàòî÷íîðàâíàêðóãîâîéñêîðîñòüþíàäàííîéâûñîòå
h
,òîòåëî
áóäåòäâèãàòüñÿïîêðóãîâîéîðáèòå.
Åñëèýòàñêîðîñòüïðåâûøàåòêðóãîâîþ,òîòåëîáóäåòäâèãàòüñÿïîýëëèïñó,ïðè÷¼ì
ïåðèãåéýòîãîýëëèïñàîêàæåòñÿâòî÷êåâûõîäàíàîðáèòó.
Åñëèæåñîîáù¼ííàÿñêîðîñòüíåñêîëüêîìåíüøåêðóãîâîé,àâûñîòà
h
äîñòàòî÷íî
áîëüøàÿ,òîòåëîòàêæåáóäåòäâèãàòüñÿïîýëëèïòè÷åñêîéîðáèòå,íîâýòîìñëó÷àå
òî÷êàâûõîäàíàîðáèòóñòàíåòàïîãååì.
Ìàññàèñêóññòâåííîãîñïóòíèêàíè÷òîæíîìàëàïîñðàâíåíèþñìàññîéÇåìëè,èåþ
ìîæíîïðåíåáðå÷ü;òîãäàêðóãîâàÿñêîðîñòü
v
c
íàðàññòîÿíèè
r
=
R
+
h
îòöåíòðàÇåìëè
áóäåò
v
c
=
r
Gm
R
+
h
=
r
gR
2
R
+
h
;
(II.17)
ãäå
m
-ìàññàÇåìëè,
R
-å¼ðàäèóñ,
g
-óñêîðåíèåñâîáîäíîãîïàäåíèÿóïîâåðõíîñòè
Çåìëè,
h
-âûñîòàòî÷êèçàïóñêàñïóòíèêàîòïîâåðõíîñòèÇåìëè.
Óâîîáðàæàåìîãîñïóòíèêà,äâèæóùåãîñÿïîîêðóæíîñòèóñàìîéïîâåðõíîñòèÇåì-
ëè,
(
h
=0)
R
=6
;
378

10
6
ìè
g
=9
;
81
ì
=
ñ
2
ñêîðîñòüäîëæíàáûòüðàâíà
v
1
k
=7
;
91
êì
=
ñ
:
Ñêîðîñòü
v
1
k
íàçûâàåòñÿ
ïåðâîéêîñìè÷åñêîéñêîðîñòüþ
îòíîñèòåëüíîÇåìëè.Îäíà-
êî,èç-çàíàëè÷èÿâîêðóãÇåìëèàòìîñôåðûñïóòíèê,äâèæóùèéñÿóñàìîéå¼ïîâåðõíî-
ñòè,ðåàëüíîñóùåñòâîâàòüíåìîæåò.ÏîýòîìóçàïóñêÈÑÇïðîèçâîäèòñÿíàíåêîòîðîé
36
II.2.ÄâèæåíèåèñêóññòâåííûõñïóòíèêîâÇåìëè
âûñîòå
h
(
h�
150
êì
)
.Êðóãîâàÿñêîðîñòüíàâûñîòå
h
ìåíüøåïåðâîéêîñìè÷åñêîéñêî-
ðîñòüþ
v
1
k
èîïðåäåëÿåòñÿèçóðàâíåíèÿ(
II.17
)èëèïîôîðìóëå
v
c
=
v
1
k
r
R
R
+
h
:
ÝëåìåíòûîðáèòûÈÑÇçàâèñÿòîòìåñòàèâðåìåíèçàïóñêà,îòâåëè÷èíûèíàïðàâ-
ëåíèÿíà÷àëüíîéñêîðîñòè.Ñâÿçüìåæäóáîëüøîéïîëóîñüþ
a
îðáèòûñïóòíèêàèåãî
íà÷àëüíîéñêîðîñòüþ
v
0
,ñîãëàñíîôîðìóëå(2.41,îïðåäåëÿåòñÿôîðìóëîé
v
0
2
=
Gm

2
r
0

1
a

;
ãäå
r
0
-ðàññòîÿíèåòî÷êèâûõîäàÈÑÇíàîðáèòóîòöåíòðàÇåìëè.
Îáû÷íîçàïóñêÈÑÇïðîèçâîäèòñÿãîðèçîíòàëüíî,òî÷íåå,ïåðïåíäèêóëÿðíîêðàäè-
àëüíîìóíàïðàâëåíèþ.Ýêñöåíòðèñèòåòîðáèòû
e
ïðèãîðèçîíòàëüíîìçàïóñêåðàâåí
e
=1

q
a
;
ãäå
q
-ðàññòîÿíèå
ïåðèãåÿ
(áëèæàéùåéòî÷êèîðáèòûîòöåíòðàÇåìëè).
Âñëó÷àåýëëèïòè÷åñêîéîðáèòû(
Ðèñ.18
)
q
=
a
(1

e
)=
R
+
h
n
ãäå
h
n
ëèíåéíàÿ
âûñîòàïåðèãåÿíàäïîâåðõíîñòèÇåìëè.Ðàññòîÿíèå
àïîãåÿ
(íàèáîëååóäàë¼ííîéòî÷êè
îðáèòûîòöåíòðàÇåìëè)
Q
=
a
(1+
e
)=
R
+
h
a
,ãäå
h
a
-âûñîòààïîãåÿíàäçåìíîé
ïîâåðõíîñòüþ.Åñëèçàïóñêïðîèçâåäåíâïåðèãåå,òî
r
0
=
q
=
R
+
h
n
:
Ðèñ.18
ÇàâèñèìîñòüôîðìûîðáèòûÈÑÇîòíà÷àëüíîéñêîðîñòè,ñêîòîðîéîíâûâåäåííà
îðáèòó,ïîêàçàíàíà(
Ðèñ.19
).Åñëèâòî÷êå
K
ñïóòíèêóñîîáùåíàãîðèçîíòàëüíàÿñêî-
ðîñòü,ðàâíàÿêðóãîâîéäëÿýòîãîðàññòîÿíèÿîòöåíòðàÇåìëè,òîîíáóäåòäâèãàòüñÿ
37
ÃëàâàII.Ìàòåìàòè÷åñêèåîñíîâûäâèæåíèÿâïîëåòÿæåñòè
ïîêðóãîâîéîðáèòå(I).Åñëèíà÷àëüíàÿñêîðîñòüâòî÷êåìåíüøåñîîòâåòñòâóþùåé
êðóãîâîé,òîñïóòíèêáóäåòäâèãàòüñÿïîýëëèïñó(II),àïðèî÷åíüìàëîéñêîðîñòè-ïî
ýëëèïñó(III),ñèëüíîâûòÿíóòîìóèïåðåñåêàþùåìóïîâåðõíîñòüÇåìëè;âýòîìñëó÷àå
çàïóùåííûéñïóòíèêóïàä¼òíàïîâåðõíîñòüÇåìëè,íåñîâåðøèâèîäíîãîîáîðîòà.Åñ-
ëèñêîðîñòüâòî÷êåáîëüøåñîîòâåòñòâóþùåéêðóãîâîé,íîìåíüøåñîîòâåòñòâóþùåé
ïàðàáîëè÷åñêîé,òîñïóòíèêáóäåòäâèãàòüñÿïîýëëèïñó(IV).Ïðèìåðíîåðàñïîëîæåíèå
Ðèñ.19
ýëëèïòè÷åñêîéîðáèòûñïóòíèêàâïðîñòðàíñòâåïîêàçàíîíà(
Ðèñ.20
).Çäåñü
i
-íàêëî-
íåíèåîðáèòûñïóòíèêàêýêâàòîðóÇåìëè,

-âîñõîäÿùèéóçåëîðáèòû,

-íèñõîäÿùèé
óçåë,

-ïåðèãåéîðáèòû,
A
-àïîãåéîðáèòû,

-òî÷êàâåñåííåãîðàâíîäåíñòâèÿíàçåì-
íîìýêâàòîðå,

-ïðÿìîåâîñõîæäåíèåâîñõîäÿùåãîóçëà,
!
-óãëîâîåðàññòîÿíèåïåðèãåÿ
îòâîñõîäÿùåãîóçëà.
ÏåðèîäîáðàùåíèÿÈÑÇîïðåäåëÿåòñÿïîòðåòüåìóçàêîíóÊåïëåðà(
II.16
).Îíðàâåí
T
=
2

p
Gm
a
3
=
2
èëè
T
=
2

R
p
g
a
3
=
2
:
Åñëèâûðàæàòüâêèëîìåòðàõ,òîïðè
R
=6370
êìè
g
=9
;
81
ì
=
ñ
2
ïåðèîäîáðàùåíèÿ
ñïóòíèêàïîëó÷èòñÿâìèíóòàõèçñëåäóþùåéôîðìóëû:
T
=1
;
659

