18 questions linked to/from What variables does the action $S$ depend on?
6k views

### What's the interpretation of Feynman's picture proof of Noether's Theorem?

On pp 103 - 105 of The Character of Physical Law, Feynman draws this diagram to demonstrate that invariance under spatial translation leads to conservation of momentum: To paraphrase Feynman's ...
5k views

### Hamilton-Jacobi Equation

In the Hamilton-Jacobi equation, we take the partial time derivative of the action. But the action comes from integrating the Lagrangian over time, so time seems to just be a dummy variable here and ...
5k views

### Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?

All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before. Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
3k views

### Is the Lagrangian density a functional or a function?

Weinberg at page 300 of The Quantum Theory of Fields - Volume I says: $L$ itself should be a space integral of an ordinary scalar function of $\Psi(x)$ and $\partial \Psi(x)/\partial x^\mu \,$, known ...
3k views

### The number of independent variables in the Lagrangian and Hamiltonian methods in Classical Mechanics

It's told in Landau - Classical Mechanics, that in the Hamiltonian method, generalized coordinates $q_j$ and generalized momenta $p_j$ are independent variables of a mechanical system. Anyway, in the ...
2k views

### Can we find the boundary conditions of fields from the stationary action principle?

First principle of stationary action Consider a real Klein-Gordon scalar field $\phi$ living in a $D$ dimensional flat spacetime. The field is considered off shell (the on shell condition is defined ...
1k views

### Poincare invariant Lagrangians

The Lagrangian density of a Poincare invariant theory should not depend explicitly on the space-time coordinates. Does this mean $$\partial_\mu \mathcal{L}=0~?$$ If this is the case doesn't the ...
453 views

### On Landau-Lifshitz's derivation of four-momentum

I'm studying the ninth section of The Classical Theory of Fields by Landau & Lifshitz, where they introduce four-momentum through the principle of least action. I can understand the derivation ...
1k views

734 views

199 views

### Conceptual problem with action considered as function of endpoints

I am having some trouble with understanding why it makes sense to consider action in classical mechanics as function of endpoints $q_{initial}, \ q_{final}$ and endtimes $t_{initial}, \ t_{final}$. ...
In Landau-Lifshitz, Classical theory of fields (second chapter), the four-momentum is defined by the equation $$-\frac{\partial S}{\partial x^i}=p_i\tag{9.12},$$ where $S$ is the action integral. The ...
### Derivation of $\partial S / \partial t = -H$ for non-classical trajectories
In classical mechanics, one can show that $$\frac{\partial S}{\partial t} = -H,\tag{1}$$ where $$S=\int_0^t L(q, \dot{q}, t')dt'\tag{2}$$ is the action associated with a trajectory and $H$ is the ...