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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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Action principle and Functional derivative in CM

I want to extremize this well known action. $$S[\phi]=\int \mathcal{L}(\phi(t),\dot{\phi}(t)) dt $$ The result is also well known. It turns out to be E-L equation. The Action principle states that the ...
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2answers
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Geodesics from variational principle with respect to coordinate?

I know you can find geodesic equations with respect to proper time $\tau $ using the variational principle, i.e. using Euler-Lagrange equations $$ \frac{\partial}{\partial x^{\mu}}L-\frac{d}{d\tau}\...
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Lagrangian Mechanics to solve arbitrary maximization problem

I've been thinking for some time about how to better find the optimal weights for neural networks and it struck me that when solving Lagrangian mechanics problem you are optimizing the action function....
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3answers
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Does light take the path of least time because it travels in straight lines or vice versa?

My question is which of these two feats is a consequence of the other? Light travels in straight lines, mostly. Does it do that as a result of Fermat's principle of least time? and if so, is there a ...
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Get the action from knowledge of all wave functions?

Say I know all the values of the wave functions at all times $\psi[\phi,t)$. Can I use this knowledge to find the action $S[\phi]$? i.e. to give the action as a function of the wave functions? $\phi$ ...
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1answer
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How to ascertain that the Rayleigh-Ritz variational method gives the exact value of the ground state energy?

So the Rayleigh-Ritz variational method can be used to calculate the ground state energy of a quantum system. If $\phi(x)$ is a suitable (square integrable) and normalised function of the coordinates ...
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1answer
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Proving that test particles in GR, follow spacetime geodesics

My question is pretty much in the title. According to this paper, this is not exactly proven rigorously yet. What I dont understand is what exactly is not proven. If I'm not too wrong, a test particle ...
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1answer
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Euler-Lagrange Equation and moment of inertia

I'm self studying and having trouble with the following question: "Consider a solid of revolution of a given height. Determine the shape of the solid if it has the minimum moment of inertia about its ...
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How principle of least action? [duplicate]

I had learned the principle of least action.But I didn't get the motive behind taking the least action. Or why should the particle follow a path where it have a least action?
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Variational formulation of Maxwell equations with interface/boundary conditions

Consider $\Omega = \Omega_1 \cup \Omega_2$, where $\Omega _1$ and $\Omega_2$ are two different media with conductivity and permeability \begin{equation} \sigma= \begin{cases} \sigma _1 & \text{in ...
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2answers
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Geodesics for FRW metric using variational principle

I am trying to find geodesics for the FRW metric, $$ d\tau^2 = dt^2 - a(t)^2 \left(d\mathbf{x}^2 + K \frac{(\mathbf{x}\cdot d\mathbf{x})^2}{1-K\mathbf{x}^2} \right), $$ where $\mathbf{x}$ is 3-...
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Insufficiency of Newton's third law to solve multiple body problems

In The Variational Principles of Mechanics Lanczos describes what he calls 'vectorial mechanics': the process of solving mechanical problems by recourse to the immediate consequences of Newton's laws, ...
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Show two Lagrangians are equivalent

I need to show that these two Lagrangians are equivalent: \begin{align} L(\dot{x},\dot{y},x,y)&=\dot x^2+\dot y + x^2-y ,\\ \tilde{L}(\dot x, \dot y, x, y)&=\dot x^2+\dot y -2y^3. \end{align} ...
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What is the source of the difficulty (within the variational approach framework) in the attempt to unify quantum mechanics with general relativity? [duplicate]

It seems to me that quantum mechanics can be formulated within the general mathematical framework of variational  principles. Derivation of the equations of nonrelativistic quantum mechanics based on ...
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Can all phase space conserving dynamics be described by a Lagrangian system? [duplicate]

Given a system described by a set of ODE's that can be shown to conserve phase space, does there necessarily exist a Lagrangian (or Action) formulation that describes my system? I'm comfortable ...
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1answer
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Choosing approximate eigenfunctions

In the variational approach of estimating the ground state energy of a system we choose an approximate eigenfunction dependent on certain parameters and then minimize the expectation value of the ...
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1answer
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Clarifications regarding the Maupertuis/Jacobi principle

I'm slightly confused regarding the Maupertuis' Principle. I have read the Wikipedia page but the confusion is even in that derivation. So, say we have a Lagrangian described by $\textbf{q}=(q^1,...q^...
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1answer
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Confusing with the equation $(2.4)$ and $(2.5)$ of Landau and Lifshitz, Mechanics, Chapter 1, The principle of Least Action

