Show the Lagrange equations can also be written on Nielsen's form Show that the Lagrange Equations can also be written on Nielsen's form
$$\frac{\partial \dot{T}}{\partial \dot{q}_j} - 2\frac{\partial T}{\partial q_j}=Q_j.$$
 A: The total time derivative is
$$ \frac{d}{dt}~=~\frac{\partial}{\partial t}+\dot{q}^j\frac{\partial}{\partial q^j}+\ddot{q}^j\frac{\partial}{\partial \dot{q}^j}+\dddot{q}^j\frac{\partial}{\partial \ddot{q}^j}+\ldots, \tag{1}$$
where dot denotes time differentiation.
Hence the commutator between a velocity differentiation and total time differentiation is a position differentiation
$$ \left[  \frac{\partial}{\partial \dot{q}^j},\frac{d}{dt}\right]
~\stackrel{(1)}{=}~ \frac{\partial}{\partial q^j}.\tag{2}$$
In particular,
$$\frac{d}{dt} \frac{\partial T}{\partial \dot{q}^j}
~\stackrel{(2)}{=}~ 
\frac{\partial\dot{T}}{\partial \dot{q}^j}-\frac{\partial T}{\partial q^j},\tag{3}$$
where $T(t,q,\dot{q})$ is the kinetic energy.
The Lagrange equations read
$$ Q_j~=~\frac{d}{dt} \frac{\partial T}{\partial \dot{q}^j}-\frac{\partial T}{\partial q^j}~\stackrel{(3)}{=}~  \frac{\partial\dot{T}}{\partial \dot{q}^j}-2\frac{\partial T}{\partial q^j}, \tag{4}$$
where $Q_j$ is the generalized force.
A note on terminology: It is customary to only refer to Lagrange equations (4) as Euler-Lagrange equations if they arise from a variational principle. This is the case if there exists a generalized potential $U(t,q,\dot{q})$ for the generalized forces $Q_j$.
A: so, Here is the Solution to your Question.
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