10

4
a
3
=
2
38
II.2.ÄâèæåíèåèñêóññòâåííûõñïóòíèêîâÇåìëè
Ðèñ.20
Îñíîâíûõïðè÷èí,èçìåíÿþùèõîðáèòóÈÑÇ,äâå:äåéñòâèåýêâàòîðèàëüíîãîóòîë-
ùåíèÿÇåìëèèâëèÿíèåñîïðîòèâëåíèÿàòìîñôåðûÇåìëè.Ïåðâàÿïðè÷èíàâûçûâàåò
âåêîâûåâîçìóùåíèÿâîñõîäÿùåãîóçëà

èïåðèãåÿ

!
,êîòîðûåëåãêîó÷èòûâàþò-
ñÿïîôîðìóëàìíåáåñíîéìåõàíèêè.Âòîðàÿïðè÷èíàâûçûâàåòóìåíüøåíèåáîëüøîé
ïîëóîñè
a
,ò.å.âûñîòû
h
,èèçìåíåíèåôîðìûîðáèòû.Ïîñêîëüêóïëîòíîñòüàòìîñôå-
ðûáûñòðîïàäàåòñâûñîòîé,îñíîâíîåñîïðîòèâëåíèåèóìåíüøåíèåñêîðîñòèñïóòíèê
èñïûòûâàåòâáëèçèïåðèãåÿ.Âñëåäñòâèåýòîãîâûñîòààïîãåÿîðáèòûñïóòíèêàñêàæ-
äûìîáîðîòîìçàìåòíîóìåíüøàåòñÿ(âûñîòàïåðèãåÿóìåíüøàåòñÿãîðàçäîìåäëåííåå).
Âðåçóëüòàòåóìåíüøàåòñÿáîëüøàÿïîëóîñüèýêñöåíòðèñèòåòîðáèòû;îðáèòàñïóòíèêà
ïîñòåïåííîîêðóãëÿåòñÿ.Êîãäàâûñîòààïîãåÿñòàíîâèòñÿñðàâíèìîéñâûñîòîéïåðè-
ãåÿ,ñïóòíèêèñïûòûâàåòòîðìîæåíèåèòåðÿåòñâîþñêîðîñòüâäîëüïî÷òèâñåéîðáèòû,
óìåíüøåíèåâûñîòûàïîãåÿèïåðèãåÿïðîèñõîäèòåù¼áûñòðåå,èñïóòíèê,ïðèáëèæàÿñü
ïîñïèðàëèêïîâåðõíîñòèÇåìëè,âõîäèòâïëîòíûåñëîèàòìîñôåðûèñãîðàåò.Òàêêàê
ñïóòíèêñêàæäûìîáîðîòîìñíèæàåòñÿ,òîåãîïîòåíöèàëüíàÿýíåðãèÿóìåíüøàåòñÿ,
÷àñòüå¼ïåðåõîäèòâêèíåòè÷åñêóþýíåðãèþ.Ýòîïðèðàùåíèåêèíåòè÷åñêîéýíåðãèèñ
èçáûòêîìïîêðûâàåòýíåðãèþäâèæåíèÿ,êîòîðàÿòåðÿåòñÿïðèòîðìîæåíèè.Ïîýòîìó
ñêîðîñòüñïóòíèêàíåóìåíüøàåòñÿ,àíàîáîðîò,óâåëè÷èâàåòñÿ,âòîâðåìÿêàêîðáèòà
óìåíüøàåòñÿ.Ñëåäîâàòåëüíî,ïîìåðåñíèæåíèÿñïóòíèêàåãîïåðèîäîáðàùåíèÿâîêðóã
Çåìëèñîêðàùàåòñÿ.
Îïèñàííîåâîçìóù¼ííîåäâèæåíèåñïóòíèêàäàíîâïåðâîìïðèáëèæåíèè.Âäåé-
ñòâèòåëüíîñòèýëåìåíòûîðáèòûñïóòíèêàèñïûòûâàþòáîëååñëîæíûåèðàçíîîáðàç-
íûåâîçìóùåíèÿ.ÑæàòèåÇåìëè,îòëè÷èåãðàâèòàöèîííîãîïîëÿîòïîëÿñôåðè÷åñêè-
ñèììåòðè÷íîéïðèòÿãèâàþùåéìàññû,âûçûâàþòíåòîëüêîâåêîâûåâîçìóùåíèÿäîëãî-
òûâîñõîäÿùåãîóçëà

,èðàññòîÿíèÿïåðèãåÿîòóçëà
!
.Îíèÿâëÿþòñÿòàêæåïðè÷èíîé
èõïåðèîäè÷åñêèõâîçìóùåíèé,àòàêæåýêñöåíòðèñèòåòà
e
(ïðàâäà,âåñüìàóìåðåííûõ)
èìàëûõêîëåáàíèéíàêëîíåíèÿîðáèòûêýêâàòîðó
i
.
39
ÃëàâàII.Ìàòåìàòè÷åñêèåîñíîâûäâèæåíèÿâïîëåòÿæåñòè
Íàëè÷èåàòìîñôåðûâûçûâàåòíåòîëüêîâåêîâîåóìåíüøåíèåáîëüøîéïîëóîñè
a
èýêñöåíòðèñèòåòà
e
.Áîêîâîåäàâëåíèåíàñïóòíèê,ñîçäàâàåìîåâðàùàþùåéàòìîñôå-
ðîé,ïðèâîäèòêìîíîòîííîìóèçìåíåíèþ
i
,çíàêêîòîðîãîîïðåäåëÿåòñÿíàïðàâëåíèåì
äâèæåíèÿñïóòíèêàíàîðáèòå.Àòìîñôåðàîáóñëàâëèâàåòòàêæåìàëûåïåðèîäè÷åñêèå
èçìåíåíèÿ

è
!
.Íàêîíåö,âîçìóùàþùèåäåéñòâèÿËóíûèÑîëíöàâûçûâàþòìàëûå
ïåðèîäè÷åñêèåâîçìóùåíèÿâñåõýëåìåíòîâîðáèòûñïóòíèêà.
40
ÃëàâàIII
Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâ
MAPLE
III.1Êðèâûåâòîðîãîïîðÿäêà
ÊðèâûåâòîðîãîïîðÿäêàâïðîãðàììåMAPLEìîæíîèçó÷àòüñïîìîùüþïàêåòà
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
Ðèñ.21
Åñëèíåîáõîäèìî,ïîëó÷åííóþôèãóðóìîæíîïîâåðíóòüíàëþáîéóãîë:
display(rotate(Ell,(1/12)*Pi),scaling=constrained);
Ðèñ.22
Ãèïåðáîëàçàäà¼òñÿàíàëîãè÷íî:
hyperbola
(
c;a;b;r
1
::r
2
;
options
)
ãäå,
ñ-öåíòðñèììåòðèèãèïåðáîëû,
a-äåéñòâèòåëüíàÿïîëóîñüãèïåðáîëû,
b-ìíèìàÿïîëóîñüãèïåðáîëû,âñïîìíèì,÷òî
a
2

e
2
=
a
2
+
b
2
;
ãäåe-ýêñöåíòðèñèòåò
ãèïåðáîëû
r
1
::r
2
-îáëàñòüîïðåäåëåíèÿãðàôèêà.
Ãðàôèêáóäåòíàðèñîâàí;
îòòî÷êè
[
x
0
+
a
cosh
r
1
;y
0
+
b
sinh
r
1
]
äîòî÷êè
[
x
0
+
a
cosh
r
2
;y
0
+
b
sinh
r
2
]
èîòòî÷êè
[
x
0

