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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Lagrangian Formulation in Non-Conservative Systems
I am working in a non-conservative system. Would it make a difference if I
Formulate the Lagrange Equation with an additional term on the right hand side of the equation to account for the Rayleigh ...
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Deriving electrostatics problems using variational methods
I was thinking, as a new method, we can use the variational method to find σ (charge density) on the surface of a metal if total charge Q in put on it. My idea was that, the total electrostatic energy,...
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Variation of Polyakov's action (Polchinski)
In Polchinski's book, he states the following equation (which is equation (1.2.27) in the book)
$$\delta S_P = \frac{1}{2\pi\alpha'}\int_{-\infty}^\infty d\tau\int_{0}^\ell d\sigma (-\gamma)^{1/2}\...
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Deriving Euler-Lagrange Equation [closed]
The author is demonstrating how you might derive the Euler-Lagrange equation by minimising the action at a certain point. He substitutes a point $x_8$ into the lagrangian and then differentiates the ...
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Identifying Lagrangian as the solution to the variational problem from the Hamilton's principle
We understand that we derive Lagrange equations,
$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_j}\right)-\frac{\partial L}{\partial q_j}=0, \tag{1}$$
starting from d'Alembert's principle of ...
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Does nature maximize or minimize spacetime curvature?
This question is based on the Einstein-Hilbert action, which states $S=\frac{1}{2\kappa}\int_{ }^{ }R\sqrt{-g}d^{4}x$, where $R$, the Ricci Scalar, is some measure of the spacetime curvature.
The ...
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One electron orbiting two protons ($H_{2}^{+}$)
When there's one electron attracted by two protons, (in a two dimensional plane), the Hamiltonian of the electron is $$H=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial \textbf{r}^{2}}-\frac{e^{2}}{|...
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How does $\dot{q}_i p_i - H = \dot{Q}_i P_i - K + \frac{d}{dt}F$ will give the same EL and EoM for corresponding coords? [duplicate]
How does $\dot{q}_i p_i - H = \dot{Q}_i P_i - K + \frac{d}{dt}F$ give the same Euler-Lagrange equations and Equations of motion (EoM) for corresponding coordinates and allow us to determine a ...
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Euler-Lagrange equations for fields
I am following the discussion presented on Hobson's book "General Relativity: An Introduction for Physicists" where he deduct the Euler-Lagrange equation for fields and I am stuck in a ...
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Lagrangian of a multi-dimensional scalar field
We know that the Lagrangian has to be a scalar. Would it be possible if this scalar is multi-dimensional (for example $m\times m$)? Let's say a field $\phi$ is represented with an $m\times m$ matrix ...
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Are there any experiments that examine Hamilton's Principle directly?
Or can it be examined?
I 'd glad if you can share some ideas about "principles" in general.
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Does quasi-symmetry preserve the solution of the equation of motion?
In some field theory textbooks, such as the CFT Yellow Book (P40), the authors claim that a theory has a certain symmetry, which means that the action of the theory does not change under the symmetry ...
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Independence of the equations resulting from the action principle $\delta (I_{\text{gravity}} + I_{\text{other fields}}) = 0$
In Dirac's "GTR", Chap. $30$, he discusses the "comprehensive action principle" and shows that variation of the combined action of the Hilbert-Einstein action plus all other matter-...
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Relating Brachistochrone problem to Fermat's principle of least time [closed]
When I came across the Brachistochrone problem, my teacher said we could relate it to Fermat's principle of least time.
So, we could make many glass slabs of high $\mathrm dx$, and every slab has a ...
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Does Hamilton's principle allow a path to have both a process of time forward evolution and a process of time backward evolution?
This is from Analytical Mechanics by Louis Hand et al. The proof is about Maupertuis' principle. The author seems to say that Hamilton's principle allow a path to have both a process of time forward ...
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What does the optical Hamiltonian mean?
So I was trying to demonstrate Snell's law with Hamilton's equations, and when I got the Hamiltonian:
$$H = -\sqrt{n^2-p_{1}^2-p_{2}^2}.$$
I had a question about what this Hamiltonian indicates. I ...
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Derivation of Noether Current in Condensed Matter Field Theory by Altland and Simons
In Section 1.6 of Condensed Matter Field Theory by Altland and Simons, they prove Noether's theorem. In order to do so, they consider an infinitesimal transformation of the coordinates and the field:
$...
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How to get a lower bound of the ground state energy?
The variational principle gives an upper bound of the ground state energy. Thus it is quite easy to get an upper bound for the ground state energy. Every variational wave function will provide one.
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Can Hartree-Fock determinant WLOG taken to be real?
For a many-electron Hamiltonian $H$, a Hartree-Fock determinant is a Slater determinant $\Psi$ that minimizes the energy $\frac{\langle\Psi,H\Psi\rangle}{\langle\Psi,\Psi\rangle}$. In general, $\Psi$ ...
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Proving (not inferring) Dirac Eq. 27.4 in "GTR" involving variation of the matter current
Dirac infers (but does not derive) Eq. 27.4 in his GTR text. Here $p^\mu$ is the matter current (momentum) four vector, and we have the conservation law $\partial_\mu p^\mu = 0$. Dirac varies the ...
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Boundary terms in the gravitational action
A correct derivation of Einstein's field equations from an action principle $\delta I=0$ is given in
Poisson, E. (2004). A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge ...
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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 ...
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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) $$S[q^i(t)] = \int dt \, L(q^i, \...
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Boundary conditions in $\delta I=0$ to derive Einstein's equations -- why the derivatives of $g_{\mu\nu}$ are held constant?
Dirac derives Einstein's field equations from the action principle $\delta I=0$ where $$I=\int R\sqrt{-g} \, d^4x$$ ($R$ is the Ricci scalar). Using partial integration, he shows that $$I=\int L\sqrt{-...
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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 ...
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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 ...
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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 ...
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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 ...
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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:
$$\displaystyle I = \int_{...
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Applying Fermat's principle in Fraunhofer's diffraction
The following set up with a source, 2 convex lenses, a slit and a screen is of that of Fraunhofer's diffraction:
*Correction S is not on the common optical axis, but above it.
$\theta$ = Angle ...
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Is there a proof that a physical system with a *stationary* action principle cannot always be modelled by a *least* action principle?
I'm aware that with Lagrangian mechanics, the path of the system is one that makes the action stationary. I've also read that its possible to find a choice of Lagrangian such that minimization is ...
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Minimality of the action [duplicate]
How is it proved that the extremal of the action obtained with Hamilton's principle in classical mechanics or classical field theory is in fact a minimum of the action and not just a stationary point? ...
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Which block reaches the floor first?
There are two blocks, each starting at the top of an incline. The particular inclines are depicted in the image below.
The height through which the blocks fall is the same, the table lengths are the ...
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What is the boundary action need for topological massive gravity (TMG)?
For pure Einstein gravity with Dirichilet boundary conditions, Gibbons-Hawking-York boundary action is needed to make the variational principle well defined. I am considering the case for topological ...
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Energy momentum tensor of a scalar field by variation of metric
For the scalar field $\phi$,
$$ 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} ...
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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 $...
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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 ...
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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{\...
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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 ...
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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 ...
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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$ ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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).
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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 ...
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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-...
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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 $\psi$ ...
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