Linked Questions

53 votes
7 answers
11k views

Can Noether's theorem be understood intuitively?

Noether's theorem is one of those surprisingly clear results of mathematical calculations, for which I am inclined to think that some kind of intuitive understanding should or must be possible. ...
Gerard's user avatar
  • 6,841
21 votes
4 answers
6k views

Gelfand-Yaglom theorem for functional determinants

What is the 'Gelfand-Yaglom' Theorem? I have heard that it is used to calculate Functional determinants by solving an initial value problem of the form $Hy(x)-zy(x)=0$ with $y(0)=0$ and $y'(0)=1$. ...
Jose Javier Garcia's user avatar
20 votes
3 answers
8k views

The Planck constant $\hbar$, the angular momentum, and the action

Is there anything interesting to say about the fact that the Planck constant $\hbar$, the angular momentum, and the action have the same units or is it a pure coincidence?
Isaac's user avatar
  • 2,890
7 votes
4 answers
6k views

Intuitive explanation for why time symmetry implies conservation of energy?

According to Noether's Theorem, every physical symmetry leads to a conservation law. For example, time-translation symmetry (the laws of physics don't change over time) implies conservation of energy,...
BlueRaja - Danny Pflughoeft's user avatar
9 votes
2 answers
3k views

Variation of Action with time coordinate variations

I was trying to derive equation (65) in the review by László B. Szabados in Living Reviews in Relativity (2002, Article 4) This slightly unusual then usual classical mechanics because it includes a ...
user50482's user avatar
16 votes
1 answer
3k views

Invariance of action $\Rightarrow$ covariance of field equations?

Invariance of action $\Rightarrow$ covariance of field equations? Is this statement true? I have only seen examples of this, like the invariance of Electromagnetic action under Lorentz ...
SuperCiocia's user avatar
  • 24.9k
9 votes
1 answer
9k views

Hamilton's characteristic and principal functions and separability

Just hoping for some clarity regarding Hamilton's characteristic function $W$. When we take a time independent Hamiltonian we can separate the Principal function $S$ up into the characteristic ...
AngusTheMan's user avatar
  • 2,441
7 votes
1 answer
2k views

What variables does the action $S$ depend on?

Action is defined as, $$S ~=~ \int L(q, q', t) dt,$$ but my question is what variables does $S$ depend on? Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \frac{dq}{dt}$? In Wikipedia I've ...
user5198's user avatar
  • 173
8 votes
4 answers
623 views

Connection between Noether's Theorem and classical definitions of energy / momentum

In classical mechanics, change in momentum $\Delta \mathbf p$ and change in kinetic energy $\Delta T$ of a particle are defined as follows in terms of the net force acting on the particle $\mathbf F_\...
Trevor Kafka's user avatar
  • 1,826
6 votes
1 answer
1k 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}\...
renyhp's user avatar
  • 430
3 votes
1 answer
647 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 ...
yellon's user avatar
  • 660
5 votes
2 answers
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... $...
Brian Bi's user avatar
  • 6,591
7 votes
1 answer
1k views

Intuition for Hamilton-Jacobi equation derived from least action

I am trying to understand the Hamilton-Jacobi equation without the framework of the canonical transformations. Even on the case of a 1D free particle I'm getting stuck. The system starts at fixed ...
Alex's user avatar
  • 866
2 votes
2 answers
547 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 ...
Ruslan's user avatar
  • 29k
3 votes
1 answer
664 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 ...
physguy's user avatar
  • 649

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