Краткий курс теоретической механики. Яковенко Г.Н. - 74 стр.

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W = W
r
+ W
ϕ
W
r
= ¨r − r ˙ϕ
2
W
ϕ
= ¨ϕr + 2 ˙ϕ ˙r
O
r
W
j
W
r
mW = F = fr/r
r
m(¨r − r ˙ϕ
2
) = f(t, r, ˙r, ϕ, ˙ϕ). (20.5)
t
0
r
0
˙r
0
ϕ
0
˙ϕ
0
r(t) ϕ(t)
r(ϕ)
t ϕ
˙r =
dr
dt
=
dr
dϕ
dϕ
dt
(20.4)
=
dr
dϕ
c
r
2
= −
d
dϕ
µ
1
r
c = −u
0
c, (20.6)
u =
1
r
, u
0
=
du
dϕ
, (20.7)
¨r =
d ˙r
dt
=
d ˙r
dϕ
dϕ
dt
(20.4)(20.6)
= −u
00
c
2
r
2
. (20.8)
u
00
+ u = −
r
2
mc
2
f(t, r, ˙r, ϕ, ˙ϕ). (20.9)
∂f/∂t = 0
u
00
+ u = −
1
mc
2
u
2
f(1/u, −u
0
c, ϕ, u
2
c) (20.10)
r(ϕ) = 1/u(ϕ)
Óñêîðåíèå òî÷êè â ïîëÿðíûõ êîîðäèíàòàõ âûðàæàåòñÿ ñëåäóþùèì îáðàçîì
((8.5), (8.6)): W = Wr + Wϕ , Wr = r̈ − rϕ̇2 , Wϕ = ϕ̈r + 2ϕ̇ṙ (ðèñ. 20.3).



                                      Wj
                                                                      Wr
O                            r
                                        Ðèñ. 20.3
Ïðîåêöèÿ óðàâíåíèÿ Íüþòîíà mW = F = f r/r íà íàïðàâëåíèå ðàäèóñ
âåêòîðà r ïðèâîäèò ê óðàâíåíèþ

                           m(r̈ − rϕ̇2 ) = f (t, r, ṙ, ϕ, ϕ̇).                (20.5)
Ñèñòåìà óðàâíåíèé (20.4), (20.5) åñòü çàìêíóòàÿ ñèñòåìà îáûêíîâåííûõ äèô-
ôåðåíöèàëüíûõ óðàâíåíèé òðåòüåãî ïîðÿäêà äëÿ îïðåäåëåíèÿ ïðè íà÷àëüíûõ
äàííûõ t0 , r0 , ṙ0 , ϕ0 , ϕ̇0 äâèæåíèÿ r(t), ϕ(t) òî÷êè. Äëÿ îïðåäåëåíèÿ òðàåêòîðèè
r(ϕ) òî÷êè â ïîëÿðíûõ êîîðäèíàòàõ ñäåëàåì ñ ó÷¼òîì (20.4) â óðàâíåíèè (20.5)
ïåðåõîä îò íåçàâèñèìîé ïåðåìåííîé t ê íåçàâèñèìîé ïåðåìåííîé ϕ:
                                                           µ ¶
                         dr      dr dϕ (20.4) dr c       d 1
                   ṙ =       =          =         2
                                                     =−         c = −u0 c,      (20.6)
                         dt      dϕ dt        dϕ r      dϕ r
ââåäåíû îáîçíà÷åíèÿ
                                   1                  du
                                 u= ,          u0 =      ,                     (20.7)
                                   r                  dϕ
                                                            2
                           dṙ   dṙ dϕ (20.4)(20.6)   00 c
                      r̈ =     =             =       −u 2 .             (20.8)
                           dt    dϕ dt                    r
Ïîäñòàíîâêà (20.4), (20.8) â óðàâíåíèå (20.5) è î÷åâèäíûå ïðåîáðàçîâàíèÿ ïðè-
âîäÿò ê óðàâíåíèþ Áèíå
                                         r2
                          u00 + u = −        f (t, r, ṙ, ϕ, ϕ̇).              (20.9)
                                         mc2
 àâòîíîìíîì ñëó÷àå (∂f /∂t = 0) óðàâíåíèå (20.9) ïîñëå ïîäñòàíîâêè (20.4),
(20.6), (20.7) ñòàíîâèòñÿ çàìêíóòûì óðàâíåíèåì
                                       1
                       u00 + u = −          f (1/u, −u0 c, ϕ, u2 c)           (20.10)
                                     mc2 u2
äëÿ íàõîæäåíèÿ òðàåêòîðèè r(ϕ) = 1/u(ϕ). Â ëèòåðàòóðå óðàâíåíèå (20.9) íà-
çûâàþò òàêæå âòîðîé ôîðìóëîé Áèíå (ïåðâàÿ ôîðìóëà çäåñü íå ðàññìàòðèâà-
åòñÿ).

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