Substitution of variables

Lagrangian mechanics

Lagrange examined how these equations of motion change under an arbitrary substitution of variables ${\displaystyle x=\Psi (t,y)}$, ${\displaystyle v={\frac {\partial \Psi (t,y)}{\partial t}}+{\frac {\partial \Psi (t,y)}{\partial y}}\cdot w.}$