Введение в аналитическую динамику. Кирсанов А.А. - 138 стр.

138                                                          Ãëàâà ïÿòàÿ

      ∂L                             pr
          = mr& = pr ,        r& =      ,
      ∂r&                            m
      ∂L                       p
        & = mr 2θ& = pθ , θ& = θ 2 ,
      ∂θ                      mr
      ∂L                             pz
          = mz& = p z ,       z& =      .
      ∂z&                            m
      Ñîñòàâèì ôóíêöèþ Ãàìèëüòîíà
      H = ∑ pi q&i − L =

          1 2 1 2         2    1  2 1 2        2
      =     pr + 2 pθ + p z  −    pr + 2 pθ + pz  + V (r ,θ , z ) .
          m     r            2m       r          
      Îêîí÷àòåëüíî ïîëó÷èì
              1  2 1 2         2
      H=         pr + 2 pθ + p z  + V (r,θ , z ).              (5.4.20)
             2m      r           
      Ñîñòàâèì êàíîíè÷åñêèå óðàâíåíèÿ Ãàìèëüòîíà
      ∂H pr        ∂H   p          ∂H pz
         =  = r& ,    = θ 2 = θ& ,     =  = z&
      ∂pr m        ∂pθ mr          ∂p z m
è

      ∂H   pθ2 ∂V             ∂H ∂V            ∂H ∂V
         =− 3 +    = − p& r ,   =   = − p& θ ,    =    = − p& z .
      ∂r   mr   ∂r            ∂θ ∂θ            ∂z   ∂z

     3. Ñôåðè÷åñêèå êîîðäèíàòû.
     Âûðàçèì äåêàðòîâû êîîðäèíàòû ìàòåðèàëüíîé òî÷êè ÷åðåç ñôå-
ðè÷åñêèå êîîðäèíàòû (ñì. çàäà÷ó ¹11)

      x = r sin θ cosϕ 
                        
      y = r sin θ sin ϕ 
                          .                                      (5.4.21)
      z = r cosθ        
                        