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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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### Why are there only derivatives to the first order in the Lagrangian?

Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, jerk,...) excluded? Is there a good reason for ...
16k views

### Calculus of variations -- how does it make sense to vary the position and the velocity independently?

In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat ...
37k 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 ...
19k views

### Why should an action integral be stationary? On what basis did Hamilton state this 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 ...
53k 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 $L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$... 4k 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 Euler-...
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. ...
7k views

### Entropy and the principle of least action

Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other ?
4k views

### Can Lagrangian be thought of as a metric?

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
3k views

### Do "typical" QFT's lack a lagrangian description?

Sometimes as a result of learning new things you realize that you are incredibly confused about something you thought you understood very well, and that perhaps your intuition needs to be revised. ...
5k 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 ...
2k views

### Is there some connection between the Virial theorem and a least action principle?

Both involve some 'averaging' over energies (kinetic and potential) and make some prediction about their mean values. As far as the least action principles, one could think of them as saying that the ...
14k views

### Noether's current expression in Peskin and Schroeder

In the second chapter of Peskin and Schroeder, An Introduction to Quantum Field Theory, it is said that the action is invariant if the Lagrangian density changes by a four-divergence. But if we ...
3k views

### Why does the curve of a hanging chain not minimize the area below it?

If we have a chain of fixed length hanging from two points we know that it will form a curve that minimizes the chain's potential energy. If we imagine the chain as having many small segments, then ...
4k views

### Confusion regarding the principle of least action in Landau & Lifshitz "The Classical Theory of Fields"

Edit: The previous title didn't really ask the same thing as the question (sorry about that), so I've changed it. To clarify, I understand that the action isn't always a minimum. My questions are in ...
2k views

### Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt}$

I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are ...
3k views

### Is Fermat's principle only an approximation?

Fermat's principle says that the path taken between two points by a ray of light is the path that can be traversed in the least time. It occurred to me today that maybe the path is actually the one ...
8k 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 ...
10k views

### How does light know which path is fastest?

We know from Fermat's principle of least time that light follows the fastest path. But how does light know which path is the fastest?
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### On a trick to derive the Noether current

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$. The trick to get the Noether current consists in making the ...
6k views

### Variational Derivation of Schrodinger Equation

In reading Weinstock's Calculus of Variations, on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles. Unfortunately I don't ...
2k views

