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 derivative of a Lagrangian

I would like to know how exactly to calculate the variational derivative of: $$L = \oint dx \: \frac{m}{a} (\dot{u}(x,t))^2 - ca(u'(x,t)^2\tag{1}$$ with respect to $u$, ($\delta L / \delta u$). The ...
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Is there a Lagrangian $L$ (equivalently an action functional $S$) which yields the Navier-Stokes equation?

The Navier-Stokes equation or the Euler equation are usually derived as the conservation laws. However, I wonder if there exists a Lagrangian $L$ or equivalently, an action functional $S[\...
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Magnitude of the variations $\delta q_i$ in the principle of stationary action

To determine the equation of motion using the principle of stationary action, one has to consider the variation of the action due to variations $\delta q_i$ in all the generalized coordinates $q_i$. ...
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Proof of principle of stationary action when the Lagrangian is not $L=T-V$

The principle of stationary action claims that the action $S$ takes a stationary value in a real system, where: $$S = \int_{t_1}^{t_2} L dt\tag{1}$$ and $L$ is the Lagrangian of the system. It can be ...
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Why does the trajectory of a relativistic particle "minimises its Minkowski distance"?

The action of a relativistic free particle is $$\mathcal{S}=\int^{t_{1}}_{t_{0}} L dt\tag{1},$$ for $$L=-\frac{mc^{2}}{\gamma}.\tag{2}$$ I understand that a particle will follow the trajectory of ...
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How to derive the Lagrangian for a system of first order equations of motion?

Please be lenient if this question has been asked before in a similar manner. My background is in CS, but I am working on the physics-based modeling of complex systems and I am studying physics as an ...
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How does the boundary term matter in scalar field and in more general cases?

People always say that boundary terms don't change the equation of motion, and some people say that boundary terms do matter in some cases. I always get confused. Here I want to consider a specific ...
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Why doesn't the Hydrogen atom, as described by the Dirac equation, collapse?

In Griffiths quantum mechanics, it's noted that the exact energies for the Dirac equation, involving fine structure, are $$E_{nj} = mc^2 \left\{ \left[1 + \left(\frac{\alpha}{n-(j+1/2)+\sqrt{(j+1/2)^2 ...
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Variational operator confusion

Let $L=L(X, \dot X)$ such that the first variation of $L$ is given by $$\delta L=\frac{\partial L}{\partial X}\delta X+\frac{\partial L}{\partial \dot X}\delta \dot X.\tag{1}$$ This is pretty standard ...
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Confusion with the variational operator $\delta$ and finding variations

I have recently started studying String Theory and this notion of variations has come up. Suppose that we have a Lagrangian $L$ such that the action of this Lagrangian is just $$S=\int dt L.$$ The ...
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Euler-Lagrange solution of $L=\ddot{q}^2$?

I'm new to Calculus of variations and have a very basic question. Suppose we want to solve the Lagrangian $L=\ddot{q}^2$ using the Euler-Lagrange equation. My intuition tells me that the solution ...
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Equations of Motion and Minimization of Spacetime Interval

I'm trying to show that the extrema of a path in spacetime, as given by the spacetime interval (or length if just considering space) is the one that solves the equations of motion. Let a path be given ...
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Time dependent Schrodinger equation through variation principle - questions about derivation

I'm reading a text which discusses time dependent variation principle (Geometry of the Time-Dependent Variational Principle in Quantum Mechanics by Kramer and Saraceno), and there is some part of a ...
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Generalization of the Hessian to field theory

For the Ostrogradski theorem to apply, in classical mechanics (finite number of degrees of freedom) the Lagrangian needs to be non-degenerate. This means that the Hessian matrix (second derivatives of ...
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How are rigid solids approached in the context of Lagrangian formalism?

Maybe it's my own fault, but neither in my classical mechanics class nor within any book I've read on the subject I have found an extensive use of analytical mechanics to discuss the motion of solids. ...
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Variation of the metric determinant

I know this question has been answered here for example but I want to make emphasis in a new aspect. Consider the variation of the metric determinant $\delta g$ with respects to variations of the ...
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Confused about relativistic point particle action in MTW

In MTW page 179 exercise 7.2 they give the following action for an particle in an electromagnetic potential. $$I = -\frac{1}{16 \pi} \int F_{\mu \nu} F^{\mu \nu} d^4x + \frac{1}{2}m \int \frac{dz^{\mu}...
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Is action ever stationary in quantum theory, or is it always minimal? [duplicate]

The question says it all. Is there an example from quantum theory of quantum field theory where action is only stationary, but not minimal? (There are such examples in classical physics - but do they ...
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Is it possible to built a variational principle for this first-order system?

