In some texts on classical mechanics and not only, the Euler--Lagrange equations of motion are directly obtained as solution of variational problems.
On the other side, sometimes reading about hamiltonian mechanics, one find the expression that this latter formulation is preferred to the lagrangian one because of it does completely avoid the appeal to variational principles.

About this

This observation , I have a suggested to myself the following question:

Is the variational approach to the Euler--Lagrange equations the only one viable?
If not, is there some reason that explain why the geometry of the Euler-Lagrange eqns is much more hidden than the geometry of the Hamilton eqns?

I was searching for suggestion of reading for best tackle this question.

As usual any feedback is welcome.

In some texts on classical mechanics and not only, the Euler--Lagrange equations of motion are directly obtained as solution of variational problems.
On the other side, sometimes reading about hamiltonian mechanics, one find the expression that this latter formulation is preferred to the lagrangian one because of it does completely avoid the appeal to variational principles.

About this observation, I have a question:

Is the variational approach to the Euler--Lagrange equations the only one viable?
If not, is there some reason that explain why the geometry of the Euler-Lagrange eqns is much more hidden than the geometry of the Hamilton eqns?

As usual any feedback is welcome.