a
cosh
r
1
;y
0

b
sinh
r
1
]
äîòî÷êè
[
x
0

a
cosh
r
2
;y
0

b
sinh
r
2
]
.
Ðàññìîòðèìïðîñòîéïðèìåð:
with(plottools):
with(plots):
42
III.1.Êðèâûåâòîðîãîïîðÿäêà
a:=4:b:=3:x[0]:=0:y[0]:=0:
Hyp_1:=hyperbola([x[0],y[0]],a,b,-2..2,color=black,thickness=3):
display(Hyp_1,scaling=constrained);
Ðèñ.23
Ìûïîëó÷èëèãèïåðáîëóñöåíòðîìâíà÷àëåêîîðäèíàò,ôîêóñûíàðàññòîÿíèè
c
=
p
a
2
+
b
2
=5
îòöåíòðà,ñýêñöåíòðèñèòåòîì
e
=
c
a
=
5
4
:
III.1.2Êðèâûåâòîðîãîïîðÿäêàâïàêåòå
plots
Âïàêåòå
plots
ñïîìîùüþêîìàíäû[implicitplot]ìîæíîïîëó÷èòüâñåâèäûêðèâûõ
âòîðîãîïîðÿäêà,çàäàâóðàâíåíèÿâíåÿâíîìâèäå.Ñòðóêòóðàêîìàíäû
implicitplot
âûãëÿäèòñëåäóþùèìîáðàçîì:
implicitplot(expr,
x
=
a::b;y
=
c
(
x
)
::d
(
x
)
,options)
implicitplot(ineq,
x
=
a::b;y
=
c
(
x
)
::d
(
x
)
,options)
implicitplot(
f
,
a::b;c::d
,options)
implicitplot([expr1,expr2,t],
x
=
a::b;y
=
c
(
x
)
::d
(
x
)
,options)
ãäå:
expr
-âûðàæåíèåèëèóðàâíåíèå,çàâèñÿùååîò
x
è
y
,
ineq
-íåðàâåíñòâî,çàâèñÿùååîò
x
è
y
,
f
-óðàâíåíèå,ñîäåðæàùååïðîöåäóðûèëèîïåðàòîðû,ïðåäñòàâëÿþùèåôóíêöèèäâóõ
ïåðåìåííûõ,
expr1,expr2
-óðàâíåíèÿèëèâûðàæåíèÿ÷åðåç
x
,
y
èëè
t
,èëèïîëèíîìâ
t
x,y,t
-ïåðåìåííûå,
a,b,c,d
-äåéñòâèòåëüíûåïîñòîÿííûå,
c(x),d(x)
-âûðàæåíèÿ,îïðåäåëÿþùèåäåéñòâèòåëüíûåïîñòîÿííûåäëÿäàííîãîçíà÷å-
íèÿx,Íà÷í¼ìñýëëèïñà:
with(plots):
43
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
a:=10:b:=5:x[0]:=0:y[0]:=0:
Ell_3:=(x-x[0])^2/a^2+(y-y[0])^2/b^2=1:
implicitplot(Ell_3,x=-12..12,y=-6..6,scaling=constrained,
filledregions=true,
coloring=["WhiteSmoke",white],thickness=3);
Ðèñ.24
Îáðàòèìâíèìàíèå,÷òîâêîìàíäå
implicitplot
îïöèèíåìíîãîîòëè÷àåòñÿîòîáû÷-
íûõîïöèé(plot/options).Íàïðèìåð,îïöèÿ
filledregions
ðàáîòàåòòîëüêîñêîìàíäàìè
contourplot
,
listcontplot
è
implicitplot
.
Åñëèïåðâûéàðãóìåíòÿâëÿåòñÿâûðàæåíèåìèëèïðîöåäóðîé
f;
òîîíàçàêðàøèâàåò
îáëàñòü
f
=0
.Îïöèÿ
coloring=[c1,c2]
çàêðàøèâàåòîáëàñòü
f
0
ñîöâåòîì
c
1
è
îáëàñòü
f�
0
ñîöâåòîì
c
2
.Åñëèïåðâûéàðãóìåíòÿâëÿåòñÿóðàâíåíèåì,êàê
f
=
g
,òî
îïöèÿ
filledregions
=
true
çàêðàøèâàåòîáëàñòü
f

g
.
Ïðîäîëæàåìñïàðàáîëîé:
with(plots):
p:=3:
Par:=implicitplot(y^2=2*p*x,x=-3..10,y=-8..8,filledregions=true,
coloring=["WhiteSmoke",white],thickness=3):
display(Par);
44
III.1.Êðèâûåâòîðîãîïîðÿäêà
Ðèñ.25
Òàêæåìîæíîçàäàòüèãèïåðáîëó:
with(plots):
a:=4:b:=3:
Hyp_2:=implicitplot(x^2/a^2-y^2/b^2=1,x=-10..10,y=-10..10,
color=black,thickness=3):
display(Hyp_2,scaling=constrained);
Ðèñ.26
III.1.3Êðèâûåâòîðîãîïîðÿäêàâïàêåòå
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
âàòüñåìüþðàçíûìèñïîñîáàìè:
ellipse(p,[A,B,C,E,F],n)
ellipse(p,['directrix'=dir,'focus'=fou,'eccentricity'=ecc],n)
ellipse(p,['foci'=foi,'MajorAxis'=lma],n)
ellipse(p,['foci'=foi,'MinorAxis'=lmi],n)
ellipse(p,['foci'=foi,'distance'=dis],n)
ellipse(p,['MajorAxis'=ep1,'MinorAxis'=ep2],n)
ellipse(p,eqn,n)
ãäå;
p
-Íàçâàíèåýëëèïñà,
A,B,C,E,F
-ïÿòüðàçëè÷íûõòî÷åêýëëèïñà,
'directrix'=dir
-äèðåêòðèñàýëëèïñà,
'focus'=fou
-ôîêóñýëëèïñà,
'eccentricity'=ecc
-ýêñöåíòðèñèòåòýëëèïñà,
'foci'=foi
-îáàôîêóñàýëëèïñà,
'MajorAxis'=lma
-äëèíàáîëüøîéïîëóîñèýëëèïñà,
'MinorAxis'=lmi
-äëèíàìàëîéïîëóîñèýëëèïñà,
'distance'=dis
-ñóììàðàññòîÿíèéòî÷êèíàýëëèïñåäîôîêóñîâ,
'MajorAxis'=ep1
-ïàðàìåòðû(òî÷êè)êîíöîâáîëüøîéïîëóîñè,
'MinorAxis'=ep2
-ïàðàìåòðû(òî÷êè)êîíöîâìàëîéïîëóîñè,
eqn
-óðàâíåíèåýëëèïñà,
n
-íàçâàíèÿãîðèçîíòàëüíîéèâåðòèêàëüíîéîñåé.
Ïîñëåîïðåäåëåíèÿýëëèïñàñïîìîùüþêîìàíäû
7
;
0]
;
[2
p
7
;
0]]
lengthofthemajoraxis:16
lengthoftheminoraxis:12
equationoftheellipse:
x
2
64
+
x
2
36

1=0
FF:=map(coordinates,foci(Ell_5));
point(F1,FF[1]);point(F2,FF[2]);
46
III.1.Êðèâûåâòîðîãîïîðÿäêà
FF:=
[[

2
p
7
;
0]
;
[2
p
7
;
0]]
F1
F2
CC:=center(Ell_5);
CC:=center_Ell_5
draw([Ell_5,F1(printtext=true),F2(printtext=true),CC],scaling=constrained,
axes=normal,thickness=3,color=black,labels=[x,y]);
Ðèñ.27
Òåïåðüîïðåäåëèìýëëèïñïîåãîôîêóñàìèäëèíåáîëüøîéïîëóîñè:
restart;
33
equationoftheellipse:
132
_
x
2
+1156
_
y
2

4224
_
x
+24255=0
47
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
FF:=map(coordinates,foci(Ell_6));
point(F_1,FF[1]);point(F_2,FF[2]);
[[8,0],[24,0]]
F_1
F_2
CC:=center(Ell_6);
center_Ell_6
draw([Ell_6,F_1,F_2,CC],scaling=constrained,axes=normal,
thickness=3,color=black);
Ðèñ.28
Ïàðàáîëóâïàêåòå
III.1.Êðèâûåâòîðîãîïîðÿäêà
nameoftheobject:
par
_
1
formoftheobject:
parabola2d
vertex:
[3
;
0]
focus:
[4
;
0]
directrix:

