Nielsen form

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

<math>

{d \over dt}{\partial{T}\over \partial{q'_j}}-{\partial{T}\over \partial q_j} = Q_j </math>

The Nielsen Form is an alternative formulation written as

<math>

{\partial{T'}\over \partial{q'_j}}-2{\partial{T}\over \partial q_j} = Q_j </math>

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

<math>T = T(q_i, q'_i)</math>

then

<math>

 \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}\\ & = & {d \over dt} \left( {\partial T \over \partial q'_j} \right) -{\partial T \over \partial q_j}\\

 \end{matrix}

</math> As desired.