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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Is there a minimization principle for Hamiltonian? [duplicate]

Consider a point particle in $n$ dimensions. For a Lagrangian $\mathcal L(\mathbf{q, \dot q}, t)$, we have that $\mathbf q(t)$ is a feasible trajectory for times $t_0<t<t_f$ iff it extremizes ...
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How is the relationship between the old and the new canonical variables justified?

In Classical Hamiltonian Mechanics, a canonical transformation of the phase-space coordinates $(p,q,t) \to (P,Q,t)$ is such that the general form of Hamilton's equations is followed and Hamilton's ...
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Hamilton's principle for fields

According to Goldstein, Hamilton's principle can be summerized as follows: The motion of the system from time $t_{1}$ to time $t_{2}$ is such that the line integral (called the action or the action ...
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How to approach stationarity in Hamilton mechanics?

The analogue of the action in Hamiltonian mechanics is $$ S [ q, p] =\int_{t_1}^{t_2} [p_\alpha (t^\prime) \dot{q}_\alpha (t^\prime) - H (q_\alpha (t^\prime),p_\alpha (t^\prime),t^\prime)]d t^\prime. $...
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Calculating the streamline that minimizes velocity shear and directional divergence

For an arbitrary 2D flow field, I would like to compute the streamline with minimal path-integrated directional divergence and velocity shear. In natural coordinates (flow-oriented) the divergence is ...
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Euler-Lagrange equation in a differential form notation

Treating the Lagrangian density as a $d$-form in $d$-dimensional spacetime, how can one write the Euler-Lagrangian equation basis independently in the form notation? If possible, can you also apply ...
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Proper way of adding parameters to a test wave function

So experimenting with the variational method, I thought of a test wave function for an infinite deep well system, $$\Psi(x) = N(a^2-x^2) \text{ for $-a$ < $x$ < $a$}$$ and $0$ everywhere else. ...
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Why does introducing more variational parameters improve the accuracy of our approximation

If we consider the variational method in quantum mechanics as outlined in this wikipedia article: https://en.wikipedia.org/wiki/Variational_method_(quantum_mechanics) It is stated that 'Some choices ...
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Solving the matrix Schrödinger equation

One can solve the Schrödinger equation by diagonalizing the Hamiltonian $H$. Due to limited memory, we truncate $H$ up to $N$. Now I red here on slide 8, that increasing $N$ cant lead to higher ...
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Numerical minimization of the action in python [migrated]

I want to find the trajectory $x(t)$ which minimizes the action $S = \int_{t_i}^{t_f} L(x(t), \dot{x}(t)) \mathrm{d}t$ numerically. I am trying to do it by discretizing the action so it is more of a ...
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Does the space of Slater determinants or bosonic permanents have any nice mathematical structure?

(This is a soft question.) In the Hartree-Fock approximation, you approximate the ground state of a many-body fermionic system by performing a variational minimization of the expected energy of a ...
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In what cases can we show that the ground state wavefunction from the variational principle is at all similar to the true wavefunction?

The variational principle gives a ground state that minimises the expected energy from the TISE, while from this we can be sure that we have an upper bound on the ground state energy, it is not always ...
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Does Hartree-Fock method always converge to global energy minimum?

I'm not asking about whether the Hartree-Fock method will always converge, but, if it does, it seems like Wikipedia is saying that it will always converge to the global minimum energy of the TISE. ...
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Beltrami identity and E-L equation incompatible?

$\newcommand{\dd}[2]{\frac{\partial {#1}}{\partial {#2}}}$ $\newcommand{\DD}[2]{\frac{d {#1}}{d {#2}}}$ Begin with the Euler-Lagrange equation $\dd Ly = \DD \ x [\dd L {y'}]$ where $L=L(y,y',x)$, (...
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Can light travel in closed loops indefinitely?

For simplicity, consider the two dimensional space. My question is that can there be a refractive index $(x, y)\mapsto n(x, y)$ such that the there is at least one closed permissible path (which can ...
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What are the applications of calculus of variations, if any, to the subject of thermodynamics?