I'm a 12th Grader and I'm interested in Lagrangian Mechanics and having a bit of knowledge about the Newtonian Mechanics. So, I found a book of Landau and Lifshitz's Mechanics and started reading from ...
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2answers
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Electric potential energy of charges localized to the surface of an object

For a 3D object with a uniform density of surface charge, is there a particular shape which minimizes the total electrical potential energy? For instance, one could consider an object of fixed volume ...
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0answers
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Invariance with respect to time dilations of the free particle

Consider the action of a free particle in the space $$ s=\int_{t_1}^{t_2} \frac{m v^2}{2} d t.\tag{*} $$ The change of time coordinates $t'=\alpha t$, where $\alpha\in(0,1]$, preserves the form of the ...
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Einstein Field Equation for a Garfinkle-Horowitz-Strominger theory

When deriving the Einstein field equation from a theory with a dilaton field, we have: GHS's calculation However, in my calculation: My calculation So I found that GHS has missed $R$ and $(▽φ)^2$. ...
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1answer
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Is Hilbert-Einstein action just the leading order of some kind of series?

Introducing the action for the gravitational field my GR professor stated that, in principle, one could write it as $$S = k\int d^4x\sqrt{g}(\sum_n\sum_m a_{nm} R_n^m - 2\Lambda), \space \space \...
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Time-independent Schrödinger equation Lagrangian derivation

Recently I was taking a calculus of variations class and our professor casually obtained the time-independent Schrödinger equation for a free particle from the integral (constants dropped) and it's ...
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How general is the Lagrangian formulation? [duplicate]

Haven't seriously tackled this problem myself because it's been awhile since I've done any hard mathematics and I'm a bit rusty. However, you needn't spare the math in your answers. I've been ...
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2answers
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Euler-Lagrange equations from a complex Lagrangian

I'm looking for generalizations of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” and “maximum” don't have obvious meaning for a ...
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2answers
139 views

Is Action Always “Locally” Least?

In general, I know it's true that the Principle of Least Action is more properly called the Principle of "Stationary" Action. However, there are results which seem to suggest that for sufficiently ...
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1answer
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Proving that the Euler-Lagrange Equation has no solution [closed]

I'm trying to show that the Euler - Lagrange equation for the functional $$I(y)=\int_{a}^{b} y\:dx$$ subject to $y(0)=y(1)=0$ has no solutions. The Euler - Lagrange equation states that: $$\frac{d}{...
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Intuition behind the use of the Principle of Stationary Action in Classical Field Theory [duplicate]

Whilst studying Field Theory and after checking numerous sources it appears that people always just state the action without providing some sort of motivation/intuition as to why we should/can use the ...
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49 views

Equations of motion from action variation

I was reading about dilaton gravity in 2D, and I was trying to reproduce the equations of motion of a related theory. If I consider the following action: $$S = \int d^4x \sqrt{-g} e^{-2\phi}(R+4(\...
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2answers
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How does Hamilton's Principle give us the path taken?

We defined the action as: $$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$ where $q_i(t_1)$ and $q_i(t_2)$ are known and fixed. Hamilton's principle states that the path that is ...
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1answer
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How can Lagrangian method work whenever the Lagrangian is not convex?

Let $$L(x,\dot x)=\frac{1}{2}m\dot x^2-\frac{1}{2}k(x-x_0)^2-mgx$$ the Lagrangian of a system. Euler Lagrange theorem says that a necessary condition to be a minimizer is to satisfy Euler-Lagrange ...
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2answers
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History of Principle of Least Action [closed]

I am interested in how Lagrange came up with Principle of Least Action. Is it derived from some experimental data or by mathematical deduction? And how? Either seems hard. I hope this might ...
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1answer
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Lagrange's equation for system not having time translation

While we are deriving Lagrange's equation from D'Alembert's principle, when we argues as; $$\delta \vec r_\alpha = \sum_i \frac{\partial \vec r_\alpha}{\partial q_i }\delta q_i + \frac{ \partial \...
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Deriving Canonical Transformation from Generating Function using Principle of Stationary Action