2+
x
=0
equationoftheparabola:
y
2

4
x
+12=0
point(Vertex,3,0):
draw([Dir,par_1,F,Vertex],axes=normal,thickness=3,scaling=constrained);
Ðèñ.29
Ãèïåðáîëóòàêæåìîæíîîïðåäåëèòüðàçëè÷íûìèå¼ïàðàìåòðàìè:
hyperbola(p,[A,B,C,E,F],n)
hyperbola(p,['directrix'=dir,'focus'=fou,'eccentricity'=ecc],n)
hyperbola(p,['foci'=foi,'vertices'=ver],n)
hyperbola(p,['foci'=foi,'distancev'=disv],n)
hyperbola(p,['vertices'=ver,'distancef'=disf],n)
hyperbola(p,eqn,n)
,ãäå:
p
-Íàçâàíèåãèïåðáîëû,
A,B,C,E,F
-ïÿòüðàçëè÷íûõòî÷åêãèïåðáîëû,
'directrix'=dir
-äèðåêòðèñàãèïåðáîëû,
'focus'=fou
-ôîêóñãèïåðáîëû,
'eccentricity'=ecc
-ýêñöåíòðèñèòåòãèïåðáîëû(åñòåñòâåííîáîëüøååäèíèöû),
'vertices'=ver
-âåðøèíûãèïåðáîëû,
'foci'=foi
-ôîêóñûãèïåðáîëû,
'distancev'=disv
-ðàññòîÿíèåìåæäóâåðøèíàìèãèïåðáîëû,
'distancef'=disf
-ðàññòîÿíèåìåæäóôîêóñàìèãèïåðáîëû½
eqn
-óðàâíåíèåãèïåðáîëû,
49
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
n
-íàçâàíèÿãîðèçîíòàëüíîéèâåðòèêàëüíîéîñåé.
Îïðåäåëèìãèïåðáîëóïîåãîóðàâíåíèþ,ïîëó÷èìå¼ïàðàìåòðûèíàðèñóåìå¼:
restart;
4
=0
;y

3
x
4
=0

equationofthehyperbola:
x
2
64

y
2
36

1=0
CC:=center(Hyp_3);
center_Hyp_3
FF:=map(coordinates,foci(Hyp_3));
point(F_1,FF[1]);point(F_2,FF[2]);
[[-10,0],[10,0]]
F_1
F_2
VER:=map(coordinates,vertices(Hyp_3));
point(VER_1,VER[1]);point(VER_2,VER[2]);
[[-8,0],[8,0]]
VER_1
VER_2
AS:=map(Equation,asymptotes(Hyp_3));
line(AS1,AS[1],[x,y]):line(AS2,AS[2],[x,y]):

y
+
3
4
x
=0
;y

3
4
x
=0

draw([Hyp_3(color=black,thickness=2),AS1(color=red,linestyle=dash),
AS2(color=red,linestyle=dash),CC,F_1(color=blue),F_2(color=blue),
VER_1,VER_2],axes=normal,labels=["x","y"],labelfont=[times,14]);
50
III.2.Äâèæåíèåíåáåñíûõòåë
Ðèñ.30
III.2Äâèæåíèåíåáåñíûõòåë
III.2.1Äâèæåíèåïëàíåòñîëíå÷íîéñèñòåìû
Îðáèòûïëàíåòñîëíå÷íîéñèñòåìûèìåþòôîðìóýëëèïñà.Îíèîòëè÷àþòñÿñâîèìè
ýêñöåíòðèñèòåòàìèèðàäèóñàìè(âåðíååäëèíàìèáîëüøîéèëèìàëåíüêîéïîëóîñåé).
Ñîçäàäèìïðîöåäóðó,êîòîðàÿáóäåòðèñîâàòüýëëèïñïîçàäàííîìóýêñöåíòðèñèòåòóè
äëèíåáîëüøîéïîëóîñè.×òîáûëó÷øåâèäåòü,êàêâëèÿåòýêñöåíòðèñèòåòíàôîðìó
ýëëèïñà,âîçüì¼ìýëëèïñû,óêîòîðûõäëèíûáîëüøèõïîëóîñåéðàâíûå,íîçíà÷åíèÿ
ýêñöåíòðèñèòåòàðàçíûå.
restart:
with(plots):
Ellipse:=proc(epsilon,a,c1)localc:
c:=a*epsilon:
Ïîîïðåäåëåíèþ
c
=
a

e;
ãäåñ-ïîëîæåíèåôîêóñîâîòíîñèòåëüíîöåíòðàýëëèïñà.À
öåíòðíàøèõýëëèïñîâñîâïàäàåòñíà÷àëîìêîîðäèíàò.
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
E3:=Ellipse(0.7,10,green):
E4:=Ellipse(0.9,10,blue):
E5:=Ellipse(0.99,10,black):
display([E1,E2,E3,E4,E5],frames=10,scaling=constrained,
title=`Èçìåíåíèåôîðìûýëëèïñàñýêñöåíòðèñèòåòîì`,titlefont=["ROMAN",15]);
Ðèñ.31
:Èçìåíåíèåôîðìûýëëèïñàñýêñöåíòðèñèòåòîì
Çäåñüìûèñïîëüçîâàëèóðàâíåíèåýëëèïñàâïîëÿðíûõêîîðäèíàòàõ:
r
=
p
(1

e
cos

)
ãäå,
p
=
b
2
a
;b
=
a
p
(1

e
2
)
;x
=
r
cos
;y
=
r
sin
:
Êàêìûâèäèìèçðèñóíêà,ïðèå=0,
ýëëèïñïðåâðàùàåòñÿâîêðóæíîñòü.È÷åìáîëüøåýêñöåíòðèñèòåòýëëèïñà,òåìñèëü-
íååîíîòëè÷àåòñÿîòîêðóæíîñòè.Òåïåðüïîâòîðèìòàêóþæåïðîöåäóðó,íîïîñòàâèì
íà÷àëîêîîðäèíàòíàîäèíèçôîêóñîâýëëèïñà:
Ellipse1:=proc(epsilon,a,c1)localc:
c:=a*epsilon:
III.2.Äâèæåíèåíåáåñíûõòåë
E9:=Ellipse1(0.9,10,blue):
E10:=Ellipse1(0.99,10,black):
display([E6,E7,E8,E9,E10],frames=10,
title=`Ëåâûéôîêóñâñåõýëëèïñîâñîâïàäàåòñíà÷àëîìêîîðäèíàò`,
titlefont=["ROMAN",14]);
Ðèñ.32
:Èçìåíåíèåôîðìûýëëèïñàñýêñöåíòðèñèòåòîì
Òåïåðüïðèñòóïèìêèçó÷åíèþäâèæåíèÿïëàíåòâîêðóãÑîëíöà.ÍàðèñóåìÑîëíöåè
ïëàíåòó,ïîñòàâèâöåíòðÑîëíöà,ñîãëàñíîïåðâîìóçàêîíóÊåïëåðà,íàîäèíèçôîêóñîâ
ýëëèïñà,ïîêîòîðîìóäâèãàåòñÿïëàíåòà.
Sun:=proc(a,epsilon,r,c2)localF,SolarDisc:
F:=[a*epsilon,0]:
F

