Show the Lagrange equations can also be written on Nielsen's form

Question

Show that the Lagrange Equations can also be written on Nielsen's form

$$\frac{\partial \dot{T}}{\partial \dot{q}_j} - 2\frac{\partial T}{\partial q_j}=Q_j.$$

Answer 1

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 $Q_j$.

Answer 2

so, Here is the Solution to your Question. (Please click on Image to view it clearly)