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.