If we apply calculus of variations to Newtonian mechanics, we can discuss of functionals such as the lagrangian and how optimizing it leads to the equations of motion. However, does there exist ...
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Overdamped dynamics and minimization principles

Most physicists are familiar with Hamilton's principle, which allows us to derive the equations of classical mechanics from the principle of stationary action, $$\delta S=0.\tag{1}$$ At the same time, ...
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Normalize the variational wave function

I am trying to normalize the following variational wave function: $$\psi(x,\alpha)= |x|^{\alpha} + L^{\alpha}$$ and I'm using this: $$1= \int_{-L}^{L} |\psi(x,\alpha)|^2 dx$$ Solving the integral gave ...
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Is the lagrangian density convex if the lagrangian is convex?

Let $L = \dot{q}^T M(q) \dot{q} + V(q)$, i.e., the lagrangian has a quadratic form and hence is convex w.r.t to the velocities, considering that $V(q)$ plays the role of a constant. And now let the ...
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Boundary term for Chern-Simons action

As discussed in David Tong's lecture series on the edge modes in the quantum Hall effect (http://www.damtp.cam.ac.uk/user/tong/qhe.html) (page 203), varying the 2+1D Chern-Simons action yields: $$\...
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Variational principles for approximating thermodynamic potentials in the inverse Ising problem: How to go from double to single extremum?

I'm trying to wrap my head around section 2.2.6 (on variational principles) in the following paper (on the inverse Ising problem): https://arxiv.org/abs/1702.01522 Here the authors explain how to use ...
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When is variational method minising condition equivalent to solving TISE on part of Hamiltonian in reduced basis?

In the tight binding model, my book says that we are using a variational method with $|\psi \rangle = \sum_{n} \phi_n |n\rangle$ (with $ \langle n|m\rangle = \delta_{n,m}$, and claims that the ...
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Equality modulo equations of motion [closed]

What does Qmechanic mean by “equality modulo equations of motion” when talking about Lagrangian formulation/formalism and so on?
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Application of the Cartan Structure Equations seems to imply the Einstein-Palatini action is zero?

The Einstein-Palatini action can be written as $$ S = M_{pl}^2\int\varepsilon_{abcd}\left(e^a\wedge e^b\wedge R^{cd}\right), $$ where $e^a={e^a}_\mu\text{dx}^\mu$ is the basis one-form and $R^{ab}=\...
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What's the logic behind light's following the longest path according to Fermat's Principle? [duplicate]

In our textbook, it's written that light rays will follow the minimum or maximum distance after being reflected from the plane surface and thus the path will be fixed. At somewhere, I read that light ...
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Dirichlet boundary condtions in Nambu-Goto string action

The Nambu-Goto action for an open string with parameter domain $[0,\tau_1]\times[0,\sigma_1]$ is given by \begin{equation} S_{NG} = \int_{0}^{\tau_1} d\tau \int_{0}^{\sigma_1} \ d\sigma \ \mathcal{L}(\...
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Does light always take the shortest path?

Does light always take the shortest path? And is it possible to change the probability of a photon travelling to a point by only disturbing the paths that are far away from the shortest path?
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Maxwell equations as Euler-Lagrange equation without electromagnetic potential

The standard way to write the Maxwell equations (say in vacuum in absence of charges) as Euler-Lagrange (EL) equations is to take the first pair of the Maxwell equaitons and to deduce from it ...
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Is it possible to think an example of refraction in which Fermat principle involve a maximum without using reflection?

In a question What is incorrect about the original statement of Fermat's principle? is showed an example of reflection in which Fermat principle involve a maximum, and in comments is said that it ...
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Is there an accepted Lagrangian for the transport equation?

Perhaps because it is so simple, I have not seen a lagrangian form of the transport equation $$(\partial_t + a \partial_x)q = 0.$$ This equation is first order, which makes obtaining it from the Euler-...
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Path of light ray through varying refractive index

Suppose light ray passing through a medium with refractive index $n=n(y)$. In the case of an inhomogeneous medium in which $n$ varies continuously in the $y$-direction, We have curved rays that ...
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Can we determine the order of the equations of motion simply by looking at the action?

