any of several principles that find the physical trajectory of a system by minimizing or maximizing some value computed over the proposed path (for instance geometric optics can be reproduced by insisting on a minimum time principle).
15
votes
5answers
2k views
Why does calculus of variations work?
How does it make sense to vary the position and the velocity independently?
Edit:
Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics ...
22
votes
5answers
1k views
Hamilton's Principle
Hamilton's principle states that a dynamic system always follows a path such that its action integral is stationary (that is, maximum or minimum).
Why should the action integral be stationary? On ...
2
votes
1answer
206 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 ...
12
votes
6answers
2k views
Why the Principle of Least Action?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action ...
14
votes
4answers
455 views
Is the principle of least action a boundary value or initial condition problem?
Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating:
In analytic (Lagrangian) mechanics, the derivation of the ...
5
votes
4answers
726 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 ...
8
votes
1answer
1k views
Explicit Variation of Gibbons-Hawking-York Boundary Term
Are there any references that present the explicit variation of the Hilbert-Einstein action plus the Hawking-Gibbons-York boundary term, and demonstrate the cancellation of the normal derivatives of ...
2
votes
2answers
197 views
How do I show that there exists variational/action principle for a given classical system?
We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's ...
3
votes
3answers
1k views
Derivation of Maxwell's equations from field tensor lagrangian
I've started reading Peskin and Schroeder on my own time, and I'm a bit confused about how to obtain Maxwell's equations from the (source-free) lagrangian density $L = ...
4
votes
1answer
448 views
Lagrangian for Relativistic Dust derivation questions
In most GR textbooks, one derives the stress energy tensor for relativistic dust:
$$
T_{\mu\nu} = \rho v_\mu v_\nu
$$
And then one puts this on the right hand side of the Einstein's equations. I ...
8
votes
2answers
76 views
More general invariance of the action functional
I will formulate my question in the classical case, where things are simplest.
Usually when one discusses a continuous symmetry of a theory, one means a one-parameter group of diffeomorphisms of the ...
4
votes
1answer
86 views
What is the action for an electromagnetic field if including magnetic charge
Recently, I try to write an action of an electromagnetic field with magnetic charge and quantize it. But it seems not as easy as it seems to be. Does anyone know anything or think of anything like ...
6
votes
4answers
244 views
Least-action classical electrodynamics without potentials
Is it possible to formulate classical electrodynamics (in the sense of deriving Maxwell's equations) from a least-action principle, without the use of potentials? That is, is there a lagrangian which ...
3
votes
2answers
273 views
What are some interesting calculus of variation problems? [closed]
That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks (catenary, brachistochrone, Fermat, etc..)
3
votes
3answers
364 views
Type of stationary point in Hamilton's principle
In this question it is discussed why by Hamilton's principle the action integral must be stationary. Most examples deal with the case that the action integral is minimal: this makes sense - we all ...
1
vote
2answers
226 views
Can cos(x) or sin(x) be the function of stationary action?
Is there a way to express $\cos(x(t))$ (or $\sin(x(t))$) as the solution to the Euler-Lagrange equation, in other words is there a sense in which this function is the path of stationary action?
4
votes
3answers
393 views
Understanding boundary conditions on slices of AdS5
This is a thing Iïve seen on many papers dealing with Warped Extra Dimensions, specifically on slices of AdS5. But the one where it appears more clearly is a lecture by Tony Gherghetta:
...
2
votes
1answer
157 views
How light know which path is smallest?
We know from fermat's principle that light follows the smallest path. But how light know that which path is smallest?
2
votes
1answer
320 views
Snell's Law of Refraction
I was told that "Snell's law of refraction implies that a light ray in an isotropic medium travels from point a to point b in stationary time." Why is this true?
Thanks.
2
votes
2answers
202 views
Must the action for a relativistic charge be a Lorentz scalar?
Page 580, Chapter 12 in Jackson's 3rd edition text carries the statement:
From the first postulate of special relativity the action integral must be a Lorentz scalar because the equations of ...
2
votes
1answer
484 views
Does Action in Classical Mechanics have a Interpretation? [duplicate]
Possible Duplicate:
Hamilton's Principle
The Lagrangian formulation of Classical Mechanics seem to suggest strongly that "action" is more than a mathematical trick. I suspect strongly ...
1
vote
1answer
51 views
Why vary the action with respect to the inverse metric?
Whenever I have read texts which employ actions that contain metric tensors, such as the Nambu-Goto, Polyakov or Einstein-Hilbert action, the equations of motion are derived by varying with respect to ...
