Nielsen form

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Lagrange's Equations in Lagrangian mechanics are usually written in the form

${\mathrm{d} \over \mathrm{d}t}{\partial{T}\over \partial{q'_j}}-{\partial{T}\over \partial q_j} = Q_j$

The Nielsen Form is an alternative formulation written as

${\partial{T'}\over \partial{q'_j}}-2{\partial{T}\over \partial q_j} = Q_j$

These two forms are equivalent; this can easily be shown by the Chain rule. Notice that if

T = T (q i, q' i)

then

$\begin{matrix} Q_j = {\partial{T'}\over \partial{q'_j}}-2{\partial{T}\over \partial q_j} & = & {\partial \over \partial q'_j} \sum_{i} \left [ {\partial T \over \partial q_i}q'_i + {\partial T \over \partial q'_i}q''_i \right ]-2{\partial{T}\over \partial q_j}\\ & = & \sum_{i} \left [ {\partial \over \partial q'_j} {\partial T \over \partial q_i} q'_i + {\partial \over \partial q'_j} {\partial T \over \partial q'_i}q''_i \right ] + {\partial T \over \partial q_j} -2{\partial{T}\over \partial q_j}\\ & = & \sum_{i} \left [ {\partial \over \partial q_i} {\partial T \over \partial q'_j} q'_i + {\partial \over \partial q'_i} {\partial T \over \partial q'_j}q''_i \right ] -{\partial{T}\over \partial q_j}\\ & = & {\mathrm{d} \over \mathrm{d}t} \left( {\partial T \over \partial q'_j} \right) -{\partial T \over \partial q_j}\\ \end{matrix}$