In Hamill's "A Student's Guide to Lagrangians and Hamiltonians", section 5.2, the equations for a canonical transformation $(q,p) \to (Q,P)$, induced by the generating function $F(q,Q,t)$ are derived ...
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1answer
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Attaining extrema when a stationarity condition has no solution

I was wondering if someone could shed some light on the following for me: If a stationarity (maximizing or minimizing) condition has no solution inside a particular domain, then how do we reason that ...
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Paths of least action and loops in time

In the book Quantum Field Theory for the Gifted Amateur link: https://books.google.ca/books?hl=en&lr=&id=nIk6AwAAQBAJ&oi=fnd&pg=PP1&ots=JZjwG_qDt5&sig=...
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Why relativistic Lagrangian doesn't simply equal kinetic minus potential energy $L=T-V$?

As the question above, I wonder why the relativistic Lagrangian is written as: $$L=-mc² \sqrt{1-\frac{v²}{c²}} - V ~=~-\frac{mc^2}{\gamma} -V~?$$ I know that the kinetic energy of a relativistic ...
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0answers
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Transformation of ADM parameters under diffeomorphisms

I am trying to prove the invariance of the ADM formalism under (infinitesimal) diffeomorphisms. I have checked Wald and other textbooks on the subject but have been unable to find expressions for how ...
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Lagrangian for first order equation of motion? [duplicate]

Let us have the following equation of motion (it might not necessarily correspond to a physical system): $$\dot{x} + a \cdot x + b \cdot x^2 + c=0.$$ I would like to deduce the corresponding ...
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How can we predict how a system evolves using the stationary action principle even though we need to specify the final state? [duplicate]

The stationary action principle states that a system evolves between a fixed initial and fixed final configuration in such a way that the action is stationary. But isn't the final configuration what ...
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Rigid Body Equations in terms of Body Coordinates by Hamilton's Principle

I sought-for the equations of motion of an unrestrained rigid body. The equations of motion are readily available in the literature, but my concern is to derive them by Hamilton's principle. ...
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2answers
84 views

Approximating ground-state energy without using variational principle

Given the Hamiltonian for one dimension harmonic oscillator: $$H=-\frac{\hbar^2}{2\mu}\frac{d^2}{dx^2}+\frac{\mu\omega}{2}x^2 ,$$ I need to calculate the approximate ground state energy using the ...
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2answers
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Fermat's principle: when does light actually travel along the local maximum of accumulated phase? [duplicate]

In class we learned that Fermat's principle dictates that light travels either along a local minimum or a local maximum of the accumulated optical phase, but the professor only gave examples of local ...
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0answers
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Is Entropy a monotonically increasing function of Gibbs Free Energy/ Helmholtz free energy/ Enthalpy?

Entropy can be axiomatically taken as a monotonically increasing function of internal energy $(E) ,$ from where "energy minimum principle" can be deduced, and this can be stated using variational ...
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1answer
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D'Alembert's principle and equation of motion

Is obtaining proper equation of motion from D'Alembert's principle a mere coincidence or there is some logic behind this? This is asked because while we are finding the equations in a regime where ...
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4answers
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What is the physical content of the principle of least action?

Say the world is governed by the Principle of Least Action (or Hamiltonian mechanics) and let's not worry about quantum mechanics too much. Independently of any Lagrangian or Hamiltonian, does that ...
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1answer
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Scaling Problem with Variational Method

$\def\braket#1{\langle#1\rangle}$ I am attempting to solve a particular Hamiltonian by variational method. The wavefunction that I have selected is as follows: $$ \Psi = Ne^{\frac{-kr}{2}}\sum_{i=0}...
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0answers
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Physical Meaning of the Gutzwiller Constraints

I have a doubt on the constraints for the expecation values obtained by Bünemann et all. First i want to introduce my notation To analytically solve a tight-binding model, \begin{equation} \hat{H}= ...
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1answer
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Variation of action for massive point particle (pp)

So I'm pretty sure I'm missing something obvious, but for the life of me I cannot replicate the step between 1.2.2 and 1.2.3 in Polchinski Vol 1. Basically, I'm trying to find the variation of: $$S_{...
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1answer
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How can the action can describe a movement? What is the argument behind? [duplicate]

We define the action of a system as $$S(q)=\int_{t_1}^{t_2}L(t,q(t),q'(t))dt,$$ where $q(t)$ is the evolution of the system and $L$ is the Lagrangien. How can a stationary point of $S$ can describe ...