# Questions tagged [variational-principle]

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).

765 questions
Filter by
Sorted by
Tagged with
55 views

### Why do the eigenvalues minimize a variational problem?

Could anyone recommend a source where they prove or explain the following claim at an undergraduate level? "More generally, it follows immediately from the properties of Hermitian eigensystems ...
30 views

### Why must the action be minimized? [duplicate]

In mechanics, the only physical route a particle can take is the one where action is minimized. Why is this true? Is there a proof?
38 views

20 views

### Does a free particle always follow the trajectory of shortest distance? [duplicate]

Some context for the above question is warranted. While reading Hartle's Gravity, the following statements made recurring appearances: "Gravity is not a force, it is the geometry of 4D spacetime. ...
42 views

123 views

### About variational methods, renormalization and $a$, $c$-theorems

Variational approximation Variational methods are an important technique, frequently applied for the approximation of complicated probability distributions, with the applications in statistical ...
145 views

### Calculus of variations: meaning of infinitesimal variation $\delta$ and action minimum

So I am studying classical mechanics through the MIT 8.223 notes, and encountered the derivation of the Euler Lagrange equation. There is a part I don't quite understand, which resides in the actual ...
32 views

### What happens when the same action extremal value can be obtained in more than one path in the configuration space?

I'm trying to understand the logic underneath the concept of action and lagrangian. I know this kind of questions have been asked many times, but I was unable to find an answer to this one. I've ever ...
4k views

### How does Fermat's principle make light choose a straight path over a short path?

This is a thought experiment where I have made a "C" shaped hole inside diamond. The refractive index $(\mu)$ of diamond is 2.45. Say we shine a laser from top of the "C" as shown. ...
47 views

22 views

### Fermat's Principle of Least Time and Non-Smooth Bends in Light Paths

Consider a light ray eminiating from a point within a square medium where it has an infintesimally slow velocity. My understanding using Fermat's principle of least time is that if I was to observe ...
59 views

31 views

### Fields that lend themselves to variational principles? [duplicate]

In physics, we often describe the dynamic properties of fields using variational principles like defining an action or a Lagrangian. A field however is simply some function of space $\phi(x)$ so I ...
A free particle Lagrangian in a 3D curvilinear coordinate system can be written as an inner product with the metric $g$: $$L = \frac{1}{2}m\sum_{i,j=1}^3v^ig_{ij}v^j.$$ This equation was taught to ...