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There are 2 ideas of symmetry I have seen in Classical Mechanics:

  1. Noetherian symmetry: Here they discuss infinitesimal point transformations where only position coordinates (and their derivatives ) get perturbed. They define symmetry to be those infinitesimal transformations where Lagrangian is not changed. Noether's theorem then relates these symmetric transformations to a conserved quantity $Q = \sum_i p_i f_i(q)$

  2. Hamiltonian symmetry: Here they discuss the idea infinitesimal canonical transformations (ICT), which are actually in (q,p) phase space. It is then shown that the ICT generated by $G$ is a symmetry of Hamiltonian iff $G$ is conserved in time as system evolves by $H$.

I wanted to know if these 2 are equivalent in some sense or not. There are some obvious differences such as the first one is in $q$ space and second in $(q,p)$ space and ICT will allow $G$ (and consequently the transformations) to be dependent on time. But we see that in both cases a translation symmetry leads to a conserved linear momentum and a rotational symmetry leads to a conserved angular momentum.

My query is that is there any formal result which combines the 2 ideas of symmetry?

Specifically is one of these symmetries a subset of the other?

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    $\begingroup$ I'm pretty sure both phenomena are considered cases of Noether's theorem. You might enjoy this paper $\endgroup$
    – rschwieb
    Nov 5, 2020 at 19:17

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The trick is to consider the Hamiltonian version of Noether theorem, see e.g. this Phys.SE. Then the Noether symmetries correspond to Hamiltonian vector fields (HVF), or infinitesimal symplectomorphisms, which in turn correspond to infinitesimal canonical transformations (ICT).

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