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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115 views

Lagrangian for relativistic massless particle

For relativistic massive particle, the action is $$S ~=~ -m_0 \int ds ~=~ -m_0 \int d\lambda ~(\dot x ^\mu \dot x_\mu)^{\frac{1}{2}} ~=~ \int d\lambda \ L,$$ where $ds$ is the proper time of the ...
1
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0answers
67 views

Why does Principle for least action hold for classical fields [duplicate]

Let $\mathscr L (\phi(\mathbf x), \partial \phi(\mathbf x))$ denote the Lagrangian density of field $\phi(\mathbf x)$. Then then actual value of the field $\phi(\mathbf x)$ can be computed from the ...
2
votes
1answer
34 views

Evolution of minimization of surface tension

What are governing equations (or/and variational principles) for evolution of a simply connected body of water in vacuum? Initial state - for time $t=0$ we have a bounded simply connected set ...
9
votes
1answer
115 views

Is it possible to prove that planets should be approximately spherical using the calculus of variations?

Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical? Lets assume that a planet is 'born' when lots of ...
0
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2answers
88 views

Derivation of Lagrangian?

I know that the Lagrangian $L$ is defined to be $T-V$, i.e. the difference between kinetic energy and potential energy. Also the Action $S$ is defined to be $\int Ldx$ and from this we can derive ...
3
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1answer
55 views

Field equations in extended EH-GHY action. Is Schwarzschild a solution?

When taking the EH action, $$S_{EH} = \frac{1}{16\pi G}\int_M d^4x \sqrt{-g}R$$ and making a small variation in the metric while ignoring boundary terms, we obtain $$\delta S_{EH} = \frac{1}{16\pi ...
13
votes
4answers
281 views

The Lagrangian as a metric

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
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0answers
61 views

Variational principle

In the LMTO method, the interstitial region is approximated by plane waves and the muffin tin region of the potential by solutions to the radial Schrodinger equation. In using the variational method ...
4
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1answer
102 views

Variational calculus problem

I've been wondering lately about a problem that comes from, among other places, an old video game "Lunar Lander". In the game there is a spaceship that has a small tank of fuel, and you're supposed to ...
6
votes
1answer
94 views

Intuition for actions written as integrals over spacetime

Right now I'm simply looking for an intuitive explaination of actions that integrate over a 4-volume element, $d^4x$ rather than a parameter say $\lambda$. More specifically I'm well versed in action ...
3
votes
2answers
183 views

Fermat's principle to prove the law of reflection

Fermat's principle tells that a light ray will follow a path from point $A$ to point $B$ so that the optical path length of this path is an extremum over neighboring paths. I wanted to use this ...
3
votes
1answer
69 views

weak solution of Schroedinger equation .. are they useful?

We know that Schroedinger equation can be deduced from a variational principle (non relativistic Schroedinger equation). Assuming this I have 2 questions: a) Using variational methods, could we ...
4
votes
1answer
89 views

How Hamilton's Principle was found?

Hamilton's principle states that the actual path a particle follows from points $p_1$ and $p_2$ in the configuration space between times $t_1$ and $t_2$ is such that the integral $$S = ...
3
votes
1answer
137 views

Is there a Lagrangian whose Euler-Lagrange equation is the gradient?

I am trying to recast a problem I am working on in terms of Lagrangian mechanics. I am in the following situation. Suppose I have a function $f:X \rightarrow \mathbb{R}$ (a field). In the its ...
2
votes
0answers
59 views

equation of motion for the scalar field via variational principle in general relativity

I would like to find the equation of motion for the scalar field $\phi$ by varying the following action in General Relativity. Special Relativity: $$ S = -\tfrac{1}{2}\int d^4\xi\, \eta^{ab} ...
4
votes
2answers
108 views

Geodesics equations via variational principle

I would like to recover the (timelike) geodesics equations via the variational principle of the following action: $$ \mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}} $$ ...
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0answers
39 views

Helmholtz free energy minimization using Lagrange [closed]

can you please help me with that one: Minimize free energy for a liquid crystal: $F = \int (K_{11} (div(n))^2 + K_{22} (n*rot(n) + q)^2 + K_{33} (n*rot(n))^2 ) dV$ in the case $n = cos(\alpha)*e_z ...
10
votes
2answers
333 views

Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $

I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are ...
1
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1answer
145 views