### Shape of a rotating rope with one free-end [closed]

One end of a uniform rope (with total mass $M$) is fixed on the edge of a cylinder. The cylinder has a radius $R$ and rotates with angular velocity $\omega$. The axis is vertical in a gravitational ...
It is well known that geodesics on some manifold $M$, covered by some coordinates ${x_\mu}$, say with a Riemannian metric can be obtained by an action principle . Let $C$ be curve $\mathbb{R} \to M$, $... 20 votes 1 answer 3k views ### Why is it so coincident that Palatini variation of Einstein-Hilbert action will obtain an equation that connection is Levi-Civita connection? There are two ways to do the variation of Einstein-Hilbert action. First one is Einstein formalism which takes only metric independent. After variation of action, we get the Einstein field equation. ... 20 votes 1 answer 1k views ### What makes a Lagrangian a Lagrangian? I just wanted to know what the characteristic property of a Lagrangian is? How do you see without referring to Newtonian Mechanics that it has to be$L=T-V? People constructed a Lagrangian in ... 19 votes 3 answers 10k views ### How to derive Maxwell's equations from the electromagnetic Lagrangian? In Heaviside-Lorentz units the Maxwell's equations are: $$\nabla \cdot \vec{E} = \rho$$ $$\nabla \times \vec{B} - \frac{\partial \vec{E}}{\partial t} = \vec{J}$$ $$\nabla \times \vec{E} + \frac{\... 19 votes 4 answers 4k views ### Derivation of the Polyakov Action As is usually done when first presenting string theory, the Nambu-Goto Action,$$ S_{\text{NG}}:=-T\int d\tau d\sigma \sqrt{-g} (g:=\det (g_{\alpha \beta}) is the induced metric on the world-... 19 votes 2 answers 5k views ### Functional derivative in Lagrangian field theory The following functional derivative holds: \begin{align} \frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t') \end{align} and \begin{align} \frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t') \end{... 18 votes 3 answers 18k views ### Derivation of Euler-Lagrange equations for Lagrangian with dependence on second order derivatives Suppose we have a Lagrangian that depends on second-order derivatives:L = L(q, \dot{q}, \ddot{q},t).\tag{1}$$If we're working on the variational problem for this Lagrangian, then I know that we'... 18 votes 7 answers 2k views ### When/why does the principle of least action plus boundary conditions not uniquely specify a path? A few months ago I was telling high school students about Fermat's principle. You can use it to show that light reflects off a surface at equal angles. To set it up, you put in boundary conditions, ... 18 votes 2 answers 961 views ### Is it possible to prove that planets should be approximately spherical using the calculus of variations? Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical? Let's assume that a planet is 'born' when lots of ... 18 votes 4 answers 3k views ### D'Alembert's Principle: Necessity of virtual displacements Why is the d'Alembert's Principle$$\sum_{i} ( {F}_{i} - m_i \bf{a}_i )\cdot \delta \bf r_i = 0$$stated in terms of "virtual" displacements instead of actual displacements? Why is it so necessary ... 18 votes 2 answers 338 views ### What's the lowest nuclear charge Z < 1 that will support a bound two-electron ion (Z,2e^-)? In my programming project I calculate the minimal energy of an atom with 2 electrons in the L=0, S=0 state, using a Hylleraas wave function. The values I find for Z=2 (He) and Z=1 (H^-) are ... 17 votes 1 answer 7k views ### How do non-conservative forces affect Lagrange equations? If we have a system and we know all the degrees of freedom, we can find the Lagrangian of the dynamical system. What happens if we apply some non-conservative forces in the system? I mean how to deal ... 17 votes 2 answers 2k views ### Another Solution To Brachistochrone Problem Recalling the statement of the problem : Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the ... 15 votes 3 answers 4k views ### When is the principle of stationary action not the principle of least action? I've only had a very brief introduction to Lagrangian mechanics. In a physics course I took last year, we briefly covered the principle of stationary action --- we looked at it, derived some equations ... 15 votes 2 answers 6k views ### Geodesics equations via variational principle I would like to recover the (timelike) geodesics equations via the variational principle of the following action:$$ \mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}} $$... 15 votes 4 answers 3k views ### Least-action classical electrodynamics without potentials Is it possible to formulate classical electrodynamics (in the sense of deriving Maxwell's equations) from a least-action principle, without the use of potentials? That is, is there a lagrangian which ... 15 votes 4 answers 7k views ### Lagrangian for relativistic massless point particle For relativistic massive particle, the action is$$\begin{align}S ~=~& -m_0 \int ds \cr ~=~& -m_0 \int d\lambda ~\sqrt{ g_{\mu\nu} \dot{x}^{\mu}\dot{x}^{\nu}} \cr ~=~& \int d\lambda \ L,\... 15 votes 3 answers 2k views ### Is Einstein-Hilbert action the unique action whose variation gives Einstein's field equations? I know that scaling the action with a non-zero multiplicative constant, or adding a total divergence term to the Lagrangian density do not change the Euler-Lagrange equations, cf. e.g. this Phys.... 15 votes 1 answer 1k views ### Physical Interpretation of EM Field Lagrangian Using differential forms and their picture interpretations, I wonder if it's possible to give a nice geometric & physical motivation for the form of the Electromagnetic Lagrangian density? The ... 14 votes 6 answers 2k views ### Does universal speed limit of information contradict the ability of a particle to pick a trajectory using Principle of Least Action? I'm doing some self reading on Lagrangian Mechanics and Special Relavivity. The following are two statements that seem to be taken as absolute fundamentals and yet I'm unable to reconcile one with the ... 14 votes 6 answers 2k views ### Trouble understanding Caroll's explanation on why geodesics maximize proper time I'm reading Caroll's Lectures on GR 2 on pages 71-72, he states: Let’s now explain the earlier remark that timelike geodesics are maxima of the proper time. The reason we know this is true is ... 14 votes 5 answers 3k views ### What is the best path for a given initial and final state? I am trying to calculate an efficient acceleration curve given starting and final positions and velocities. I'm assuming no friction, and that the acceleration can be applied in any direction at any ... 14 votes 4 answers 3k views ### What is the actual form of Noether current in field theory? Let us considerN$independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by$\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a region$\Omega$in a$D$-... 14 votes 5 answers 6k views ### Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)? All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before. Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ... 14 votes 3 answers 824 views ### Why boundary terms make the variational principle ill-defined? Let me start with the definitions I'm used to. Let$I[\Phi^i]$be the action for some collection of fields. A variation of the fields about the field configuration$\Phi^i_0(x)$is a one-parameter ... 14 votes 0 answers 221 views ### About variational methods, renormalization and$a$,$c$-theorems Variational approximation Variational methods are an important technique, frequently applied for the approximation of complicated probability distributions, with the applications in statistical ... 13 votes 3 answers 2k views ### Optics: Derivation of$\vec\nabla{n} = \frac{d(n\hat{u})}{ds}$I have been given this formula from optics here, with no background: $$\vec\nabla{n} = \frac{d(n\hat{u})}{ds}$$ Where$n$is the refractive index and$\hat{u}$is a unit vector tangent to the path$...
TL;DR: Why can't we write $\mathcal{L} = E - 2V$ where $E = T + V =$ Total Energy? Let us consider the case of a particle in a gravitational field starting from rest. Initially, Kinetic energy $T$ is ...