I have been trying to understand the Legendre transformation (in mechanics, in the hyperregular case: when the Legendre transformation is one-to-one) and the correspondence between symmetry $\to$ conserved current on the Lagrangian side and conserved charge $\to$ symmetry on the Hamiltonian side. I found a lot of interesting elements here on Physics Stack Exchange:
- Is the converse of Noether's first theorem true: Every conservation law has a symmetry?
- A kind of Noether's theorem for the Hamiltonian formalism
- Invariance of action ⇒ covariance of field equations?
- Why do the Lagrangian and Hamiltonian formulations give the same conserved quantities for the same symmetries?
among others. Let's say we have a configuraiton space $E$. So here is the picture I get:
- From a (quasi-)symmetry $Y$ of the Lagrangian $L:TE\to\mathbb{R}$, one gets the conserved Noether charge $Q_Y$. As a conserved charge, from the Hamiltonian formulation we obtain a corresponding (infinitesimal) symmetry $X_{Q_Y}$ acting on the phase space $T^*E$.
- From the Hamiltonian $H:T^*E\to\mathbb{R}$ we construct the Hamiltonian Lagrangian $L_H(q,\dot q,p) = p\dot q - H(q,p)$. Its quasi-symmetries are in one-to-one correspondance with the conserved charge of the Hamiltonian.
- A quasi-symmetry $X$ of the Hamiltonian Lagrangian provides a quasi-symmetry of the initial Lagrangian directly (by pullback) through the Legendre transformation $(q,\dot q)\mapsto (\mathcal{L}(q,\dot q),\dot q)=(q,\partial_{\dot q}L(q,\dot q),\dot q)$.
But I found myself unable to show that the quasi-symmetry of the initial Lagrangian then obtained matches with the initial one. From the second point I know they have the same Noether charge. I'd be happy with a more direct argument but I would be already satisfied if one can show that it does imply that the (Lagrangian) quasi-symmetry are the same. To show that $X_{Q_Y} = \mathcal{L}_*Y$ or that $\delta_{\mathcal{L}_*Y} L_H = f$ would be enough for what I am looking for.
Another point that is not clear to me are the so called generalized variational symmetries $Y$ defined here. They are symmetries depending on the first order derivative, e.g. the symmetry associated with the Laplace-Runge-Lenz vector. Geometrically, should they be viewed as a vector field of $E$ defined on $TE$ (section of the pullback $TE\times_E TE\to TE$)? Or a vector field of $TE$?