Imagine there is a mechanical system described in unitary units by the equation: $$\dot{x} = -\text{sgn}(x)\sqrt{|x|},\quad x(0)=1 \tag{Eq. 1}$$ such it has a finite duration solution: $$x(t) = \frac{...
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What is the difference between variational principle, principle of stationary action and Hamilton's principle?

In advanced mechanics, we learn about the variational principle, the principle of stationary action, and the Hamilton's principle. I feel that the difference between them is not very clearly organized ...
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Question about the Weiss variational of gravitational action and related equations of motion

I was reading The Weiss Variation of the Gravitational Action by Feng and Matzner, where the authors take the variations of the gravitational action with respect to the bulk metric $g$, the induced ...
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Understanding some implications of Fermat principle

I am modelling the propagation of light rays from one point to another, with some object in between, in this case a disk to simplify matters, and I compared the travel times of a refracted ray and a ...
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1 answer
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Numerical Solution of least action principle

I am trying to numerically find the path of least action between two points (ignoring the time step normalizing factor): $$S=\sum_i (x_{i+1}-x_{i})^2/2-V(x_i)$$ I don't have the potential in explicit ...
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Why is d'Alembert's principle not as applicable in physics as the principle of stationary action?

Any textbook in classical mechanics will tell you that there are two different routes one can follow to derive the Euler-Lagrange equations: Route 1: Write d'Alembert's principle in the form $\sum_{i=...
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Principle of least action breaking down

We know Newton's law doesn't hold in very small distances and very high speeds. Do you know any conditions under which the principle of least action doesn't hold/breaks down?
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Proof of the Variational Theorem

I have trouble understanding the proof of the Variational Theorem. I'll recall quickly the proof to show my problems (see also this post and the answer by Mateus Sampaio for a detailed proof). Let $H$ ...
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D'Alembert Principle and Euler-Lagrange. Virtual displacement

I have a little trouble with d'Alembert Principle and with virtual displacement. Imagine that with the d'Alembert Principle: $$ \sum_i \boldsymbol{\mathrm{F_i}} \; \cdot \delta \boldsymbol{\mathrm{...
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QFT: Possible typo in Schwartz problem 3.5?

I have a doubt about the problem 3.5 in Schwartz's "QFT and the Standard Model". The problem states: Spontaneous symmetry breaking is an important subject, to be discussed in depth in ...
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Is there a universal algorithm that allows to find the Lagrangian/Hamiltonian, given **any** physical system that admits them?

Every time I have seen someone ask what the universally valid definition of Lagrangian/Hamiltonian is, the answer has often been one of the following two kinds: Partially or fully incorrect, i.e. not ...
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What if the minimising path for the least action isn't unique? [duplicate]

Using the Lagrangian method, we find the path which minimises the action. But what if multiple paths each have the same minimum action? How do we resolve such a situation?
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Why principle of least action is true? [duplicate]

Recently I am studying lagrangian mechanics where I came across the topic "principle of least action" which states that a system always takes the path of least action or when the action is ...
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Why do we maximize according to the values of Lagrange multipliers?

In some Lagrangian problems, when we use the lagrange multipliers to minimize a function $f(x)$ they write: \begin{equation} \max_{\lambda,\mu} \min_{x}\mathcal{L} = \max_{\lambda,\mu}\min_{x} \Big( ...
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Why do we decompose the boundary surface in Initial boundary-value problems?

Just started to learn about initial boundary-value problem, and I have came across the concept of decomposing the boundary surface into to disjoint parts, one for stress tensor field and another for ...
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Variational approach to the Laplace and Poisson equation in Jackson's book

In Section 1.12, Chapter 1 in Jackson's Classical Electrodynamics, he considered an "energy-like" functional, $$ I[\psi]=\frac{1}{2}\int_V\nabla\psi\cdot\nabla\psi\ d^3x - \int_V g\psi\ d^3 ...
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Lagrangian mechanics and geodesics in configuration space? [duplicate]

In lagrangian mechanics Is the path that take a System in the configuration space between initial and final state is identical to the geodesic which connect the two points?
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What is the shape made by a half-filled ball?

