Введение в аналитическую динамику. Кирсанов А.А. - 89 стр.

Óðàâíåíèÿ Ëàãðàíæà                                                    89

              n
                          ∂x j ∂x j ∂y j ∂y j ∂z j ∂z j 
      ari =   ∑ m  ∂q
              j =1
                     j         ⋅
                             r ∂qi
                                    +    ⋅    +    ⋅
                                      ∂qr ∂qi ∂qr ∂qi 
                                                               (3.7.26)

èëè


         ∂x  2  ∂y  2  ∂z  2             
a11 = m   +   +    = ml 2 ,              
         ∂θ   ∂θ   ∂θ                  
                                                 
                 ∂x ∂x   ∂y ∂y   ∂z ∂z   
a12 = a 21 = m  ⋅  +  ⋅  +  ⋅  = 0,              (3.7.27)
                 ∂θ ∂ϕ   ∂θ ∂ϕ   ∂θ ∂ϕ   
          ∂x   ∂y   ∂z  
                 2       2      2                
a 22 = m   +   +    = ml sin θ.
                                       2   2     
          ∂ϕ   ∂ϕ   ∂ϕ                 
                                                 


      Òàêèì îáðàçîì

      T=
           2
               (
           m 2&2 2 2 2
             l θ + l sin θϕ& .        )                         (3.7.28)

      Òîãäà
      ∂T                          ∂T
          = ml 2 θ& ,                = ml 2 sin θ cos θϕ& 2 ,
      ∂θ&                         ∂θ
      ∂T                     ∂T
          = ml 2 sin 2 θϕ& ,    =0.
      ∂ϕ&                    ∂ϕ
      Òåïåðü ñ ó÷¸òîì (3.7.25) ñîñòàâèì óðàâíåíèÿ Ëàãðàíæà (3.2.1)

      d  ∂ T  ∂T                        d  ∂T  ∂T
              −   = Q1 ,                          −   =0
      dt  ∂θ&  ∂θ                       dt  ∂ϕ&  ∂ϕ
èëè

      &θ& − sin θ cos θϕ& 2 + g sin θ = 0 ,     && = 0 .
                                                ϕ               (3.7.29)
                              l