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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Tensionless string in Nambu-Goto action

I am studying string theory from the book "String theory and M-theory", written by Becker, Becker and Schwartz. My question is: We are taught that one of the advantages of introducing a ...
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Nambu-Goto action and the World-Sheet Area

I am studying string theory from the book "String theory and M-theory", written by Becker, Becker and Schwartz. My question is: We are told that the Nambu-Goto action is simply the one that ...
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Integration by parts on generic tensors

I try to rephrase here a my question (https://math.stackexchange.com/q/4661784/), explaining more specifically the case. Given a lagrangian $L=L(\theta_{\mu\nu},\phi)$ , the conserved Noether current ...
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Why do we put factors of zero in a Lagrangian that is to be extremized?

According to the Wikipedia page on Lagrange multipliers under the section - Example 3: Entropy, it is written that: $$f(p_1,p_2,\ldots,p_n) = -\sum_{j=1}^n p_j\log_2 p_j$$ For this to be a ...
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Variational principle and the medium equation for photon paths in general relativity

I am reading a 2015 paper by Rogers on Frequency-dependent effects of gravitational lensing within plasma. He gives the relation between the components of the photon four-momentum and the refractive ...
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Variation of Action and Border Terms

I need to compute the following very general (piece of) variation: \begin{equation} \int d^4x \delta (\sqrt{-g} R ) f \tag{1} \end{equation} where $R$ is Ricci scalar and $f$ a generic scalar ...
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Does there exist a square root of Euler-Lagrange equations of a field? (Factorization)

Does there exist a square root of Euler-Lagrange equations $\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}-\frac{\partial \mathcal{L}}{\partial \phi} = 0$ in the sense that $(x+...
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Feynman physics on least time of light

What does light checks all paths mean by Feynman? Especially the statement is labeled by yellow. Why there is only one path that leads radiowaves to D’? And how wave check all paths, that is, why it ...
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Simplicity constraints from $SO(4)$ Plebanski action

The $SO(4)$ Plebanski action yields a first order formulation of Euclidean General Relativity as a constrained (topological) BF-theory. It depends on a $so(4)$ connection 1-form $\omega^{IJ} = \omega_{...
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Geodesics: Energy functional vs length functional

The Wikipedia article about geodesics talks about the equivalence of obtaining the geodesic by either minimizing the length functional $L$, or by minimizing the energy functional $L^2/2$, cf. the Phys....
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From EH action to Newtonian Mechanics action? [closed]

How does one start from the Einstein Hilbert action go to the action of a (point particles + some field) in special relativity and then Newtonian mechanics for a local neighborhood around a point for ...
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Different relativistic actions [duplicate]

I am slightly confused about different action integrals in relativity. When you work through some introductions to general relativity, you usually get in contact with the relativistic action integral \...
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Is calling it "The Principal of Extremal/Stationary Action" pedantry? [duplicate]

I understand that the equations appear to permit paths of maximal action, but is there any real physical case where this actually occurs? Would it not be more sensible to refer to this as the ...
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Gauge Symmetry of the Lagrangian

My teacher told the following statement to me during office hours. Is it correct and if so, how could one go about proving it? Given a material system subject to holonomic and smooth constraints ...
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Why does a complete four-divergence term in a Lagrangian density not affect the equation of motion in special relativity? [duplicate]

The classical theory of fields by Landau and Lifshitz, page 68 says: As for the quantity $\epsilon^{iklm}F_{ik}F_{lm}$ (§ 25), as pointed out in the footnote on p. 63, it is a complete four- ...
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The principle of stationary action from the principle of inertia?

Can we infer the principle of stationary action from the principle of inertia that causes a mass particle to resist changes in its momentum? The following is my own speculation. When a mass particle ...
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Is action an extensive quantity - always?

Action, the integral over time of the kinetic minus the potential energy seems to be an extensive quantity. (There is nothing serious coming up in Google on this issue. Neither on Google Scholar.) In ...
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Do solutions of the Schrödinger equation for multiple particles automatically obey spin-statistics?