Naively, one would expect the EL equations arising from an action to contain derivatives (of the dynamics field) of an order that is twice the order of the highest-order derivative (of the dynamic ...
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Help deriving optical path

I have to derive the path of a light ray traveling in a stratified medium with linear variation of its refractive index. For context the first two exercises ask you to derive Snell's law using ...
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How to know if a spinor $\Psi\left(x\right)$ is the ground state of the system?

Suppose we have time independent one-dimensional single particle Schrödinger-like equation$$-\frac{d}{dx}\left(A\left(x\right)\frac{d}{dx}\psi\left(x\right)\right)+V\left(x\right)\psi\left(x\right)=E\...
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Is there some alternatives to least action principle

The principle of least action seems to be one of the most fundamental of high-energy/fundamental interactions physics. But is there some other possibility ton construct a theory of interactions? Or, ...
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What is the proper way to understand Fermat's principle?

I am studying Fermat's principle for the first time and the basic knowledge that I could gather said that it states that "that the path taken by a ray between two given points is the path that ...
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How to compute the variation of the Lagrangian density correctly?

I'm studying Schwinger's action principle in Classical Field Theory, but I have some doubts about the correct variation of the Lagrangian density. Let's consider an infinitesimal transformation of ...
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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 ...
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How to solve this minimization problem in classical mechanics? [closed]

One should minimize the distance between two points x1=(x1,y1) and x2=(x2,y2). The holonomic constraint states that $f(x)=x^2-2x+5$ where x1 is an element of this graph and x2 is an element of the ...
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Blackbody radiation from principle of least action

I was recently studying Blackbody Radiation and the principle behind it (as far as used in the Plancks original paper) is to find the energy distribution which maximizes the number of ways in which ...
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Adding a total time derivative to a Lagrangian [duplicate]

I'm aware there are many posts regarding this topic, yet I haven't found an answer to this specific question there. Say we have a Lagrangian $$\mathcal{L(q,\dot q, \ddot q, t)}$$ We know that adding ...
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Find function which minize variance of intensity on target plane

I am trying to find continuous function $\sigma(x)$ for uniform intensity. I set two parallel horizontal lines with length $W$ and with distance each other $H$. The upper line is a target plane ...
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Vanishing action integral for Gravitation Field

For a Gravitation Field Action Integral looks like: \begin{equation}\label{1} S_{gravity} = \frac{c^3}{16\pi G}\int R\sqrt{-g} d^4x. \end{equation} А Least Action Principle says the $\delta S_{...
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Application of Lagrange Multipliers in action principle

In Goldstein's Classical Mechanics, he suggests the use of Lagrange Multipliers to introduce certain types of non-holonomic and holonomic contraints into our action. The method he suggests is to ...
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Equivalence Of Newton's Law and Leact action principle Without Math (Intuition): initial vs. boundary problem

First Let me write both the laws So Newton's law says that $$\mathbf{F}=m\mathbf{a}$$ and least action principle says that a particle occupy, at the instants $t_1$ and $t_2$, positions defined by two ...
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Is It Possible to Express all fundamental forces in the form of generalized potentials? [duplicate]

I have Started reading Hamilton's Principle or Principle of Least Action In first course of Undergraduate classical mechanics. So, I think it becomes easier to apply the Variational principles if ...
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Convert a quasi-symmetry of the action into a strict symmetry

A quasi-symmetry of an action $S$ is a transformation of the fields that leaves the action invariant up to a boundary term (see e.g. the answer to this question). In contrast, let us call a ...
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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 ...
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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?
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Euler Lagrange equations from discrete to continuum

Given $$\frac{d}{dt} \frac{\partial L}{\partial \dot \phi_a^{(i j k)}} - \frac{\partial L}{\partial \phi_a^{(i j k)}} = 0\tag{1}$$ $$\partial_{\mu} \frac{\partial \mathscr{L}}{\partial(\partial_{\mu} \...

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