Forma di Nielsen

La Forma di Nielsen è una rappresentazione alternativa delle equazioni di Lagrange del I tipo, che vengono scritte come

${\displaystyle {\partial {\dot {T}} \over \partial {{\dot {q}}_{j}}}-2{\partial {T} \over \partial q_{j}}=Q_{j}}$

La dimostrazione dell'equivalenza si ottiene utilizzando la regola della catena. Infatti, detta ${\displaystyle \delta _{ij}}$ la delta di Kronecker, dalle equazioni di Lagrange del I tipo:

${\displaystyle Q_{j}={\mathrm {d} \over \mathrm {d} t}\left({\partial T \over \partial {\dot {q}}_{j}}\right)-{\partial T \over \partial q_{j}}=}$

${\displaystyle =\sum _{i}\left[{\partial \over \partial q_{i}}{\partial T \over \partial {\dot {q}}_{j}}{\dot {q}}_{i}+{\partial \over \partial {\dot {q}}_{i}}{\partial T \over \partial {\dot {q}}_{j}}{\ddot {q}}_{i}\right]-{\partial {T} \over \partial q_{j}}=}$

${\displaystyle =\sum _{i}\left[{\partial \over \partial {\dot {q}}_{j}}\left({\partial T \over \partial q_{i}}\right){\dot {q}}_{i}+{\partial \over \partial {\dot {q}}_{j}}\left({\partial T \over \partial {\dot {q}}_{i}}\right){\ddot {q}}_{i}\right]+{\partial T \over \partial q_{j}}-2{\partial {T} \over \partial q_{j}}=}$

${\displaystyle =\sum _{i}\left[{\partial \over \partial {\dot {q}}_{j}}\left({\partial T \over \partial q_{i}}\right){\dot {q}}_{i}+{\partial T \over \partial q_{i}}\delta _{ij}+{\partial \over \partial {\dot {q}}_{j}}\left({\partial T \over \partial {\dot {q}}_{i}}\right){\ddot {q}}_{i}+{\partial T \over \partial {\dot {q}}_{i}}{\partial {\ddot {q}}_{i} \over \partial {\dot {q}}_{j}}\right]-2{\partial {T} \over \partial q_{j}}=}$

${\displaystyle ={\partial \over \partial {\dot {q}}_{j}}\sum _{i}\left[{\partial T \over \partial q_{i}}{\dot {q}}_{i}+{\partial T \over \partial {\dot {q}}_{i}}{\ddot {q}}_{i}\right]-2{\partial {T} \over \partial q_{j}}=}$

${\displaystyle ={\partial {\dot {T}} \over \partial {{\dot {q}}_{j}}}-2{\partial {T} \over \partial q_{j}}}$

utilizzando le relazioni:

${\displaystyle {\partial {\ddot {q}}_{i} \over \partial {\dot {q}}_{i}}=0}$

${\displaystyle {\partial \over \partial {\dot {q}}_{j}}{\partial T \over \partial q_{i}}={\partial \over \partial q_{i}}{\partial T \over \partial {\dot {q}}_{j}}}$

${\displaystyle {\partial \over \partial {\dot {q}}_{j}}{\partial T \over \partial {\dot {q}}_{i}}={\partial \over \partial {\dot {q}}_{i}}{\partial T \over \partial {\dot {q}}_{j}}}$.