Consider the Hamilitonian for a general two-electron system subject to an external potential $V_\mathrm{ext}$ and an interaction potential $V_\mathrm{ee}$. In this case $$H\psi(x, y) = -\frac{1}{2} \...
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How to derive the free rigid body equations from Euler-Lagrange? [closed]

I'm trying to retrieve the equations of motion for a free rigid body: $$ I(t)\dot{\omega}(t)+\omega(t)^T \times (I(t)w(t)) = 0 $$ where $$ I(t)=R(t)I_{0}R(t)^T $$ I know that Euler-Lagrange equations ...
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Thermodynamic interpretation of the cosmological constant?

So I'm going through this paper of emergent gravity where T. Padmanabhan provides a thermodynamic interpretation of Gravity. What I'm failing to understand is what exactly is the thermodynamic ...
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Understanding this Lagrangian calculation

I was trying to understand this section of a Wikipedia article: $$0 = \delta \int \sqrt{2T} d\tau = \int \frac{\delta T}{\sqrt{2T}} d\tau = \frac{1}{c} \delta \int T d\tau$$ For the life of me, ...
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The action principle, uncertainty principle and entropy [duplicate]

This is more or less a philosophical question about the connection between entropy driven (spontaneous) phenomena and fundamental (random) quantum fluctuations. It´s well known that the Principle of ...
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Coulomb force from a variational principle

See the attached discussion from Zangwill's Modern Electrodynamics, and in particular footnote 9. The point of this question is to understand how to recover Coulomb’s force law from an assumed form ...
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Are interfering rays real in explaining the principle of least time in optics?

It is known that a ray of light chooses the path along which it will most quickly reach from point A to point B. Here there are questions about how light can choose something. To explain this, we say ...
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Maupertuis' principle by variational method

On p.464 of Spivak's mechanics book, the author proves the equivalence of Maupertuis' principle and stationary action principle by considering variation of some path $c(t)$, such that other paths in ...
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Physical meaning of functional derivative of Coulomb potential energy

I am considering the problem of a conductor of arbitrary shape, and I want to prove (I reckon it's possible) that the field inside the conductor is zero assuming only that the charges are in a ...
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Variational formulation for the Kerr solution

The critical points of the Einstein-Hilbert action, if one allows axial symmetry, are Kerr-type solutions of the Einstein field equations, in which a parameter $\alpha$ interpreted as the angular ...
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Variational Principle for Relativistic Action

I'm going through p. 27 in Landau & Lifshitz Classical Field Theory (vol 2), and I'm confused as to why only the contravariant part of the proper time is varied? They start with $$\delta S=-mc\...
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Variational Method for A Symmetric double well Potential

I am given a set of trial wave functions of the form $$ Φ_n^{\pm}(x)=Ψ_{n}(x-α)\pm Ψ_{n}(x+a) $$ Where $Ψ_n$ are the $n$th Harmonic oscillator wavefunctions. in order to approximate the energy levels ...
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Brachistochrone problem with drag

If we didn't consider friction, this would just be the brachistochrone curve. But what would the optimal path be if there was also a drag force that was in proportion to the object's speed?
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"In spacetime, a straight path yields the longest elapsed time between two events". Could someone explain this please?

I know this may appear to be a duplicate question but the other question Straight lines and longest distance doesn't seem to explain in laymans terms. So... I'm trying to understand this but have read ...
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What is the lagrangian density of Joos-Weinberg field?

It's frequently said, that Joos-Weinberg equation describes fields with any spin. But what is the lagrangian from which the equation could be derived?
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Newtonian Gravity from curved space?

Imagine you have the arc-length of a curve, in spherical, coordinates: $$ s = \int_{\mathcal C}{d\tau \; \sqrt{f(r)^2 \left (\frac{dr}{d \tau} \right )^2 + r^2 \left (\frac{d \theta}{d \tau} \right )^...
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Lagrangian formulation of Maxwell's equations with magnetic monopole

If we set $\nabla \cdot {\bf B}=\rho_m$ where pm is the density of magnetic charges we lose the ability to write ${\bf B}=\nabla \times{\bf A}$ . Can we get a new Lagrangian that leads to the new ...
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Most stable shape if Newtonian gravity was proportional to $r^\alpha$

Consider lots of mass in isolated 3D space, close to each other. Consider that only the gravitational force (Newtonian) exists. Also consider that there is no rotational motion. It is evident that a ...
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Why is the full Hamiltonian used instead of the approximate Hamiltonian for determining the effective nuclear charge using the variational principle?

