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# 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).

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### Energy-Momentum tensor of Polyakov action vanishes

In the lecture notes of David Tong on String Theory he defines the energy momentum tensor of the polyakov action as \begin{align*} T_{\alpha\beta}=-\frac{2}{T}\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta ...
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### Hamilton's Principle extended to deal with non-conservative forces [closed]

I have seen both authors stating that there is an extension of Hamilton's Principle to systems with non-conservative forces, and others stating that there is no variational principle available for ...
1 vote
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### Deriving Hamilton's Principle from Lagrange's Equations

I'm trying to derive Hamilton's Principle from Lagrange's Equations, as I've heard they're logically equivalent statements, and am stuck on a final step. For simplicity, assume we're dealing with a ...
1 vote
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### Why is the time taken for light propagation between two points in anisotropic media independent of $y$?

Background Light propagating in an anisotropic medium does not (in general) take a straight-line path between two points. The propagation time between those points, then, is dependent on the total ...
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### Numerically approximate the ground state wave function of the finite potential well problem [closed]

I am trying to numerically approximate the ground state wave function of an embedded square well, but when I plot the wave function, I get an unreasonable result. Problem Setup Here is the set up of ...
1 vote
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### How can I derive the equations of motion with the least action principle from the action of $p$-Form Electrodynamics? [closed]

I know this is the correct formula for the action for a arbitrary $p$. I know how to obtain the equations of motion for $p=1$, but I struggle to find a way to do this with an arbitrary $p$. I also ...
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### 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 Euler-...
1 vote
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### Directly solving Hamilton's principle as an initial-value problem? [duplicate]

Can Hamilton's principle, i.e. the principle of stationary action, be posed as an initial-value problem instead of the usual boundary-value problem and still produce the correct equations of motion? ...
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### The einbein in the action of a relativistic massive point particles [closed]

The action of a relativistic massive point particle moving in space-time is S=-m\int d\tau \sqrt{g _{\nu \rho}\frac{dx^{\nu}}{d\tau}\frac{dx^{\rho}}{d\tau}}\tag{1} \label{eq1} \end{...
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### Fermat's principle and a non-physical conclusion

Fermat's Principle is the statement that a ray will follow a minimum-time path between a point, A, to a point, B. So, if I have a block of material of high refractive index, so that it slows the light ...
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### How to distinguish a trivial gauge transformation from a non-trivial one?

Two days ago I posted a post that discusses a very generic gauge transformation. I repeat it here. Suppose we have an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. We ...
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### Question about Trivial Gauge Transformation

Suppose we have an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. We denote the variation of $S$ wrt to a given field, say $a$, i.e. $\frac{\delta S}{\delta a}$, by $E_a$....
1 vote
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### Virtual Work - Is the presentation in Cornelius Lanczos wrong?

Book: The Variational Principles of Mechanics by Cornelius Lanczos Edition: 4th Chapter: 3, The Principle of Virtual Work I am on the second page of the 3rd chapter (pg 75; it has the Eqn. 31.1). ...
The addition of $$\mathcal{L}' = \epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma} \propto \vec{E}\cdot\vec{B}$$ to the electromagnetic Lagrangian density leaves Maxwell's equations unchanged (shown ...