In Lagrangian formalism, we consider point transformations $Q_i=Q_i(q,t)$ because the Euler-Lagrange equation is covariant only under these transformations. Point transformations do not explicitly depend on $\dot{q}_i$ (except when we include change of time $t$). A point transformation that satisfies $$L(Q,\dot{Q},t)=L(q,\dot{q},t)+\frac{d}{dt}\chi(q,t)$$ is a symmetry of the system. When it is continuous its infinitesimal version implies the existence of a conserved quantity (Noether theorem).
In Hamiltonian formalism, we consider a larger class of transformations, called canonical transformations (definition is summarized in this post), $Q_i=Q_i(q,p,t)$, $P_i=P_i(q,p,t)$. A generator $G(q,p,t)$ that satisfies $$ \{G,H\}+\frac{\partial G}{\partial t}=0 $$ is a symmetry generator and is conserved (Noether theorem).
Does this mean we could get a larger class of conserved charges in Hamiltonian formalism? Are there any concrete examples? I was thinking about the Runge–Lenz vector in the Kepler-type problem but not sure if it can be discussed in the Lagrangian when the change of time is allowed.
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I now think we need to make the following clear distinction.
[A] A coordinate transformation that leaves the Euler-Lagrange equation covariant is point transformation $Q_i(q,t)$. The ambiguity in the Lagrangian that does not affect EL eq takes the form $\frac{d}{dt}\chi(q,t)$.
[B] To derive a conserved quantity, we can consider wider class of transformation $Q_i(q,\dot{q},t)$. The change in the Lagrangian is allowed to take the form $\frac{d}{dt}\chi(q,\dot{q},t)$. This is not usually considered as a symmetry because the EL eq is not covariant (at least without modification).
Examples of type [B] include the Runge-Lenz vector.