My question is in regards to the variational principle in approximating the wavefunction of Helium. Some Background: $$\hat{H}=-\frac{\hbar^2}{2m_{e}}\nabla_{1}^{2}-\frac{\hbar^2}{2m_{e}}\nabla_{2}^{2}...
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Why is it possible to neglect higher order terms in the variation of the action?

In order to get the Euler-Lagrange equations, we should find the variation of the action $\delta S$ and to neglect higher-order terms: $$\delta S=\int L(q+\delta q,\,q'+\delta q',\,t)dt-\int L(q ,\,q',...
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What gives us the equations of motion in GR?

Maybe stupid question, but to my understanding, the Einstein equation tells us the differential equation governing the Geometry of spacetime. That's all good and fine , but suppose I had an actual ...
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Schwinger action principle derivation in Parker-Toms

I'm reading "Quantum Field Theory in Curved Spacetime" by Parker, Toms and I'm stuck in the very last part of the demonstration of the Schwinger action principle. I arrived at eq. 1.34 $$ \...
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Finding upper bound for the first excited energy eigenstate in quartic potential using variational principle [closed]

I'm solving problem number 5 from https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/resources/mit8_05f13_ps2/. (a) Here I got: $$ \beta = \frac{\hbar^{\frac{1}{3}}}{(\alpha m)^\frac{1}{6}} ...
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Variational principle for a particle with varied mass [closed]

I can't quite understand how the variational principle works. For a relativistic free particle we can find the extremum of the action $$ S=-m\int ds $$ and find the equations of motion. But what if ...
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Exact function for the brachistochrone

I have watched videos on the brachistochrone problem and how to find the quickest path a particle can take between two points. However they never gave an exact function for the path. I thought of ...
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1 answer
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Using EOM in QED Lagrangian [duplicate]

Let's have the QED Lagrangian. $$\mathcal{L} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} + \bar{\Psi}(i\partial_\mu \gamma^\mu - m)\Psi + g\bar{\Psi}A_\mu \gamma^\mu \Psi.\tag{1}$$ The equations of motion are:...
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Noether's theorem with or without boundary term in the classical mechanics

I'm recently reading both a variational calculus textbook and a classical mechanics textbook. I found that Noether's theorem is stated differently within those two backgrounds. I'm wondering about the ...
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Use of Lagrange multipliers in derivation of the Bose-Einstein distribution

My main question in regards to this is an explanation on why/how you can use Lagrange multipliers when you have a function of infinite variables, what is the justification behind this? So to derive ...
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Inquiry about applying stationary action to field lagrangian [closed]

I am reading David Tong's lecture notes on quantum field theory. There is a part where he says: \begin{align} \delta S & = \int d^4x \left[\frac{\partial \mathcal{L}}{\partial \phi_a}\delta \...
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Can ground states computed through the variational method not be eigenvectors of a Hamiltonian?

Suppose you have to apply the variational principle to compute the ground state of a system whose Hamiltionian is $H_s$ and your ansatz is a linear combination of ground states of a simpler system ...
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Variation of the square of the Weyl tensor

After reading a bit about Conformal gravity, I came across the Lagrangian of the form: $$L=C_{abcd}C^{abcd} \sqrt{-g}$$ Where C is the Weyl tensor, I am interested in finding the field equations that ...
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Spacetime dimension by variational principle

I'm asking it out of curiosity: if hypothetically speaking spacetime had the freedom to "choose" its dimension, and it can be described by an action, is it possible to do an analytic ...
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In principle, "how much" of the path integral is required to match the electron $g$-factor experiment?

As Dirac was the first to realize (Dirac 1933, page 69), the reason the quantum path integral converges to the classical action principle as $h\rightarrow 0$ is that The only important part in the ...
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