Êîîðäèíàòû
(
x;y
)
îäíîãîèçôîêóñîâ,âäàííîìñëó÷àåïðàâîãîôîêóñà.
SolarDisc:=plottools[disk](F,r,color=c2):
display(SolarDisc,scaling=constrained)
endproc:
Ell:=(a,epsil�on,r,c1,c2)-plots[display](Ellipse(epsilon,a,c1),
Sun(a,epsilon,r,c2)):
Ell(10,0.8,0.5,green,yellow):
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
Ðèñ.33
Ìûïîëó÷èëèòðàåêòîðèþïëàíåòûñäëèíîéáîëüøîéïîëóîñè
10
åäèíèöèýêñöåí-
òðèñèòåòîì
0
:
8
.Âäåéñòâèòåëüíîñòèâñîëíå÷íîéñèñòåìåíåòïëàíåòñòàêèìáîëüøèì
ýêñöåíòðèñèòåòîìîðáèòû.Ìûñïåöèàëüíîâûáðàëèòàêóþâåëè÷èíó,÷òîáûëó÷øåïðî-
äåìîíñòðèðîâàòüýëëèïòè÷íîñòüîðáèò.
Òåïåðü,÷òîáûáîëååíàãëÿäíîïîêàçàòüäâèæåíèåïëàíåòû,ñîçäà¼ìàíèìàöèþ:
Anim_Orbit:=proc(a,epsilon,r,c1,c2,r3,c3,T,N)locali,ti,Ori:
�ti:=(i)-T/N*i:
�Ori:=(i)-Orbit(a,epsilon,r,c1,c2,r3,c3,ti(i)):
plots[display](seq(Ori(i),i=0..N),insequence=true,scaling=CONSTRAINED):
endproc:
Çàäàäèìïàðàìåòðûàíèìàöèè:
AAnim_Orbit(10,0.8,0.5,black,yellow,0.2,red,6*Pi,100);
54
III.2.Äâèæåíèåíåáåñíûõòåë
èïîëó÷èì:
Ðèñ.34
Òåïåðüáîëååäåòàëüíîèçó÷èìòðàåêòîðèèïëàíåòñîëíå÷íîéñèñòåìû.Äëÿýòîãî
ñîçäàäèìïðîöåäóðó,êîòîðàÿáóäåòðèñîâàòüòðàåêòîðèþïëàíåòûïîçàäàííûìçíà÷å-
íèÿìýêñöåíòðèñèòåòàèàôôåëèéíîãîðàññòîÿíèÿ.
Orbits:=proc(ra,epsilon,n,col)locala,b,p,c,r,fn,orb:
a:=ra/(1+epsilon):
c:=a*epsilon:
b:=a*sqrt(1-epsilon^2):
p:=b^2/a:
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
Òàáëèöà.2
:Ïëàíåòûçåìíîéãðóïïû[
4
]
Ìåðêóðèé
Âåíåðà
Çåìëÿ
Ìàðñ
Àôôåëèéíîåðàññòîÿíèå(à.å)
0.456
0.7286
1.0767
1.6651
Ýêñöåíòðèñèòåò
0.20564
0.0068
0.01670
0.09346
Ââîäèìäàííûå
Òàáë.2
îáýêñöåíòðèñèòåòàõèàôôåëèéíèõðàññòîÿíèèïëàíåòçåì-
íîéãðóïïûñîëíå÷íîéñèñòåìûèïîëó÷èìèõîðáèò:
plots[display](Orbits(0.4560,0.20564,Mercury,"Fuchsia"),
Orbits(0.7286,0.00680,Venus,"Maroon"),
Orbits(1.0767,0.01670,Earth,blue),
Orbits(1.6651,0.09346,Mars,red),scaling=CONSTRAINED)
Ðèñ.35
:Îðáèòûïëàíåòçåìíîéãðóïïû
Òàêæåââîäèìäàííûå
Òàáë.3
ïëàíåò-ãèãàíòîâèÏëóòîí:
plots[display](Orbits(5.4525,0.04890,Jupiter,red),
Orbits(10.0735,0.05538,Saturn,"DeepSkyBlue"),
Orbits(20.0802,0.04756,Uranus,green),
Orbits(30.3275,0.00859,Neptune,blue),
Orbits(49.2981,0.24880,Pluto,brown),scaling=CONSTRAINED);
56
III.2.Äâèæåíèåíåáåñíûõòåë
Òàáëèöà.3
:Ïëàíåòû-ãèãàíòûèÏëóòîí[
4
]
Þïèòåð
Ñàòóðí
Óðàí
Íåïòóí
Ïëóòîí
Àôôåëèéíîåðàññòîÿíèå(à.å)
5.4525
10.0735
20.0802
30.3275
49.2981
Ýêñöåíòðèñèòåò
0.04845
0.05538
0.04756
0.00859
0.248
Ðèñ.36
:Îðáèòûïëàíåò-ãèãàíòîâèÍåïòóíà
Çàäà÷àÊåïëåðà
Çàäà÷åéÊåïëåðàïðèíÿòîíàçûâàòüçàäà÷óîäâèæåíèèäâóõòåë,âçàèìîäåéñòâó-
þùèõñîãëàñíîçàêîíóâñåìèðíîãîòÿãîòåíèÿÍüþòîíà,ñïðîèçâîëüíîçàäàííûìèíà-
÷àëüíûìèêîîðäèíàòàìèèñêîðîñòÿìè.Äåìîíñòðèðóåì÷èñëåííîåðåøåíèåóðàâíåíèÿ
Íüþòîíàäëÿçàäà÷èÊåïëåðàâäåêàðòîâîéñèñòåìå.
Äëÿíà÷àëàçàïèøåìîñíîâíûåâåëè÷èíû,êîòîðûåáóäåìèñïîëüçîâàòü.Âîçüì¼ì
ìàññóÑîëíöàî÷åíüáîëüøóþïîñðàâíåíèþñìàññîéÇåìëè,ïîýòîìóáóäåìäâèæåíèå
Çåìëèðàññìàòðèâàòü,êàêäâèæåíèåòåëàâîêðóãíåïîäâèæíîéòî÷êèïðèòÿæåíèÿ.
restart;
G:=6.67*10^(-11);ãðàâèòàöèîííàÿïîñòîÿííàÿ,
Ms:=1.99*10^(30);ìàññàÑîëíöà,
Me:=5.97*10^(24);ìàññàÇåìëè,
au:=1.5*10^(11);àñòðîíîìè÷åñêàÿåäèíèöà,
Fe:=G*Ms*Me/au^2;ñèëàïðèòÿæåíèÿìåæäóÑîëíöåìèÇåìëåé.
57
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
-11
G:=6.67000000010
30
Ms:=1.99000000010
24
Me:=5.97000000010
11
au:=1.50000000010
22
Fe:=3.52184893310
Òàêæåíàéäåìöåíòðîñòðåìèòåëüíîåóñêîðåíèå,ïåðèîäîáðàùåíèÿâîêðóãÑîëíöàèëè-
íåéíóþñêîðîñòüÇåìëè:
Acc:=G*Ms/au^2;
Te:=evalf(365.26*24*60*60);
ve:=evalf(2*Pi*au)/Te;
Acc:=0.005899244444
7
Te:=3.15584640010
ve:=29864.50152
Êàêìûâèäèì,ñèëàïðèòÿæåíèÿìåæäóÇåìëåéèÑîëíöåìâåëèêà,íîíåñìîòðÿíàýòî
óñêîðåíèåÇåìëèâðåçóëüòàòåíåáîëüøîå.ÑêîðîñòüäâèæåíèÿÇåìëèâîêðóãÑîëíöà
÷óòüìåíüøå30êì/÷!
ÄîýòîãîìûðàáîòàëèâñèñòåìåÑÈ.Òåïåðüïåðåõîäèìâíîâóþñèñòåìóîòñ÷¼òàäëÿ
òîãî,÷òîáûðàáîòàòüäàëüíåéøåìáûëîóäîáíåå.Ââåäåìñèñòåìóåäèíèö,÷òîáûîáåç-
ðàçìåðèòüâñåâåëè÷èíûñêîòîðûìèáóäåìðàáîòàòü.Äëÿýòîãîâûáèðàåìòðèîñíîâíûå
åäèíèöû:
Êàêåäèíèöóäëèíûâûáèðàåìàñòðîíîìè÷åñêóþåäèíèöóèëèñðåäíååðàññòîÿíèåîò
ÇåìëèäîÑîëíöà
(àu)
;
ÊàêåäèíèöóìàññûâûáåðåììàññóÑîëíöà
(Ms)
;
ÊàêåäèíèöóâðåìåíèïåðèîääâèæåíèÿÇåìëèâîêðóãÑîëíöà
(ãîä)
.
unitLength:=au;
unitTime:=Te;
unitMass:=Ms;
11
unitLength:=1.50000000010
7
unitTime:=3.15584640010
30
unitMass:=1.99000000010
Ñîîòâåòñòâåííî,îñòàëüíûåïðîèçâîäíûååäèíèöûáóäåìâûðàæàòü÷åðåçâûáðàííûå
íàìèîñíîâíûååäèíèöû:
unitVel:=unitLength/unitTime;
58
III.2.Äâèæåíèåíåáåñíûõòåë
unitAcc:=unitVel/unitTime;
unitForce:=unitMass*unitAcc;
unitVel:=4753.083040
unitAcc:=0.0001506119892
26
unitForce:=2.99717858510
Ïîñêîëüêóäëèíàîêðóæíîñòèðàâíà
2
R;
ñêîðîñòüäâèæåíèÿâîêðóãÑîëíöàáóäåòêàê
ìíîæèòåëè
2
:
ÏîñìîòðèìêàêâûãëÿäèòóñêîðåíèåÇåìëèèñèëàïðèòÿæåíèÿñîñòî-
ðîíûÑîëíöàâýòèõåäèíèöàõ:
(G*Ms/au^2)/unitAcc;
(G*Ms*Me/au^2)/unitForce;
39.16849167
0.0001175054750
Ìîæåìòàêæåîïðåäåëèòüïîñòîÿííóþâñåìèðíîãîòÿãîòåíèÿ:
unitG:=unitForce*(unitLength)^2/unitMass^2;
G/unitG;
-12
unitG:=1.70289937510
39.16849168
Âîçüì¼ìòî÷êó,ãäåñêîðîñòüïëàíåòûïåðïåíäèêóëÿðíàêðàäèóñâåêòîðó(òàêáóäåò
êîãäàïëàíåòàíàõîäèòñÿâïåðèãåëèèèëèàôåëèè).Òîãäàìîäóëüìîìåíòàèìïóëüñà
ìîæíîçàïèñàòüââèäå;
L0:=(Me*au*ve);
ïåðåâîäèìâíàøóñèñòåìó;
L0/(unitMass*unitVel*unitLength);
öåíòðîáåæíàÿïîòåíöèàëüíàÿýíåðãèÿÇåìëèçàïèñûâàåòñÿêàê:
L0^2/(2*m*r^2);
Äèôôåðåíöèðóÿïîòåíöèàëüíóþýíåðãèþïî
r
,ìîæåìíàéòèöåíòðîáåæíóþñèëó:
subs(r=au,-diff(L0^2/(2*Me*r^2),r));
àñèëàïðèòÿæåíèÿðàâíà:
59
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
-G*Ms*Me/au^2;
40
2.67436611110
0.00001884955591
22
3.54971603610
22
-3.52184893310
Êàêìûâèäèì,öåíòðîáåæíàÿñèëàèñèëàïðèòÿæåíèÿÑîëíöàíàäÇåìëåéíåòî÷íî
îäèíàêîâûïîìîäóëþ;ýòîèç-çàòîãî,÷òîîðáèòàÇåìëèíåîêðóæíîñòüàýëëèïñ.Òåïåðü
âñåýòîïðèâîäèìâíàøóñèñòåìó:
subs(r=au,diff(L0^2/(2*Me*r^2),r))*
(unitMass*unitLength^3/(unitMass*unitLength*unitVel)^2);
(-G*Ms*Me/au^2)*(unitLength^2/(unitMass^2*unitG));
-0.0001184352528
-0.0001175054750
Ññèñòåìîéåäèíèöèçìåðåíèéðàçîáðàëèñü.Ìîæåìèçó÷èòüäâèæåíèåÇåìëèèëè
äðóãîéïëàíåòûñîëíå÷íîéñèñòåìû.Ñíà÷àëàçàïèñûâàåìäàííûåâíîâîéñèñòåìå:
restart;
G:=39.17:
M:=1:
m:=6*10^(24)/(1.99*10^(30)):
Ââåäåìïîòåíöèàëüíóþýíåðãèþèïåðåâîäèìå¼âäåêàðòîâóñèñòåìó
U:=-G*M*m/r;
U:=subs(r=sqrt(x^2+y^2),U);
U
:
=
0
:
0001181005025
r
U
:
=
0
:
0001181005025
p
x
2
+
y
2
ïðîäèôôåðåíöèðîâàâïîòåíöèàëû,íàéäåìïðîåêöèèñèëûïîîñÿìxèy:
Fx:=-diff(U,x);
Fy:=-diff(U,y);
Fx
:
=

