Введение в аналитическую динамику. Кирсанов А.А. - 70 стр.

70                                                                                 Ãëàâà     òðåòüÿ
                  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 
                                                              ,                            (3.1.7)*


                        j = 1,2,..., n .
      Òàêèì îáðàçîì ñ ïîìîùüþ ðàâåíñòâà (3.1.7)* ìû óáèðàåì ñóììè-
ðîâàíèå ïî            j èç ðàâåíñòâà (3.1.7).
      T1 - îäíîðîäíàÿ ëèíåéíàÿ ôóíêöèÿ îò q& :
                      n                 n
                                            ∂x r ∂x r           n

                  ∑ m ⋅ 2∑ ∂q                                  ∑ a q&
              1
      T1 =                                            q& i =                  ,             (3.1.8)
                                               i ∂t
                              r                                       r   r
              2   r =1               i =1                      r =1
ãäå
              n
                               ∂x j ∂x j ∂y j ∂y j ∂z j ∂z j 
      ar =    ∑ m  ∂q
              j =1
                          j
                                  r
                                    ⋅    +    ⋅    +    ⋅
                                      ∂t ∂qr ∂t ∂qr ∂t 
                                                              ,                            (3.1.8)*


                                                 j = 1,2,..., n .
      T0 ÿâëÿåòñÿ ôóíêöèåé îò q1 , q2 ,.., qn , t :
                                             2
                               ∂x 
                      n

                  ∑
              1
      T0 =                 mr  r  .                                                       (3.1.9)
              2       r =1     ∂t 
       Äëÿ ñòàöèîíàðíûõ ñâÿçåé, òî åñòü ñâÿçåé, äëÿ êîòîðûõ
                                                                      ∂x r
x r = x r (q1 , q2 ,..., qn ) è, ñëåäîâàòåëüíî,                            ≡ 0 , êèíåòè÷åñêàÿ ýíåðãèÿ
                                                                       ∂t
áóäåò îäíîðîäíîé ôóíêöèåé âòîðîé ñòåïåíè îòíîñèòåëüíî îáîáù¸ííûõ
ñêîðîñòåé, òî åñòü
                               n    n

                              ∑∑ a
                          1
      T = T2 =                                   & & .
                                            ri q r qi                                       (3.1.10)
                          2   r =1 i =1
      Âåðí¸ìñÿ ñíîâà ê îñíîâíîìó óðàâíåíèþ (2.1.6)
       N

      ∑ (mx&&
      r =1
                  r    − X r )δx r = 0 .                                                    (2.1.6)