Variation of modified Einstein Hilbert Action

In general relativity one can derive the Einstein Field Equations by the principle of least action through variations with respect to the inverse of the metric tensor. In some modified theories of ...
2
votes
1answer
51 views

Classical Mechanics & Coordinates [closed]

What is the meaning generalised coordinates in Classical Mechanics? How is Lagrangian formalism different from Hamiltonian formalism? How are they related to Hamilton's Principle? How are they ...
0
votes
1answer
111 views

Can't understand the principle of least action [closed]

I tried many hours to understand the principle of least action, and those hours become days... and I still didn't understand that principle/ and how it relates to Newtonian mechanics? Could someone ...
1
vote
0answers
64 views

How to get “massless” equation of motion from the action of Nordstrom scalar field theory?

There is Nordstrom theory of the particle moving in a scalar field $\varphi (x)$: $$ S = -m\int e^{\varphi (x)}\sqrt{\eta_{\alpha \beta}\frac{dx^{\alpha}}{d \lambda}\frac{dx^{\beta}}{d ...
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0answers
68 views

Derivation of equations of motion in Nordstrom's theory of scalar gravity?

Nordstrom's theory of a particle moving in the presence of a scalar field $\varphi (x)$ is given by $$ S = -m\int e^{\varphi (x)}\sqrt{\eta_{\alpha \beta}\frac{dx^{\alpha}}{d ...
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vote
2answers
80 views

With respect to what quantities do I vary Lagrangians in field theory?

I have recently been wondering, with respect to which quantities (covariant or contravariant) one should vary QFT Lagrangians and whether there is some rule regarding this. Let me give an example ...
2
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1answer
55 views

Question about “different” equations of motion in dependence of indices

Let's have the action $$ S = \int (\partial_{\mu}h^{\mu \sigma}\partial^{\nu}h_{\nu \sigma} - \Lambda h^{\mu \nu}T_{\mu \nu}) d^{4}x. $$ For definiteness, $$ h_{\mu \nu} = h_{\nu \mu} , \quad T_{\mu ...
7
votes
1answer
209 views

Virial theorem and variational method: a question

I have an hydrogenic atom, knowing that its ground-state wavefunction has the standard form $$ \psi = A e^{-\beta r} $$ with $A = \frac{\beta^3}{\pi}$, I have to find the best value for $\beta$ ...
0
votes
1answer
55 views

Why we can set variations for the metric and its derivatives to zero at infinity?

This question is the continuation of the following one. I still don't understand why $(1)$ may be set to zero. This refers to the zero value variations of metric and its derivatives on the infinitely ...
1
vote
1answer
168 views

Einstein action and the second derivatives

I have naive question about Einstein action for field-free case: $$ S = -\frac{1}{16 \pi G}\int \sqrt{-g} d^{4}x g^{\mu \nu}R_{\mu \nu}. $$ It contains the second derivatives of metric. When we want ...
3
votes
1answer
203 views

Point of Lagrange multipliers

I am trying to understand how for a constrained system the introduction of Lagrange multipliers facilitates the incorporation of the holonomic constraints. I am using Classical Mechanics by John ...
2
votes
1answer
271 views

How to obtain the field equations in Brans-Dicke theory from the action?

The action for the Brans-Dicke-Jordan theory of gravity is $$ \\S =\int d^4x\sqrt{-g} \; \left(\frac{\phi R - \omega\frac{\partial_a\phi\partial^a\phi}{\phi}}{16\pi} + \mathcal{L}_\mathrm{M}\right). ...
3
votes
1answer
115 views

Why can we assume independent variables when using Lagrange multipliers in nonholonomic systems?

I'm studying from Goldstein's Classical Mechanics. In section 2.4, he discusses nonholonomic systems. We assume that the constraints can be put in the form $f_\alpha(q, \dot{q}, t) =0$, $\alpha = 1 ...
2
votes
1answer
128 views

Finding the approximate solution for Schrodinger equation by using variational method [closed]

I need to find the approximate solution of nonlinear Schrodinger equation $$ i\hbar \partial_{t} \Psi + \frac{\hbar^{2}}{2m}\Delta \Psi - g |\Psi|^{2}\Psi - \frac{m\omega^2 (x^2 + y^2 + z^2)}{2}\Psi = ...
3
votes
1answer
96 views

Hamilton-Jacobi formalism and on-shell actions

My question is essentially how to extract the canonical momentum out of an on-shell action. The Hamilton-Jacobi formalism tells us that Hamilton's principal function is the on-shell action, which ...
1
vote
2answers
187 views

Einstein equation and scalar field stress-energy tensor

Let's have interaction between gravitational and scalar real fields. For an action of gravitational field in vacuum I add term $S_{m} = \int d^{4}x\sqrt{-g}L_{m}$, where $$ L_{m} = \frac{1}{2}g^{\mu ...
10
votes
2answers
291 views

Is Einstein-Hilbert action the unique action whose variation gives Einstein's field equations?

