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A first integral in Lagrangian formalism is defined as a function which is constant along the solutions $(q,\dot{q})$ where $q$ are the generalized coordinates; while a first integral in Hamiltonian formalism is a function which is constant along the solutions $(q,p)$ where $p$ are the generalized momenta.

Do first integrals in Lagrangian formalism remain first integral in Hamiltonian formalism and viceversa? i.e do functions constant along solutions of Lagrange's Equations $(q,\dot{q}$ remain constant along solutions of Hamilton's Equations $(q,p)$?

At first guess I'd say yes, but I can't exactly think why; maybe it's just enough to apply the transformation rules?

$ q(q,\dot{q}) = q\\ p(q,\dot{q}) = \frac{\partial L(q,\dot{q})}{\partial\dot{q}} $

It's just a doubt I had while studying the different versions of Noether's theorem in the Lagrangian and Hamiltonian case, since we are using very different coordinates.

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  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/352393/2451 $\endgroup$ – Qmechanic Aug 19 at 12:52
  • $\begingroup$ Unfortunately it goes beyond what I studied, I'm not sure I fully understand the contents of the answers. Hopefully someone is able to explain in more elementary terms. $\endgroup$ – Roberto Gargiulo Aug 19 at 17:02

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