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 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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Is there any restriction on the Lagrangian of a system?

I have learned the calculus of variations in my previous semester, and now we are studying classical mechanics. What I found is that there is lots of lack of rigor in Lagrangian mechanics in ...
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Understanding how to derive the discrete Euler Lagrange equations

I learned how to derive Euler-Lagrange equations in classical lagrangian field theory: we start off a Lagrange density (which only depends on the fields themselves and their first order derivatives), ...
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The origin of the Lagrangian

Is it possible to explain the form of Lagrangian $L=T-V$ in the following way? The force that acts on a mass particle is due to the potential $V(x)$. This force drives the particle to minimize its ...
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Can light take a (faster) detour?

I was taught that light tends to take the 'fastest' route. However, this made me wonder about the following scenario: Suppose the start and end point (source and observer) are at the edge of a large ...
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Lagrange multiplier for Robin boundary condition in variational minimisation

Consider the partition function for a scalar field $\{\phi:\mathbb{R}_{\geq 0}\to\mathbb{R}\}$, $Z=\int D\phi D\lambda\exp(-S)$ with the action $$S=\underbrace{\int_0^\infty dx \frac{1}{2}(\partial_x\...
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Does a free particle always follow the trajectory of shortest distance? [duplicate]

Some context for the above question is warranted. While reading Hartle's Gravity, the following statements made recurring appearances: "Gravity is not a force, it is the geometry of 4D spacetime. ...
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Variational method: Why do parameters differ for two trial functions (optimization)?

Below the potential and trial functions: $$V(x)=(x^2-1)^2-x^2$$ Use the variational method with the two trial wave functions: $$\psi_{\pm}(x)=A\left(e^{-\frac{(x-x_0)^2}{2\sigma^2}}\pm e^{-\frac{(x+...
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Is a trajectory connecting two points valid for all the intermediate points too?

Suppose a particle is described by a Lagrangian $\mathcal L(q_i, \dot{q_i}, t)$. Suppose that $q_i(t)$ is a trajectory (there might be more that one) along which the action integral is stationary for ...
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What is the role of time (time interval) in principle of least action?

Action is represented by $S[Q(t)]$ where $Q(t)$ is the name of a single complete path in the configuration space of a system. The path starts at the point $q_i$ and ends at the point $q_f$. Suppose ...
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How can we form Euler-Lagrange equations for Time-Independent Schrodinger Equations?

Is it possible to form a lagrangian of the TISE using the concept of Lagrange Multipliers? I am new to this topic so any help would be much appreciated.
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Finding a “Principle of Least Action” equivalent statement for Hamiltonian Mechanics

The least-action principle is a statement in classical physics saying that all bodies in a system follow a trajectory that minimize the following functional (ignoring explicit time dependence for now):...
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On the derivation of the Euler-Lagrange equations [duplicate]

Goldstein's Classical Mechanics proposes two ways to derive the Euler-Lagrange (E-L) equations. One is by the D'Alembert's Principle of virtual work and the second is by Hamilton's Principle of Least ...
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Understanding the Schrodinger's equation by variational principle

I reviewed part of my notes in the quantum mechanics class, and still have a few questions about the variational derivation of the Schrodinger's equation: The variational principle says that the ...
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How to derive the hydrostatic equilibrium equation from the variational principle?

I was reading the book "Advanced Stellar Astrophysics" (William K. Rose, 1998) and I came across a different approach to deriving the hydrostatic equilibrium equation. First he defined the ...
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Fermat's Principle of Least Time - Analogy Confusion

Question : I was reading this analogy of Fermat's Principle of Least Time: In Figure, our problem is to go from A to B in the shortest time. To illustrate that the best thing to do is not just to go ...
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How to derive Euler-Lagrange equation for isochronous curve of Leibniz in terms of $t$, $x(t)$, $\dot{x}(t)$?

According to this source, "An isochronous curve of Leibniz is a curve such that if a particle comes down along it by the pull of gravity, the vertical component of the speed is constant, when the ...
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Does light remember its past to follow the least time path?

I read about the principle of least time. However, I think, for light to follow this principle, light would have to remember its past path before deciding where to go for its future path. Why I think ...
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Is there a “geometrical” reason for the principle of stationary action?

The principle of stationary action states that the trajectory $q(t)$ a physical system traces in configuration space is the one for which the action $$S[q]:=\int_{t_0}^{t_1}L(t,q,\dot q)\mathrm dt$$ ...
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A subtlety in the Brachistochrone problem

The following is a specific instance of the brachistochrone problem, which I first encountered in grad school, and I have occasionally used as hw problem in teaching CM. A particle is started from ...
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Interference and Fermat's principle

Fermat's principle states that light always takes the path for which the optical length is stationary. \begin{align} \delta\int n(\vec{r}) \:\mathrm ds = 0. \end{align} Furthermore, it is equivalent ...
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Is minimizing the action same as minimizing the energy?

When we differentiate the total energy with respect to the time and set it to zero (make it stationary), we get an expression as similar to what we get while we minimize action. Also putting the time ...
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Variational principle electromagnetism

In the following text of the picture, it says that the numerator is minimized if the spatial oscillation of the electrical field are minimized too. I don't understand why is this the case. Someone ...
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In a Lagrangian, why can't we replace kinetic energy by total energy minus potential energy?

