1
vote
1answer
38 views

The exact definition of conjugate momentum density

After checking various websites, I've seen the conjugate momentum density defined as either: \begin{align*} P_r ~=~ \frac{\partial \mathcal{L}}{\partial \dot{A}_r} \end{align*} or \begin{align*} P_r ...
5
votes
1answer
175 views

Why isn't $F = \frac{\partial \mathcal{L}}{\partial q}$?

If momentum is, $$p = \frac{\partial \mathcal{L}}{\partial \dot{q}}$$ and force is, $$ F = \frac{dp}{dt}$$ and by Euler-Langrange equations, $$ \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial ...
3
votes
1answer
96 views

Hamilton-Jacobi formalism and on-shell actions

My question is essentially how to extract the canonical momentum out of an on-shell action. The Hamilton-Jacobi formalism tells us that Hamilton's principal function is the on-shell action, which ...
3
votes
0answers
76 views

Lagrangian with vanishing conjugate momentum, independent variables

Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes: ...
5
votes
3answers
1k views

What is canonical momentum?

What does the canonical momentum $\textbf{p}=m\textbf{v}+e\textbf{A}$ mean? Is it just momentum accounting for electromagnetic effects?
0
votes
1answer
123 views

Non-relativistic Kepler orbits

Consider the Newtonian gravitational potential at a distance of Sun: $$\varphi \left ( r \right )~=~-\frac{GM}{r}.$$ I write the classical Lagrangian in spherical coordinates for a planet with mass ...
1
vote
1answer
143 views

Cyclic co-ordinates implying the constant velocity motion of center of mass of a system of particles

I'm reading the section on Central Force in my textbook (Goldstein's Classical Mechanics has a similar argument in the chapter titled "The Central Force Problem", first section), where we have the ...
1
vote
1answer
439 views

Relation between between linear momentum and translational kinetic energy

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 ...