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?

  • $\begingroup$ What do you mean by "geometric meaning" or "physical meaning" here? $\endgroup$ – Aaron Stevens Feb 12 at 22:43
  • $\begingroup$ @AaronStevens What does conjugate momentum represent from a geometric pov and from a physical pov? $\endgroup$ – sound wave Feb 13 at 7:50
  • $\begingroup$ What do you mean by "geometric pov" or "physical pov" here? $\endgroup$ – Aaron Stevens Feb 13 at 12:23
  • $\begingroup$ @AaronStevens pov = point of view, for example if we look at the system from a geometric perspective, what does the conj momentum represent? Same question but from a physical perspective $\endgroup$ – sound wave Feb 13 at 14:08
  • $\begingroup$ I know what pov means. You just keep saying the same unclear questions with different words. Can you please be more specific with what you are looking for? Maybe an example system you are interested in could help frame the question. $\endgroup$ – Aaron Stevens Feb 13 at 14:40