Linked Questions

50
votes
3answers
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 ...
15
votes
3answers
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 ...
12
votes
5answers
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 ...
9
votes
4answers
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 ...
7
votes
3answers
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 ...
8
votes
1answer
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 ...
2
votes
3answers
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 ...
2
votes
1answer
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 ...
5
votes
2answers
1k views

Energy and momentum as partial derivatives of on-shell action in field theory

According to L&L, if we fix the initial position of a particle at a given time and consider the on-shell action as a function of the final coordinates and time, $S(q_1, \ldots, q_n, t)$, then... $...
2
votes
1answer
134 views

Why is action a functional of $q$ only?

Action of a particle is written as $$S[q]=\int dt\hspace{0.2cm} L(q(t),\dot{q}(t),t).$$ How can I understand why $S$ is a functional of $q$, and not that of $\dot{q}$? Assuming $L=\frac{1}{2}m\dot{q}^...
3
votes
1answer
734 views

Momentum as derivative of on-shell action

In Landau & Lifshitz' book, I got stuck into this claim that the momentum is the derivative of the action as a function of coordinates i.e. $$ \begin{equation}p_i = \frac{\partial S}{\partial x_i}\...
2
votes
2answers
350 views

How to show that $\partial S/\partial q=p$ without variation of $S$?

I'm trying to get some understanding in treating action $S$ as a function of coordinates. Landau and Lifshitz consider $\delta S$, getting $\delta S=p\delta q$, thus concluding that $$\frac{\partial ...
2
votes
1answer
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 $t_{initial}, \ t_{final}$. ...
1
vote
1answer
124 views

Time component of momentum four-vector

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 ...
2
votes
2answers
108 views

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 ...

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