classical mechanics - Canonical transformations preserve Hamilton's equations. Which transformation preserve the Euler-Lagrange equations?

An important aspect in the Hamiltonian formulation of Classical Mechanics are canonical transformations which provide maps between different sets of canonical coordinates. These canonical coordinates are useful, because they represent those coordinate choices for which Hamilton's equations hold. In other words, when we want to use the Hamiltonian formalism, we need to use canonical coordinates.

For this reason, most textbooks discuss in detail which kind of transformations are canonical transformations, i.e. allowed in the Hamiltonian formalism. The main result is that a transformation is canonical if there is an appropriate generating function associated with it which acts via the Poisson bracket on the coordinates.

Now, on Wikipedia I learned that

Canonical coordinates can be obtained from the generalized coordinates of the Lagrangian formalism by a Legendre transformation, or from another set of canonical coordinates by a canonical transformation.

Thus I was wondering what the analogue to canonical transformations would be in the Lagrangian formalism. In this context, textbooks regularly discuss generalized coordinates which are coordinate choices for which the Euler-Lagrange equations hold.

What's the name of those transformations which map suitable generalized coordinates to other suitable generalized coordinates? And how are they defined? (I.e. are the similar defining conditions like for canonical transformations?)

Answer

I believe you'd just call them coordinate transformations or reparameterizations or so, and they're valid as long as they're smooth enough.

In the Hamiltonian dynamics you have three legitimately independent variables, $(q,p,t).$ But in the Euler-Lagrange dynamics you only have two, $(q, t)$, though the Lagrangian itself might contain $\dot q$ or even $\ddot q$. Hamilton is trying to track a point as it moves through phase space, while Lagrange is trying to determine a path through space given a start and end point.

Note that reparametrizattons of the space of the Lagrangian are a subset of canonical transformations, not the full set.