Assume that we have Euler-Lagrange equation for $f: \mathbb{R}^3 \to \mathbb{R}$. By solution I will mean a function $x: \mathbb{R} \to \mathbb{R}$ which satisfies the following differential equation.
$$ \frac{d}{dt} \left( \frac{\partial}{\partial \dot{x}} f(x(t),\dot{x}(t),t) \right) (t) = \frac{\partial}{\partial x} \left( f(x(t), \dot{x}(t), t) \right) (t)$$
Let us assume that if $x(t)$ is a solution then for all $s \in \mathbb{R}$ it is true that $\hat{x}(t) = x(t) + st$ is a solution.
Is there something that can be said about $f$? I would appreciate your help!