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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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Variational description of modified Einstein equations

Let us suppose that we have an Einstein equation of the form $$ R_{(\mu \nu)}-\frac{1}{2} g_{\mu \nu} R=8\pi T_{\mu \nu},$$ where $R$ is an affine connection, which differs from the Levi-Civita ...
ProphetX's user avatar
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Regarding Functionals [duplicate]

Given a function $L(x,y,y')$, how can we treat $(x,y,y')$ as set of independent variables? Aren't y and y' related. Moreover, y and y' are functions of x. So, isn't L just L(x) as y=y(x) and y'=y'(x)?...
Aditya Krishna Panickar's user avatar
2 votes
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71 views

Why can't we treat the Lagrangian as a function of the generalized positions and momenta and vary that? [duplicate]

Some background: In Lagrangian mechanics, to obtain the EL equations, one varies the action (I will be dropping the time dependence since I don't think it's relevant) $$S[q^i(t)] = \int dt \, L(q^i, \...
weirdmath's user avatar
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Boundary conditions in $\delta I=0$ to derive Einstein's equations -- why the derivatives of $g_{\mu\nu}$ are held constant?

Dirac derives Einstein's field equations from the action principle $\delta I=0$ where $$I=\int R\sqrt{-g} \, d^4x$$ ($R$ is the Ricci scalar). Using partial integration, he shows that $$I=\int L\sqrt{-...
Khun Chang's user avatar
3 votes
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Does anybody know this integral expression? [migrated]

I stumbled upon the following integral, which is a low dimensional case of a minimization problem I currently investigate. $$F(V) = \int_{\mathbb{R}^3}|W+\nabla\times V)|^2+|\nabla\times(W+\nabla\...
destinys father's user avatar
1 vote
1 answer
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How to prove that the Brachistochrone problem could be reduced to finding a curve on a plane?

Given two points in space, the 2D Brachistochrone problem could be solved to give solution of a cycloid. I am wondering how could one prove that in arbitrary dimensions ($d\geq 3$) with a 1D uniform ...
Rescy_'s user avatar
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7 votes
3 answers
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Something fishy with canonical momentum fixed at boundary in classical action

There's something fishy that I don't get clearly with the action principle of classical mechanics, and the endpoints that need to be fixed (boundary conditions). Please, take note that I'm not ...
Cham's user avatar
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In Lagrangian mechanics, do we need to filter out impossible solutions after solving?

The principle behind Lagrangian mechanics is that the true path is one that makes the action stationary. Of course, there are many absurd paths that are not physically realizable as paths. For ...
Cort Ammon's user avatar
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Deeper explanation for Principle of Stationary Action [duplicate]

The Principle of Stationary Action (sometimes called Principle of Least Action and other names) is successfully applied in a wide variety of fields in physics. It can often can be be used to derive ...
freecharly's user avatar
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Why the choice of Configuration Space in Hamilton's Principle is $(q, \dot{q}, t)$? [closed]

In most physics books I've read, such as Goldstein's Classical Mechanics, the explanation of Hamilton's principle took into consideration the equation (1) known as Action: $$\displaystyle I = \int_{...
Daniel's user avatar
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1 answer
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Applying Fermat's principle in Fraunhofer's diffraction

The following set up with a source, 2 convex lenses, a slit and a screen is of that of Fraunhofer's diffraction: *Correction S is not on the common optical axis, but above it. $\theta$ = Angle ...
soccerer's user avatar
3 votes
3 answers
120 views

Is there a proof that a physical system with a *stationary* action principle cannot always be modelled by a *least* action principle?

I'm aware that with Lagrangian mechanics, the path of the system is one that makes the action stationary. I've also read that its possible to find a choice of Lagrangian such that minimization is ...
Cort Ammon's user avatar
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Minimality of the action [duplicate]

How is it proved that the extremal of the action obtained with Hamilton's principle in classical mechanics or classical field theory is in fact a minimum of the action and not just a stationary point? ...
ErwinS's user avatar
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8 votes
7 answers
327 views

Which block reaches the floor first?

There are two blocks, each starting at the top of an incline. The particular inclines are depicted in the image below. The height through which the blocks fall is the same, the table lengths are the ...
Relativisticcucumber's user avatar
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What is the boundary action need for topological massive gravity (TMG)?

For pure Einstein gravity with Dirichilet boundary conditions, Gibbons-Hawking-York boundary action is needed to make the variational principle well defined. I am considering the case for topological ...
miranda li's user avatar
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1 answer
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Energy momentum tensor of a scalar field by variation of metric

For the scalar field $\phi$, $$ L = \frac{1}{2}\left(\partial_\mu \phi \partial^\mu \phi + m^2 \phi^2\right) $$ The energy momentum tensor calculated using noether's theorem is given by $$ T^{\mu \nu} ...
Ratul Thakur's user avatar
3 votes
0 answers
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Lagrangian for Kerr-Newman black holes

I am trying to write down the action that is extremized by Kerr-Newman solutions in General Relativity. Specifically, I am interested in parametrizing the Lagrangian by the mass $M$, angular momentum $...
Gloria's user avatar
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2 votes
2 answers
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Derivation of the geodesic equation. Why do we start with the special relativistic action?