I know that scaling the action with a non-zero multiplicative constant, or adding a total divergence term to the Lagrangian density do not change the Euler-Lagrange equations, cf. e.g. this ...
2
votes
3answers
194 views

Does a four-divergence extra term in a Lagrangian density matter to the field equations?

Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation: ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu ...
4
votes
1answer
64 views

Why does the minimum energy field configuration require the fields to be constant?

I am having a hard time in understanding a well known statement always made in the context of field theory. Background Consider a classical real scalar field theory with Lagrangian density given by ...
2
votes
2answers
164 views

Exercise about Lagrange-Euler equations

I'm solving an exercise about the Lagrange-Euler equations, that states the following: Let $\gamma (t) = \{ (t,q) : q = q(t), t_0 \leq t \leq t_1\}$ be a curve in $\mathbb{R} \times \mathbb{R}^2$. ...
1
vote
2answers
149 views

Refraction of light in medium

Given that the plane $y=0$ separates the vacuum ($y>0$) from the optical medium ($y<0$), I would like to calculate the trajectory of a light ray starting at the point $(x_1,y_1)$ and ending in ...
8
votes
4answers
391 views

D'Alembert's Principle: Necesssity of virtual displacements

Why is the D'Alembert's Principle $$\sum_{i} ( {F}_{i} - m_i \bf{a}_i )\cdot \delta \bf r_i = 0$$ stated in terms of "virtual" displacements instead of actual displacements? Why is it so necessary ...
3
votes
2answers
107 views

Non-uniqueness of solutions in Newtonian mechanics

In The Variational Principles of Mechanics by Lanczos, in section 1 of Chapter 1, Lanczos states that for a complicated situation, the Newtonian approach fails to give a unique answer to the problem, ...
4
votes
1answer
90 views

Variational principles: Meniscus

In determining the shape of a meniscus, we have to minimize the energy per unit length along the direction perpendicular to the cross-section of the meniscus: $$\frac{E}{L}=\int^L_0 dx [\gamma ...
6
votes
2answers
360 views

Functional derivative in Lagrangian field theory

The following functional derivative holds: \begin{align} \frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t') \end{align} and \begin{align} \frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t') ...
6
votes
2answers
313 views

Derivation of the Polyakov Action

As is usually done when first presenting string theory, the Nambu-Goto Action, $$ S_{\text{NG}}:=-T\int d\tau d\sigma \sqrt{-g} $$ ($g:=\det (g_{\alpha \beta})$ is the induced metric on the ...
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vote
1answer
93 views

Dimension analysis in Derrick theorem

The following image is taken from p. 85 in the textbook Topological Solitons by N. Manton and P.M. Sutcliffe: What I don't understand from the above statement: why $e(\mu)$ has minimum ...
1
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0answers
63 views

Deriving the curve of a cantilever

Essentially, there is a beam of length L and negligible mass sticking out of a wall with a mass Mg hanging at the end of it. We are given an equation for elastic energy (which I don't think needs to ...
2
votes
1answer
165 views

Motivating the Legendre Transform Mathematically

If I begin with a functional of the form $$J[y] = \int_a^b f(x,y,y')dx$$ and find its Euler-Lagrange equations $$\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'} = 0 = ...
1
vote
1answer
127 views

Does gravitational lensing violate Fermat's Principle that light must travel in straight lines?

Does bending of light due to warping of space violate Fermat's Principle or is it that in the principle light goes in a straight line with respect to space (taking space as the reference) and in ...
3
votes
1answer
265 views

Euler's equations of rigid body motion from least action principle

I would like to derive Euler's equations of rigid body motion from least action principle. Suppose we are in free space so we have no gravity so Lagrangian is equal to kinetic energy. $$ L = T = ...
5
votes
1answer
270 views

What is Maupertuis' principle good for?

The strength of Hamilton's principle is obvious to me and I see the advantage. Now, for conservative systems we also have Maupertuis' principle that says: $$ \delta \int p dq =0$$ and I am not sure ...