0
:
0001181005025
(
x
2
+
y
2
)
3
=
2
Fy
:
=

0
:
0001181005025
(
x
2
+
y
2
)
3
=
2
Çàïèñûâàåìïðîåêöèèñèëûââèäåôóíêöèèîòxèy:
60
III.2.Äâèæåíèåíåáåñíûõòåë
FX:=unapply(Fx,x,y);
FY:=unapply(Fy,x,y);
FX
:
=(
x;y
)
!�
0
:
0001181005025
(
x
2
+
y
2
)
3
=
2
FY
:
=(
x;y
)
!�
0
:
0001181005025
(
x
2
+
y
2
)
3
=
2
ÈñïîëüçóÿâòîðîéçàêîíÍüþòîíà,ìîæåìçàïèñàòü:
Eq1:=m*diff(x(t),t$2)=FX(x(t),y(t));
Eq2:=m*diff(y(t),t$2)=FY(x(t),y(t));
Eq
1
:
=0
:
000003015075377

d
2
dt
2
x
(
t
)
!
=

0
:
0001181005025
x
(
t
)

x
(
t
)
2
+
y
(
t
)
2

3
=
2
Eq
1
:
=0
:
000003015075377

d
2
dt
2
y
(
t
)
!
=

0
:
0001181005025
y
(
t
)

x
(
t
)
2
+
y
(
t
)
2

3
=
2
Çàäàäèìíà÷àëüíûåóñëîâèÿäëÿÇåìëèèðåøàåìóðàâíåíèÿÍüþòîíàñó÷¼òîìíà÷àëü-
íûõóñëîâèé:
x0:=147.09/150:y0:=0:
vx0:=0:vy0:=6.28*30.29/29.86:
IC:=x(0)=x0,D(x)(0)=vx0,y(0)=y0,D(y)(0)=vy0;
sol:=dsolve({Eq1,Eq2,IC},{x(t),y(t)},numeric,output=listprocedure):
ÌûââåëèçäåñüäàííûåÇåìëèâïåðèãåëèèïîäàííûì[
5
].ÍàéäåìêîîðäèíàòûÇåìëè
èïðîåêöèèå¼ñêîðîñòèïîîñÿìxèy:
X:=subs(sol,x(t)):
Y:=subs(sol,y(t)):
VX:=subs(sol,diff(x(t),t)):
VY:=subs(sol,diff(y(t),t)):
Ïðîâåðèìðàáîòóïðîãðàììû:
[X(5),Y(5)];
[VX(5),VY(5)];
IC
:
=
x
(0)=0
:
9806000000
;
(
D
(
x
))(0)=0
;y
(0)=0
;
(
D
(
y
))(0)=6
:
370435365
[0
:
979688608663986
;
0
:
0424888904957123]
[