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 ...
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How can we describe the path of a projectile fired from the ground using the principle of least action? [closed]

If a similar question were asked in Newtonian mechanics, the answer would be rather simple. But in Lagrangian mechanics, how can we say that the parabolic path taken by a projectile has a stationary ...
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Can a four-divergence be added pure Yang-Mills Lagrangian to alter the action? [duplicate]

A four-divergence term $\partial_\mu K^\mu$ when added to a Lagrangian, the action changes as $$S\to S^\prime=S+\int_R d^4x \partial_\mu K^\mu\tag{1}$$ where $R$ is a region of spacetime. Using Gauss' ...
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Continuum limit of Euler-Lagrange equation for Lagrangian density of 1D harmonic lattice

I'm trying to follow a derivation of the Euler-Lagrange equation at the continuum limit, and find some details hard to understand. The 1D lattice has a mono-atomic basis with atomic spacing $\mathfrak{...
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Is action maximized for a system in stable equilibrium?

Others have asked in general about cases in which the action integral is not minimized, but my question is specific: Can we show that the conventional action integral is always maximized for a system ...
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Why is $Q=p$, $P=-q$ a canonical transformation from the perspective of 2 variational principles satisfying boundary conditions? [duplicate]

This is to ask a more general question: Landau-Lifshitz say that for the variational principles $$\delta\int_{t_1}^{t_2}p\mathrm{d}q-H\mathrm{d}t =0$$$$ \delta\int_{t_1}^{t_2}P\mathrm{d}Q-H'\mathrm{d}...
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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 ...
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Calculus of variations: meaning of infinitesimal variation $\delta$ and action minimum

So I am studying classical mechanics through the MIT 8.223 notes, and encountered the derivation of the Euler Lagrange equation. There is a part I don't quite understand, which resides in the actual ...
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What happens when the same action extremal value can be obtained in more than one path in the configuration space?

I'm trying to understand the logic underneath the concept of action and lagrangian. I know this kind of questions have been asked many times, but I was unable to find an answer to this one. I've ever ...
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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. ...
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Variational problem for strings

I'm working through Zwiebach's String theory book by myself and I'm having trouble starting problem 4.7. For those that do not have a copy, I'll paraphrase the question: A string is stretched from $x=...
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What is 'covariant variation'? [closed]

What is 'covariant variation'? As opposed to the usual variation with respect to a gauge parameter?
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How is Schwinger's quantum action principle related to least action?

The principle of least action says that a body moves in such a way that the action value $S=\int L dt$ is stationary (often minimal). The principle is written as $$\delta S =0 \ .$$ In contrast, ...
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Hamilton-Jacobi equation and Action Functional

Let the action functional $S[q]$ given by \begin{equation}\label{eq16} S[q]=\int\limits^{t_2}_{t_1}L\left(q^i(t),\dot{q}^i(t)\right)dt.\tag{1} \end{equation} Also, we know that using Legendre ...
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Euler-Lagrange Equation: From boundary value to initial value problem

In the principle of stationary action, the initial and final points in configuration space are held fixed. This is a boundary value problem. However, this principle leads to the Euler-Lagrange ...
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Help understanding equations of motions for a line element in their full generality

(Note: I am crossposting this from https://math.stackexchange.com/q/3740738/) Let us work with these definitions $L=\sqrt{dAdB}$ and $S[A,B]=\int_{\Delta \lambda}\sqrt{\frac{\partial A[\lambda]}{\...
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Fermat's Principle of Least Time and Non-Smooth Bends in Light Paths

Consider a light ray eminiating from a point within a square medium where it has an infintesimally slow velocity. My understanding using Fermat's principle of least time is that if I was to observe ...
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How does posing $\sqrt{\dot{x}\eta \dot{x}}=c$ not contradict the fact that we are looking for variations?

For example, if I have a function which gives the radius of a circle dependant on $x$ and $y$: $$ R[x,y]=\sqrt{x^2+y^2}.\tag{1} $$ Then its partial derivative with respect to $x$ will be $$ \frac{\...
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How to obtain the variational of a vector (velocity vector)?

I recently started to study flight dynamics and I have to derive the equations of motion of a plane from the Hamilton's Principle. To better understand this principle, it is needed to have some ...
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Variation of EM action with respect to boundary metric

Say you have a charged black hole solution in General Relativity. I am wondering if the variation of the electromagnetic action with respect to the induced boundary metric is $0$? I am asking because ...
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Is it always the case that the square root of a lagrangian gives the same equations of motion as the lagrangian itself?

Inspired by the Phys.SE post Geodesic Equation from variation: Is the squared lagrangian equivalent? I was wondering if it is always the case that the square root of a lagrangian gives the same ...
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Derivation of the EOM for a string from the Polyakov action

This question is really a follow up regarding an older post: Derivation Of The Equation Of Motion Of String from Polyakov action In my derivation of the EOM, I am able to reproduce the expression, $$\...
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Fields that lend themselves to variational principles? [duplicate]

In physics, we often describe the dynamic properties of fields using variational principles like defining an action or a Lagrangian. A field however is simply some function of space $\phi(x)$ so I ...
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When is Newtonian physics in curved coordinates sufficient for GR?

A free particle Lagrangian in a 3D curvilinear coordinate system can be written as an inner product with the metric $g$: $$ L = \frac{1}{2}m\sum_{i,j=1}^3v^ig_{ij}v^j. $$ This equation was taught to ...
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How does Fermat's principle of least time for light apply to curved spacetime?

In a region of space which has no massive object light rays travel parallel to each other or ,simply, in a straight line. However, in a positively curved region of space (like near a planet or a star),...

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