I'm working on a derivation of the geodesic equation from the action functional. In special relativity, specifically for flat spacetime, we assume that the metric tensor is constant (not necessarily ...
Giovanni Brown's user avatar
6 votes
1 answer
202 views

Minimization over a function is equivalent to the problem of finding the minimum energy eigenstate in an infinite potential well?

I'm reading this paper [Eqs.(10,11)] and met the following problem. The author states that the following minimization problem $$ \underset{\tilde{g}\left( \mu \right)}{\min}\,\,\int_a^b{\left| \frac{\...
narip's user avatar
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Weinberg gravitation variational principle in free falling bodies [duplicate]

In weinberg's gravitation and cosmology in page 77 appears this I can't see why the equation and the symmetry of Christoffel symbols and equation 3.3.5 makes that equation 3.3.10 appears I ask my ...
Alberto Alejandro Blanco Rojas's user avatar
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1 answer
37 views

Identification of the variation on the boundary and why $\delta S_{\partial V}=0$

I recently asked this question about variational principles and how it all works. The essential answer I got was to go read a book on the calculus of variations, which I did, and this helped me make ...
Alex Byard's user avatar
1 vote
1 answer
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Regarding varying the coefficients of constraints in the pre-extended Hamiltonian in Dirac method

consider the following variational principle: when we vary $p$ and $q$ independently to find the equations of motion, why aren't we explicitly varying the Coeff $u$ which are clearly functions of $p$ ...
Spotless-hola's user avatar
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How to understand variational principles and the math underlying them? [duplicate]

I work in finance, and studied math in college. I'm trying to use QFT statistics to model some aspects of the market. (I've already made some progress by deriving the Black-Karasinski Hamiltonian for ...
Alex Byard's user avatar
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0 answers
48 views

Can anyone explain convergence of parallel rays on the focus of a parabolic reflector using Fermat's Principle?

Can anyone explain convergence of parallel rays on the focus of a parabolic reflector using Fermat's Principle? using optimization techniques from calculus?
Sachin Kalakoti's user avatar
1 vote
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75 views

Inconsistency in solving the Brachistochrone Problem. Did I make a mistake? [closed]

Background: Equation of Motion Okay. First I want to see if my "Newtonian Mechanics" lens of the problem is correct. Let the particle's path be given by $\vec{r}(t) = (x(t), y(t))$ and just ...
Lendel Deguia's user avatar
5 votes
3 answers
614 views

Is Principle of Least Action a first principle? [closed]

It is on the basis of Principle of Least Action, that Lagrangian mechanics is built upon, and is responsible for light travelling in a straight line. Is its the classical equivalent of Schrodinger's ...
megamonster68's user avatar
5 votes
1 answer
423 views

Dirac Lagrangian in Classical Field Theory with Grassmann numbers

The concept of the Grassmann number makes me confused. It is used to describe fermionic fields, especially path integral quantization. Also, it is used to deal with the classical field theory of ...
Jaeok Yi's user avatar
4 votes
1 answer
333 views

Doubts about "whether a given system has a Lagrangian" and "inverse problem of the calculus of variations"

There has been extensive discussion in the literature and on this forum regarding the question of "whether a given system has a Lagrangian" (e.g. post1, post2, post3, and paper1, paper2). ...
Luessiaw's user avatar
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1 answer
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Variation of action under coordinate transformations

I am currently studying General Relativity from M.P. Hobson's "General Relativity: An Introduction for Physicists" and I had difficulty in understanding some concepts in variational field ...
Ethan's user avatar
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3 votes
5 answers
301 views

How does light know the destination?

According to Fermat's principle, light travels the fastest path from dot A to dot B. I wondered how light knows which path is the fastest, and found out that light actually goes all path, but non-...
tneserp's user avatar
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1 vote
1 answer
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Total derivative of Grassmann variables

From page 21 of "Conformal Field Theory" by Di Francesco, Mathieu, and Sénéchal, the free Fermion Lagrangian is given by: $$L=\frac{i}{2}\psi_i T_{ij}\dot{\psi}_j-V(\psi)$$ Where the $\psi$ ...
QPhysl's user avatar
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2 votes
0 answers
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Ambiguity of Lagrangian density in field theory [duplicate]

In classical mechanics, we know $L(q,\dot{q},t)$ and $L(q,\dot{q},t)+\frac{d}{dt}\Lambda(q,t)$ give the same Euler-Lagrange equation $\frac{d}{dt}\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}_i}=\...
watahoo's user avatar
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1 vote
1 answer
80 views

Computing equations of motion for type IIB supergravity

I'm doing some introductory calculations on the D3-brane solution of type IIB supergravity. I'm considering the theory with action $$S_{\text{IIB}} = \frac{1}{8\pi G_{10}}\int d^{10}z\sqrt{g_{10}}\...
Geigercounter's user avatar
0 votes
0 answers
48 views

QFT by Schwartz Problem 3.1 Solution

I am having trouble while solving in the Problem 3.1 of the QFT book by Schwartz. Problem Find the generalization of the Euler-Lagrange equations for general higher-order Lagrangians of the form $\...
darkphysics's user avatar
1 vote
1 answer
200 views

Variational principle in thermodynamics

I am looking at how to derive the Thermodynamic free energy expressions, e.g. the Gibbs free energy or the Maximum work but from a variational principle. In analytical mechanics we can transform a ...
User198's user avatar
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8 votes
1 answer
421 views

Does it follow from Least Action Principle that particles do not go back in time, or do we stipulate this?

