Nielsen form

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

:$$egin{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 ight ] -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 ight ] + {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 ight ] -{partial{T}over partial q_j}\& = & {mathrm{d} over mathrm{d}t} left( {partial T over partial q'_j} ight) -{partial T over partial q_j}\ end{matrix}as desired.