An inflatable ball is made of a thin, inelastic, but bendable material. When fully inflated, it has a radius $R$. If the ball is filled half full with air, and is then continuously pushed underwater ...
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Variation with respect to metric in the Einstein-Cartan formalism

Consider the variation of the EH action with respect to the metric $$\delta S_{EH} = \int d^4x ~(\delta\sqrt{-g} R + \sqrt{-g}\delta g^{\mu\nu} R_{\mu\nu})$$ Now I make use of $$ \delta\sqrt{-g} = \...
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How to determine $n(x)$ when the functional depends exclusively on $n(x)$ and $x$? (Fermat's principle)

Recently I was taught an introduction to calculus of variations in reference to a course on analytical mechanics, where one problem involved Fermat's principle, stating that the path taken by a light ...
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If the Lagrangian included acceleration [duplicate]

If the Lagrangian included acceleration, then how many conditions would be needed to unambiguously define the trajectory, on what variations would the minimum action problem be considered, what would ...
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Search for maximum and minimum functionality

How can I prove that the trajectory I found after applying the principle of least action corresponds to the minimum of action, and not to the maximum?
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Euler's Equations with auxiliary conditions - "why is $\frac{\delta y}{\delta \alpha}$ and $\frac{\delta z}{\delta \alpha}$ no longer independent?"

let $J(\alpha)$ be a functional of the parameter $\alpha $ such that: \begin{equation}J(\alpha) = \int_{x_1}^{x_2}f\{y,y',z,z';x\}dx \end{equation} and let \begin{equation}f = f\{y,y',z,z';x\} \end{...
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How to understand the variation of potential energy

I have to present a way to approach the principle of virtual work from the notion of work in cases of conservative and time-independant physical systems. Here I try to approach the problem in terms of ...
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Metric variation of a four-vector

I am trying to calculate the metric variation $\frac{\delta T^{\alpha\beta}}{\delta g^{\mu\nu}}$. Here, $T^{\alpha\beta}$ is the stress-energy tensor of a perfect fluid, given by $T^{\alpha\beta} = (\...
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Test field vs backreaction of field theory in curved spacetime

Is there a way to understand test field regime as some limit of backreaction in general relativity? Consider the Einstein-Hilbert action augmented with the standard electromagnetic field coupled ...
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Euler-Lagrange equation for positron from QED lagrangian

Taking $e<0$ to be the charge of the electron, the Euler-Lagrange equation derived from taking the first variation of the $\psi$ field in the QED lagrangian $${\cal L} = - \frac 1 4 F_{\mu\nu}F^{\...
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Is refraction just an application of minimising the distance $s$ in 4D spacetime?

I apologise that I don’t know how to fully put into words the thought I’m having so it may be a bit rambly. In school we’re taught that refraction of light happens because as light enters a denser ...
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2 answers
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Refraction across two interfaces: is it correct to use Snell's law as constraint in an application of Fermat's principle?

The problem area is geometric optics, namely refraction across homogeneous media with constant speed of light. I explain the three steps of a methodological doubt that popped up. Polite request: ...
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2 votes
5 answers
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Axiomatising classical mechanics to arrive at the principle of stationary action - what are the fundamental definitions of momentum, etc.?

$\newcommand{\d}{\mathrm{d}}\newcommand{\l}{\mathcal{L}}$Throughout all my study of physics, it has never been clear what is a definition, what is an axiom, what is a law and what is a proof in ...
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How can I show that the action of a SHO is a saddle-point solution if $t_{f} - t_{0} > T/2$?

In this post and in this post, QMechanic claims that a simple harmonic oscillator with Dirichlet boundary conditions has saddle-point solutions if $t_{f} - t_{0} > T/2$ where $T$ is the ...
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Use Hamilton's principle to show expression for $L$ [closed]

I have following diagram I have here to find the kinetic energy and the potential energy. I think that kinetic energy is: $$T=\frac{1}{2} M(\dot{x_1}^2+\dot{x_2}^2)$$ and the potenitial energy must ...
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