Consider the action integral, $S[\gamma] := \int L(\gamma(t),\dot{\gamma}(t),t)dt$. We can always re-write it in terms of an arbitrary curve parameter $\tau$ which need not coincide with time $t$: $$S[...
Rochelle's user avatar
0 votes
1 answer
58 views

Euler-Lagrange confusion

Consider the action $S = \int dt \sqrt{G_{ab}(q)\dot{q}^a\dot{q}^b}.$ Now for computing the Euler-Lagrange equations, we need the time derivative of $\frac{\partial L}{\partial \dot{q}^c} = \frac{1}{\...
Geigercounter's user avatar
1 vote
0 answers
51 views

Variation of functional with respect to Lagrange multiplier in QM

So, I am reading a paper on Quantum Brachistochrome and on the second page they say that they are doing a variation w.r.p. $<\phi|$, (which is a lagrange mulriplier) of the following action: $$ S(\...
Gytis Vejelis's user avatar
1 vote
0 answers
46 views

Euler-Lagrange Equation Problem with Holonomic constraint [closed]

I am trying to solve this problem; minimizing the functional $\int\sqrt{y^2+y\prime^2+z\prime^2}dx$ with the constraint $y+z=1$ (Here $y\prime=dy/dx$, $z\prime=dz/dx$, and $x$ is the only independent ...
Init's user avatar
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16 votes
6 answers
8k views

Does classical electrodynamics have a Lagrangian that gives both the Lorentz force and Maxwell equations?

There is a Lagrangian for a particle of mass $m$ and charge $q$ $$\mathcal{L}_1 = \mathcal{L}_k(m, \vec{v}) - q\phi + q\vec{v}\cdot\vec{A}$$ where $\mathcal{L}_k(m, \vec{v})$ is either $\frac{1}{2}m\...
Chad K's user avatar
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2 votes
1 answer
109 views

Optimal position of negative charge to "neutralise" a positive distribution

This question stems from trying to understand the notion of center of charge and if the analytical definition of this center depends on what exactly is minimized (the dipole moment or the total ...
Quillo's user avatar
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0 votes
1 answer
161 views

Ansatz for wavefunction infinite square well with linear perturbation $\alpha \cdot x$

Suppose we have an infinite square well extending from $0<x<L$ and a particle in its ground state. However, the infinite square well contains a linear perturbation $\alpha \cdot x$. The ...
Roos's user avatar
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0 votes
0 answers
55 views

Least action principle for Dirac monopole

I want to find the Lorentz force formula for the magnetic charge through the principle of least action. The magnetic charge is introduced by Dirac adding inside the electromagnetic tensor a tensor $G^{...
Pietro Scapolo's user avatar
1 vote
1 answer
74 views

Shortest space-like path

I consider the geometry around a massive homogeneous static spherical object (for example, a neutron star) This system is static, so I am not interested in the space-time distance between two events. ...
BoStan's user avatar
  • 13
2 votes
3 answers
144 views

Derivation of Hamiltonian by constraining $L(q, v, t)$ with $v = \dot{q}$

I am trying to reconstruct a derivation that I encountered a while ago somewhere on the internet, in order to build some intuition both for $H$ and $L$ in classical mechanics, and for the operation of ...
Sam K's user avatar
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1 vote
0 answers
76 views

How to derive that Optical path length should be stationary from Fermat's principle?

I read above eq. (5.3) in "Optics" by Eugene Hecht Fermat's Principle maintains that the optical path length OPL will be stationary; that is, its derivative with respect to the position ...
Tong Su's user avatar
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0 votes
1 answer
77 views

Is $F=-\nabla V$ a form of the least action principle? [closed]

Only for conservative systems, of course.
Reinhold Erwin Suchowitzki Tob's user avatar
2 votes
1 answer
89 views

About variational principles

Is there any way one can modify a known functional so as to get a solution for another quantity related to it? Because this is too general I would like to make it more clear with an example. Given the ...
Velerephon's user avatar
1 vote
1 answer
141 views

Explicit calculation of boundary conditions from the Gibbons-Hawking term

I am trying to reproduce the results in the paper Holographic Dual of BCFT (cf. https://arxiv.org/abs/1105.5165) in which, performing a variation of the metric of the Hilbert action with the Gibbons-...
Ignacio Garrido González's user avatar
0 votes
1 answer
41 views

Sign of variational principle $L=K-V$

As is known to all, the variational principle in Hamiltonian mechanics has Lagrangian $L=K-V$. Equilibrium requires that (1) the variation of $L$ be 0 or (2) $L$ be minimized. $K-V$ or $V-K$ doesn't ...
feynman's user avatar
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