0
:
271690713626179
;
6
:
36456322236963]
Òåïåðüìîæåìïîëó÷èòüãðàôèêèêîîðäèíàò,ñêîðîñòèîòâðåìåíèèäàæåòðàåêòîðèþ
äâèæåíèÿ:
plot(['X(t)','Y(t)'],t=0..3,title="Êîîðäèíàòûîáúåêòà",legend=[x(t),y(t)],
style=point,color=[blue,red],thickness=3);
plot(['VX(t)','VY(t)'],t=0..3,title="Ïðîåêöèèñêîðîñòè",legend=[Vx(t),Vy(t)],
color=[blue,red],thickness=3);
plot(['X(t)','Y(t)',t=0..3],title="Òðàåêòîðèÿäâèæåíèÿîáúåêòà",color=red,
thickness=3,scaling=constrained);
61
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
à)
á)
Ðèñ.37
:ÊîîðäèíàòûöåíòðàÇåìëèèïðîåêöèèå¼ñêîðîñòè
Ðèñ.38
:ÒðàåêòîðèÿäâèæåíèÿÇåìëèâîêðóãÑîëíöà
Òåïåðüñîçäàäèìïðîöåäóðó,÷òîáûïðîäåìîíñòðèðîâàòüâòîðîéçàêîíÊåïëåðà:Ðàäèóñ-
âåêòîðïëàíåòûçàîäèíàêîâûåïðîìåæóòêèâðåìåíèîïèñûâàåòðàâíûåïëîùàäè,
N:=288;T:=1;
Lr:=[seq([X(T/N*i),Y(T/N*i)],i=0..N)]:
with(plots):
TRAJ:=plot(Lr,style=point):
62
III.2.Äâèæåíèåíåáåñíûõòåë
forifrom0toNdo:
à)
á)
Ðèñ.39
:Ïëîùàäüêîòîðóþîïèñûâàåòðàäèóñ-âåêòîðÇåìëèçàîäèíàêîâûåïðîìåæóòêè
âðåìåíè(3/288ãîäà)
63
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
Ïîëó÷èëèàíèìàöèþäâèæåíèÿÇåìëèâîêðóãÑîëíöà.Çäåñü
t
-âðåìÿäâèæåíèÿ,
V
-ñêîðîñòüöåíòðàÇåìëè,
r
-ðàññòîÿíèåÇåìëèäîöåíòðàÑîëíöàèíàêîíåö
S
-ïëîùàäü
âêâàäðàòíûõà.å,êîòîðóþîïèñûâàåòðàäèóñ-âåêòîðïëàíåòûçàêàæäûå
3
288
ãîäà.Êàê
âèäíîèçàíèìàöèè,ïëîùàäüîñòà¼òñÿïî÷òèíåèçìåííîé.Íåáîëüøàÿðàçíèöàìåæäó
ïëîùàäÿìèâîçíèêàåòêîãäàïëàíåòàíàõîäèòñÿâïåðèãåëèèèâàïîãåå.Ìûðàññ÷èòûâà-
åìïëîùàäüòðåóãîëüíèêàâåðøèíàìèêîòîðîãîÿâëÿþòñÿíà÷àëîêîîðäèíàòïîëîæåíèÿ
ïëàíåòûïðè
t
(
i
)
è
t
(
i
+1)
:
Ñïîìîùüþíàøåéïðîãðàììûìîæåìèçó÷èòüäâèæåíèåëþáîéïëàíåòûñîëíå÷-
íîéñèñòåìû.ÐàññìîòðèìäâèæåíèåÌåðêóðèé.Ïîìåíÿåìíà÷àëüíûåóñëîâèÿ,îáëàñòü
ïðîñìîòðàãðàôèêîâ(äëÿíàãëÿäíîñòè)èâðåìÿàíèìàöèè;
x0:=46.00/150:y0:=0:
vx0:=0:vy0:=6.28*58.98/29.86
plot(['X(t)','Y(t)'],t=0..0.4,title="ÊîîðäèíàòûÏëàíåòû",
legend=[x(t),y(t)],
style=point,color=[blue,red],thickness=3);
plot(['VX(t)','VY(t)'],t=0..0.4,title="Ïðîåêöèèñêîðîñòèïëàíåòû",
legend=[Vx(t),Vy(t)],
color=[blue,red],thickness=3);
plot(['X(t)','Y(t)',t=0..0.4],title="Òðàåêòîðèÿäâèæåíèÿïëàíåòû",
color=red,thickness=3,scaling=constrained);
N:=576;T:=0.241;
Âíà÷àëüíûõóñëîâèÿõìûèñïîëüçîâàëèäàííûåÌåðêóðèéâïåðèãåëèè[
6
].Ïîëó÷èëè
ñëåäóþùèåðåçóëüòàòû:
à)
á)
Ðèñ.40
:ÊîîðäèíàòûöåíòðàÌåðêóðèÿèïðîåêöèèåãîñêîðîñòè
64
III.3.ÄâèæåíèåèñêóññòâåííûõñïóòíèêîâÇåìëè
à)
á)
Ðèñ.41
:Ïëîùàäüêîòîðóþîïèñûâàåòðàäèóñ-âåêòîðÌåðêóðèÿçàîäèíàêîâûåïðîìå-
æóòêèâðåìåíè(3/576ãîäà)
III.3ÄâèæåíèåèñêóññòâåííûõñïóòíèêîâÇåìëè
Âãëàâå
II.2
ðàññìàòðèâàëèçàâèñèìîñòüòðàåêòîðèèäâèæåíèÿèñêóññòâåííûõñïóò-
íèêîâÇåìëèîòíà÷àëüíîéñêîðîñòèèîòíà÷àëüíîéâûñîòû.Âýòîìïàðàãðàôåèçó÷èì
ýòèçàâèñèìîñòè.Ñîçäàäèìïðîöåäóðó,êîòîðàÿáóäåòîïðåäåëÿòüòðàåêòîðèþñïóòíèêà
ïîçàäàííûìïàðàìåòðàìçàïóñêà.
restart;with(plots):
G:=6.67*10^(-11):
Me:=6*10^24:
R:=6.378*10^6:
h0:=600*10^3:
v1:=sqrt(G*Me/(R+h0));
v2:=sqrt(2*G*Me/(R+h0));
h0:=600000v1:=7573.088980v2:=10709.96514
Çäåñü
v
1
è
v
2
ïåðâàÿèâòîðàÿêîñìè÷åñêèåñêîðîñòèîòíîñèòåëüíîïîâåðõíîñòèÇåì-
ëè,
vh
0
êðóãîâàÿñêîðîñòüíàâûñîòå
h
0
.Ìûáóäåìçàïóñêàòüñïóòíèêñâûñîòû
h
0=600
êì
ïàðàëëåëüíîïîâåðõíîñòèÇåìëè.Ïåðåõîäèìêïðîöåäóðå:
Satellite:=proc(yy,vxx,vyy)localax,ay,Inic,N,eq,i,T,eqT,X,Y,p1,p2;
ax:=diff(x(t),t,t)=-G*Me*x(t)/(x(t)^2+y(t)^2)^(3/2):
ay:=diff(y(t),t,t)=-G*Me*y(t)/(x(t)^2+y(t)^2)^(3/2):
Inic:=x(0)=R+h0,D(x)(0)=vxx,y(0)=yy,D(y)(0)=vyy:
Digits:=30:
eq:=dsolve({ax,ay,Inic},{x(t),y(t)},numeric):
N:=700:
65
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
forifrom0toNdo;
T:=i*40;
eqT:=eq(T):
X[i]:=subs(eqT,x(t));
Y[i]:=subs(eqT,y(t));
od:
p1:=polarplot(6.378*10^6,phi=0..2*Pi,color="LightSkyBlue",transparency=0.75,filled=true);
p2:=plot([seq([X[i],Y[i]],i=0..N)],thickness=2,color=red);
display({p1,p2},labels=['x','y']);
endproc:
Ñíà÷àëàäàäèìêðóãîâóþñêîðîñòü,àïîòîìíåñêîëüêîìåíüøóþèïîñìîòðèì,êàêìå-
íÿåòñÿòðàåêòîðèÿñïóòíèêà:
Satellite(0,0,v1);
Satellite(0,0,7000);
Ïîëó÷èëè:Êàêâèäíîèçðèñóíêîâ,âñëó÷àå(à)òðàåêòîðèÿÿâëÿåòñÿîêðóæíîñòè.Âñëó-
à)
á)
Ðèñ.42
:Òðàåêòîðèèñïóòíèêàïðèíà÷àëüíîéâûñîòû600êìèñíà÷àëüíîéñêîðîñòè
ðàâíîéêðóãîâîé(à)èñíà÷àëüíîéñêîðîñòèìåíüøåéêðóãîâîé(á)
÷àå(á)òðàåêòîðèÿñïóòíèêàïåðåñåêàåòñÿñÇåìëåé,ò.å.ñïóòíèêóïàä¼òíàïîâåðõíîñòü
Çåìëè.Íàêîíåö,çàäàäèìñêîðîñòüáîëüøóþ÷åìêðóãîâàÿ,íîìåíüøóþ÷åìïàðàáîëè-
÷åñêàÿ,àòàêæå-ñêîðîñòü,ðàâíóþïàðàáîëè÷åñêîé.
Satellite(0,0,9000);
Satellite(0,0,v2);
Ðåçóëüòàòûïîëíîñòüþñîâïàäàþòñòåîðèåé,îïèñàííîéâãëàâå
II.2
.
66
III.3.ÄâèæåíèåèñêóññòâåííûõñïóòíèêîâÇåìëè
à)
á)
Ðèñ.43
:Òðàåêòîðèèñïóòíèêàñíà÷àëüíîéâûñîòîé600êìèñíà÷àëüíîéñêîðîñòüþ
áîëüøåé÷åìêðóãîâîé,íîìåíüøå÷åìïàðàáîëè÷åñêîé(à)èñíà÷àëüíîéñêîðîñòèðàâ-
íîéïàðàáîëè÷åñêîé(á)
III.3.1Ïåðåõîäîáúåêòàñîäíîéîðáèòûâäðóãóþ
Âíà÷àëåXXâåêà,êîãäàïðèíöèïèàëüíàÿâûïîëíèìîñòüêîñìè÷åñêèõïîëåòîâáûëà
íàó÷íîîáîñíîâàíà,ïîÿâèëèñüïåðâûåñîîáðàæåíèÿîáèõâîçìîæíûõòðàåêòîðèÿõ.Â
1925ãîäóíåìåöêèéèíæåíåðÂàëüòåðÃîìàí(WalterHohmann)ïîêàçàë,÷òîìèíèìàëü-
íûåçàòðàòûýíåðãèèíàïåðåëåòìåæäóäâóìÿêðóãîâûìèîðáèòàìèîáåñïå÷èâàþòñÿ,
êîãäàòðàåêòîðèÿïðåäñòàâëÿåòñîáîé¾ïîëîâèíêó¿ýëëèïñà,êàñàþùåãîñÿèñõîäíîéè
êîíå÷íîéîðáèò.Ïðèýòîìäâèãàòåëüêîñìè÷åñêîãîàïïàðàòàäîëæåíâûäàòüâñåãîäâà
èìïóëüñà:âïåðèãååèàïîãåå(åñëèðå÷üèäåòîáîêîëîçåìíîìïðîñòðàíñòâå)ïåðåõîäíîãî
ýëëèïñà.Äàííàÿñõåìàøèðîêîèñïîëüçóåòñÿ,íàïðèìåð,ïðèâûâåäåíèèíàãåîñòàöèî-
íàðíóþîðáèòó.Âìåæïëàíåòíûõïîëåòàõçàäà÷àíåñêîëüêîîñëîæíÿåòñÿíåîáõîäèìî-
ñòüþó÷èòûâàòüïðèòÿæåíèåÇåìëèèïëàíåòûíàçíà÷åíèÿñîîòâåòñòâåííîíàíà÷àëüíîì
èêîíå÷íîìó÷àñòêàõòðàåêòîðèè.Òåìíåìåíåå,ïîëåòûêÂåíåðåèÌàðñóâûïîëíÿþòñÿ
ïîîðáèòàì,áëèçêèìêãîìàíîâñêèì.ÎòïðàâëÿåìïëàíåòàðíûéçîíäñîðáèòûÇåìëèñ
ðàäèóñîì
150ìèëëèîíêì
íàîðáèòóÌàðñàñðàäèóñîì
232ìèëëèîíêì
.Äëÿíà÷àëà
ïîñìîòðèì,êàêâûãëÿäÿòýòèîðáèòû.(Äëÿïðîñòîòûìûáåðåìîðáèòûêàêîêðóæíî-
ñòè)
restart:with(plots):
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
Ðèñ.44
:ÎðáèòûäâèæåíèèÇåìëèèÌàðñàâîêðóãÑîëíöà
Äëÿíà÷àëàïîñìîòðèì,êàêáóäåòäâèãàòüñÿíàøçîíä,åñëèìûäàäèìåìóäîñòà-
òî÷íóþíà÷àëüíóþñêîðîñòü,÷òîáûîíìîãâûéòèíàîðáèòóÌàðñà.Âòàêîìñëó÷àå
çîíäáóäåòäâèãàòüñÿïîýëëèïñó;ïåðèãåéíîìðàññòîÿíèåìáóäåòðàäèóñîðáèòûÇåìëè,
ààôôåëèéíîìðàññòîÿíèåì-ðàäèóñîðáèòûÌàðñà.
rp:=150*(10^9):
ra:=232*(10^9):
R
2
=
m
v
2
i
R
,ãäå
M
-ìàññàñîëíöà,
R
-ðàäèóñîðáèòûÇåìëè.Îíàáóäåò:
vi:=sqrt(G*M/R);
ïîëíàÿìåõàíè÷åñêàÿýíåðãèÿ
E
çîíäàíàîðáèòåÇåìëèðàâíà
E:=m*v^2/2+(-GMm/R);
Âçàâèñèìîñòèîòôîðìûîðáèòû
E
ìîæåòèìåòüçíà÷åíèÿ
Ec:=-G*M*m/(2*R);#äëÿêðóãîâûõîðáèòðàäèóñîìR,
Ee:=-G*M*m/(2*a);#äëÿýëëèïòè÷åñêèõîðáèòáîëüøåéïîëóîñèa
Åñëèóâåëè÷åíèåñêîðîñòèïðîèñõîäèòî÷åíüáûñòðî(ïîêàåãîïîëîæåíèåíåñèëüíî
èçìåíèëîñü),òîïîòåíöèàëüíàÿýíåðãèÿîñòà¼òñÿíåèçìåííîé.Òîãäàèçìåíåíèåïîëíîé
ýíåðãèèèáóäåòðàâíÿòüñÿèçìåíåíèþêèíåòè÷åñêîéýíåðãèèçîíäà:
68
III.3.ÄâèæåíèåèñêóññòâåííûõñïóòíèêîâÇåìëè
Ðèñ.45
:ÂûõîäñîðáèòûÇåìëèíàîðáèòóÌàðñà
KEchange:=Ee-Ec;
È,íàêîíåö,íàéäåìêîíå÷íóþêèíåòè÷åñêóþýíåðãèþçîíäà(ñðàçóïîñëåçàïóñêà)
KEfinal:=m*(vi^2)/2+KEchange;
vi
:
=
r
GM
R
E
:
=
1
2
mv
2

