The momentum $m v$ of a particle is formally the same as the derivative its translational kinetic energy $\frac{1}{2} m v^2$ with respect to $v$.
Similarly the angular momentum $I \omega$ is the derivative of its rotational energy $\frac{1}{2}I \omega^2$ with respect to $\omega$.
Does this relation has any physical interpretation?
Indeed, in Hamiltonian Formalism this is the very definition of momentum. For example, for a free particle with one generalised coordinate $q$ and $\dot{q}\equiv v$ the Lagrangian is
$$ L=\frac{1}{2}mv^2 $$
and the momentum is (by definition) given by
$$ p =\frac{\partial L}{\partial v}$$
Check any book in classical mechanics for Lagrangian and Hamiltonian formalism for more details in the subject.
