Задачи по теоретической механике. Манаков Н.Л - 39 стр.

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E 6 E
cr
, r
min
= a
χ=
π − 2 arcsin
ρ
a
, E 6 E
cr
,
2 arcsin
Ã
ρ
a
r
E
E − U
0
!
−2 arcsin
ρ
a
, E > E
cr
.
r
min
<a
@
@
@
@
@
a
O
H
H
H
H
H
Hj
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
HH
H
H
Hj
:
r
min
=a
¥
¥
¥
¥
¥
¥
¥
-
*
©
©
©
©
©
©
©
©
χ
χ
max
= χ
Ã
ρ = a
r
E − U
0
E
!
= π − 2 arcsin
r
E − U
0
E
.
m
U(r) = −α
ln(r/a)
r
2
,
r = a exp
·
L
2
2mα
+
mα
2L
2
(ϕ − ϕ
0
)
2
¸
.
U(r) = −
α
r
6
.
r
2
=
√
2mα
L
sin 2(ϕ − ϕ
0
)
U(r) = −
α
r
+
β
r
2
.
r =
p
1 + ε cos(ϕ/A)
p =
L
2
mα
+
2β
α
, ε =
r
1 +
2Ep
α
, A =
1
p
1 + 2mβ/L
2
.
ϕ
0
= 2πA
U(r) = −
α
r
A = [v ×
L] −
αr
r
A
Åñëè E 6 Ecr , òî rmin = a, è ÷àñòèöà îò-                     H
                                                              HHHH
                                                                 HjH
                                                                  H
ðàæàåòñÿ îò ñôåðû, íå ïðîíèêàÿ âíóòðü (ñì.                        HHrmin =       :
                                                                               a  
                                                                       H    
ðèñ. 4.5). Îòêëîíåíèå îò ïåðâîíà÷àëüíîãî íà-           HH                  ¥         
                                                        j
                                                        HH
                                                          H
                                                              r min <  a  ¥-      
                                                                                 
                                                                                   *
                                                                                   
ïðàâëåíèÿ äâèæåíèÿ îïðåäåëÿåòñÿ óãëîì:                      H
                                                            @ H         ¥    ©
                                                                               ©
                  ρ
                                                             @ H       ¥ ©
   
    π −  2 arcsin   ,   E 6 Ecr ,                            a@ H©¥ H χ
             Ã r  a      !
                                                                 @ ¥         H
                                                                               H
                                                                    @¥           H
χ=              ρ      E              ρ                                 O
   
    2 arcsin               −2 arcsin   , E > Ecr .
               a E − U0              a
                                                                      ðèñ. 4.5
Ìàêñèìóì ýòîé ôóíêöèè:
             Ã      r             !                    r
                         E − U0                            E − U0
    χmax = χ ρ = a                    = π − 2 arcsin              .
                           E                                 E

Çàäà÷à 4.4. Ïîëíàÿ ýíåðãèÿ ÷àñòèöû ìàññû m, äâèæóùåéñÿ â ïîëå
           ln(r/a)
U (r) = −α         , ðàâíà íóëþ. Íàéòè òðàåêòîðèþ òî÷êè.
              r2 ·                    ¸
                    L2    mα
Îòâåò : r = a exp       + 2 (ϕ − ϕ0 )2 .
                   2mα 2L
Çàäà÷à 4.5. Ìàòåðèàëüíàÿ òî÷êà äâèæåòñÿ â öåíòðàëüíîì ïîëå
           α
U (r) = −    . Ïîëíàÿ ýíåðãèÿ ðàâíà íóëþ. Íàéòè òðàåêòîðèþ òî÷êè.
          r6 √
                2mα
Îòâåò : r2 =        sin 2(ϕ − ϕ0 ).
                L
Çàäà÷à 4.6. Íàéòè òðàåêòîðèþ è óãîë ïîâîðîòà ïåðèãåëèÿ äëÿ ìàòåðèàëüíîé
                       α   β
òî÷êè â ïîëå U (r) = −   + 2.
                       r r
                                          p
Îòâåò : Óðàâíåíèå òðàåêòîðèè: r =                ,
                              r   1 + ε cos(ϕ/A)
               L2   2β              2Ep                1
       ãäå p =    + , ε= 1+              , A=p                .
               mα    α               α             1 + 2mβ/L2
       Óãîë ïîâîðîòà ïåðèãåëèÿ ϕ0 = 2πA.

Çàäà÷à 4.7. Ïîêàçàòü, ÷òî äëÿ ìàòåðèàëüíîé òî÷êè, äâèæóùåéñÿ â
                                        α
öåíòðàëüíî-ñèììåòðè÷íîì ïîëå U (r) = − , ñîõðàíÿåòñÿ âåêòîð A = [v ×
                                        r
     αr
L] −    (âåêòîð Ðóíãå-Ëåíöà). Îïðåäåëèòü àáñîëþòíóþ âåëè÷èíó è íàïðàâ-
     r
ëåíèå âåêòîðà A.

                                          38

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