suppose there is any arbitrary function of variables q, q' and t,
L = L(q, q', t) where q' = \frac{dq}{dt}
can we always find a function $Z(q, t)$, such that,
$L = \frac{dZ}{dt}$
note:
$L, q, t \in$ {$ C^{\infty}$}
suppose there is any arbitrary function of variables q, q' and t,
L = L(q, q', t) where q' = \frac{dq}{dt}
can we always find a function $Z(q, t)$, such that,
$L = \frac{dZ}{dt}$
note:
$L, q, t \in$ {$ C^{\infty}$}
I assume you mean the total differential, and you want a $Z$ that depends on the form of $L$ but not the form of $q$ (i.e. $q$ is merely treated as an argument). In which case the chain rule gives L=\frac{dZ}{dt}=\frac{\partial Z}{\partial q}q'+\frac{\partial Z}{\partial t}. So a necessary condition is that $L$ is linear in q' (formally). Furthermore, if we write L=uq'+v, it is necessary that $(u,v)$ have a potential function (namely $Z$), so (and this is locally sufficient), $\frac{\partial u}{\partial t}=\frac{\partial v}{\partial q}.$