GMm
R
Ec
:
=

1
2
GMm
R
Ee
:
=

1
2
GMm
a
KEchange
:
=

1
2
GMm
a
+
1
2
GMm
R
Òåïåðüââåäåìíóæíûåïîñòîÿííûå,íàéäåìêîíå÷íóþñêîðîñòüèóâåëè÷åíèåñêîðîñòè
ïîñëåïåðâîãîèìïóëüñà.
G:=6.673*10^(-11);M:=1.99*10^(30);
m:=6000;R:=150*10^9;
a:=(150*(10^9)+232*(10^9))/2;
vfinal:=sqrt(2*KEfinal/m);
Delta[v1]:=vfinal-vi;
G
:
=6
:
67300000010

11
M
:
=1
:
99000000010
3
0
m
:
=6000
R
:
=150000000000
69
ÃëàâàIII.Ìàòåìàòè÷åñêîåìîäåëèðîâàíèåâMAPLE
a
:
=191000000000
vfinal
:
=32792
:
06604

v
1
:
=3038
:
33238
Åñëèìûäàëüøåíåáóäåìâìåøèâàòüñÿ,òîçîíäáóäåòäâèãàòüñÿïîýëëèïñó,ïåðèãåëèé
êîòîðîãîíàõîäèòñÿíàîðáèòåÇåìëè,ààôåëèé-íàîðáèòåÌàðñà.Åñëèìûõîòèì÷òîáû
îíîñòàëñÿíàîðáèòåÌàðñà,ìûäîëæíûåù¼ðàçóâåëè÷èòüåãîñêîðîñòü.Íàéäåìýòîãî
óâåëè÷åíèå:
KE[f]:=KEfinal+(-G*M*m/R)-(-G*M*m/(ra));#êèíåòè÷åñêàýíåðãèÿçîíäà
ïðèäîñòèæåíèèîðáèòóÌàðñà,
v[f]:=sqrt(2*KE[f]/m);#ñêîðîñòüçîíäàïðèäîñòèæåíèèîðáèòóÌàðñà,
vf:=sqrt(G*M/ra);#êðóãîâàÿñêîðîñòüâäàííîéîðáèòå,
Delta[v2]:=vf-v[f];
Ïîëó÷èëèñëåäóþùèåðåçóëüòàòû:
KEfinal
:
=3
:
22595878610
12
v
f
:
=21201
:
76685
vf
:
=23924
:
51311

v
2
:
=2722
:
74626
Ïîëó÷åííûåðåçóëüòàòûõîðîøîñîâïàäàþòñöèôðàìèâðàáîòå[
10
].
70
Çàêëþ÷åíèå
Âíàñòîÿùåéäèññåðòàöèîííîéðàáîòåáûëèðåøåíûñëåäóþùèåçàäà÷è:
Èçó÷åíûîñíîâíûåâîçìîæíîñòèïàêåòàMapleäëÿêîìïüþòåðíîãîìîäåëèðîâàíèÿ
Èçó÷åíûçàêîíûäâèæåíèÿíåáåñíûõòåë
ÑîçäàíàïðîãðàììàïîñòðîåíèÿêðèâûõâòîðîãîïîðÿäêàâïàêåòåMaple
Ïîñòðîåíàãåîìåòðè÷åñêàÿìîäåëüÑîëíå÷íîéñèñòåìûñèñïîëüçîâàíèåìèçâåñòíûõ
àñòðîíîìè÷åñêèõäàííûõ
Ðàçðàáîòàíûêîìïüþòåðíûåìîäåëèäâèæåíèÿíåáåñíûõòåë,èëëþñòðèðóþùèåðàç-
ëè÷íûåçàêîíûÊåïëåðà
ÑîñòàâëåíàïðîãðàììàäëÿìîäåëèðîâàíèÿäâèæåíèÿèñêóññòâåííûõñïóòíèêîâÇåì-
ëè
Ïîäâîäÿèòîã,ñëåäóåòñêàçàòü,÷òîâñåçàäà÷è,ïîñòàâëåííûåâäèññåðòàöèîííîé
ðàáîòå,áûëèâûïîëíåíûïîëíîñòüþ.
71
Ëèòåðàòóðà
[1]
À.Ñ.Áîðòàêîâñêèé,À.Â.Ïàíòåëååâ.Àíàëèòè÷åñêàÿãåîìåòðèÿâïðèìåðàõèçàäà-
÷àõ.Ìîñêâà:¾Âûñøàÿøêîëà¿.2005.496ñ.
[2]
Ä.Â.Ñèâóõèí.Îáùèéêóðñôèçèêè,Òîì.I.Ìåõàíèêà.Ìîñêâà:¾ÌÔÒÈ¿.2005.
496ñ.
[3]
Ý.Â.Êîíîíîâè÷,Â.È.Ìîðîç.Îáùèéêóðñàñòðîíîìèè.Ìîñêâà:¾Åäèòîðèàë
ÓÐÑÑ¿,2004.544ñ.
[4]
http://www.astronom2000.info/àñòðîíîìèÿ/ñîëíå÷íàÿ-ñèñòåìà
[5]
Ëèòåðàòóðà
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Z
1
1
c

Îôîðìëåíèå:LaTeX-ñòèëü
B
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ïðîôåññîðàÞ.Ã.Èãíàòüåâà
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