Let us assume we are doing classical one point particle mechanics. Assume that the least action principle holds. Also, assume that Lagrangian $L$ is a function only of coordinate $x$, its derivative $\dot{x}$ and time $t$. Then by the principle, we know that the following differential equation gives the equation of motion for a particle.
$$ \frac{\partial}{\partial x} L(x, \dot{x}, t) = \frac{d}{dt} \frac{\partial}{\partial \dot{x}} L(x, \dot{x}, t) $$
Now assume that whenever $\phi(t)$ is a solution for this differential equation then $\phi(t) + vt$ is a solution as well. Here, $v$ is assumed to be constant but arbitrary.
What can be said about functional dependence of $L$ on $x, \dot{x}, t$?
I would appreciate your help!
