I'm studying mechanics and wondering about what are the physical and geometric meanings/interpretations of the conjugate momentum.
My study started with the Newtonian mechanics, and from what I understood it is mainly settled on a manifold, denote it with $Q$, and the coordinates are $(x,\dot x)$, which of course can be changed via change of coords.
Then with the Lagrangian mechanics $Q$ is parametrized and the coordinates are lifted/extended to the tangent bundle $TQ$, and are denoted by $(q,\dot q)$.
Then with the Hamiltonian mechanics the coordinates are lifted again, to the cotangent bundle $T^*Q$ and denoted by $(q,p)$.
The concept of conjugate momentum is firstly introduced in the Lagrangian mechanics (but it is used in the Hamiltonian mechanics too): given a Lagrangian $L(q,\dot q)$, the conjugate momentum of the coordinate $q^i$ is the function $p_i(q,\dot q) = \dfrac{\partial L(q,\dot q)}{\partial \dot q^i}$. Moreover, if $q^i$ is a cyclic coordinate, ie if $L$ does not depend on it, then $p_i$ is a first integral of the system, ie it is constant along the motions, hence the solutions starting in a given level set of $p_i$ remain on that level set.
Using these informations I'm trying to figure out:
- what is the geometric meaning of the conjugate momentum?
- what is the physical meaning of the conjugate momentum?
But I feel like I'm missing something. Moreover, since the word momentum is used, I guess the concept of conjugate momentum is related to the ones of linear momentum and angular momentum, but how?
