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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Calculus of variations -- how does it make sense to vary the position and the velocity independently?
In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat ...
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Why are there only derivatives to the first order in the Lagrangian?
Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, jerk,...) excluded?
Is there a good reason for ...
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How do I show that there exists variational/action principle for a given classical system?
We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's ...
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Why the Principle of Least Action?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action ...
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Why should an action integral be stationary? On what basis did Hamilton state this principle?
Hamilton's principle states that a dynamic system always follows a path such that its action integral is stationary (that is, maximum or minimum).
Why should the action integral be stationary? On ...
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Geodesic Equation from variation: Is the squared lagrangian equivalent?
It is well known that geodesics on some manifold $M$, covered by some coordinates ${x_\mu}$, say with a Riemannian metric can be obtained by an action principle . Let $C$ be curve $\mathbb{R} \to M$, $...
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How do non-conservative forces affect Lagrange equations?
If we have a system and we know all the degrees of freedom, we can find the Lagrangian of the dynamical system. What happens if we apply some non-conservative forces in the system? I mean how to deal ...
user58143
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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 A^\...
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Example in motivation for Lagrangian formalism
I started reading Quantum Field Theory for the Gifted Amateur by Lancaster & Blundell, and I have a conceptual question regarding their motivation of the Lagrangian formalism. They start by ...
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Derivation of Maxwell's equations from field tensor lagrangian
I've started reading Peskin and Schroeder on my own time, and I'm a bit confused about how to obtain Maxwell's equations from the (source-free) lagrangian density $L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$...
amc
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Lagrangian for relativistic massless point particle
For relativistic massive particle, the action is
$$\begin{align}S ~=~& -m_0 \int ds \cr
~=~& -m_0 \int d\lambda ~\sqrt{ g_{\mu\nu} \dot{x}^{\mu}\dot{x}^{\nu}} \cr
~=~& \int d\lambda \ L,\...
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Is the principle of least action a boundary value or initial condition problem?
Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating:
In analytic (Lagrangian) mechanics, the derivation of the Euler-...
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Why one should follow Snell's law for shortest time?
whenever two media and two velocities are involved, one must follow Snell's law if one wants to take the shortest time.
Why snells law must be followed to travel diffrent media in shortest time?
...
user102917
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On a trick to derive the Noether current
Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$.
The trick to get the Noether current consists in making the ...
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Confusion regarding the principle of least action in Landau & Lifshitz "The Classical Theory of Fields"
Edit: The previous title didn't really ask the same thing as the question (sorry about that), so I've changed it. To clarify, I understand that the action isn't always a minimum. My questions are in ...
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Is it possible to prove that planets should be approximately spherical using the calculus of variations?
Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical?
Let's assume that a planet is 'born' when lots of ...
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What variables does the action $S$ depend on?
Action is defined as,
$$S ~=~ \int L(q, q', t) dt,$$
but my question is what variables does $S$ depend on?
Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \frac{dq}{dt}$?
In Wikipedia I've ...
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Variational principle for a point particle (massive or massless) in curved space
We know that for a point particle, the action is
$$ S[x,e] ~=~ \frac{1}{2}\int_{\lambda_A}^{\lambda_B} d\lambda\left[e^{-1}(\lambda)~g_{\mu\nu}(x(\lambda))~\dot{x}^\mu(\lambda)~\dot{x}^\nu(\lambda) -(...
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Hamilton's principle with nonholonomic constraints in Goldstein
I am studying from Goldstein's Classical Mechanics, 3rd intl' edition, 2013. In section 2.4, he discussed Hamilton's principle with nonholonomic constraints. The constraints can be written in the form ...
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Type of stationary point in Hamilton's principle
In this question it is discussed why by Hamilton's principle the action integral must be stationary. Most examples deal with the case that the action integral is minimal: this makes sense - we all ...
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Can Lagrangian be thought of 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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Noether's current expression in Peskin and Schroeder
In the second chapter of Peskin and Schroeder, An Introduction to Quantum Field Theory, it is said that the action is invariant if the Lagrangian density changes by a four-divergence.
But if we ...
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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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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 ...
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Proof that total derivative is the only function that can be added to Lagrangian without changing the EOM
So I was reading this: Invariance of Lagrange on addition of total time derivative of a function of coordiantes and time and while the answers for the first question are good, nobody gave much ...
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How to derive Maxwell's equations from the electromagnetic Lagrangian?
In Heaviside-Lorentz units the Maxwell's equations are:
$$\nabla \cdot \vec{E} = \rho $$
$$ \nabla \times \vec{B} - \frac{\partial \vec{E}}{\partial t} = \vec{J}$$
$$ \nabla \times \vec{E} + \frac{\...
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When/why does the principle of least action plus boundary conditions not uniquely specify a path?
A few months ago I was telling high school students about Fermat's principle.
You can use it to show that light reflects off a surface at equal angles. To set it up, you put in boundary conditions, ...
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When is the principle of stationary action not the principle of least action?
I've only had a very brief introduction to Lagrangian mechanics. In a physics course I took last year, we briefly covered the principle of stationary action --- we looked at it, derived some equations ...
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Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?
All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before.
Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
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Several stationary points of the action functional
In QFT the principle of stationary action states that we choose fields that will make the action stationary but what if the action has many stationary points (for a fixed choice of boundary conditions)...
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What canonical momenta are the "right" ones?
I'm doing some classical field theory exercises with the Lagrangian
$$\mathscr{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}$$
where $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. To find the ...
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Functional Derivative in the Linear Sigma Model
In the linear sigma model, the Lagrangian is given by
$$ \mathcal{L} = \frac{1}{2}\sum_{i=1}^{N} \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) +\frac{1}{2}\mu^2\sum_{i=1}^{N}\left(\...
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Retrieving Maxwell's equations from the minimum action principle
I'm currently working at the start of Alexei Tsvelik's book Quantum Field Theory in Condensed Matter Physics. I'm kinda stumped on a few essential steps.
Starting with the action:
$$S = \int dt \int ...
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Functional derivative in Lagrangian field theory
The following functional derivative holds:
\begin{align}
\frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t')
\end{align}
and
\begin{align}
\frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t')
\end{...
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Least-action classical electrodynamics without potentials
Is it possible to formulate classical electrodynamics (in the sense of deriving Maxwell's equations) from a least-action principle, without the use of potentials? That is, is there a lagrangian which ...
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What is the actual form of Noether current in field theory?
Let us consider $N$ independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a region $\Omega$ in a $D$-...
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When can we add a total time derivative of $f(q, \dot{q}, t)$ to a Lagrangian?
The other day, I was listening to this lecture on the Lagrangian for a charged particle in an electromagnetic field, and at one point in the video, the lecturer mentions that we can add any total time ...
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Derivation of Euler-Lagrange equations for Lagrangian with dependence on second order derivatives
Suppose we have a Lagrangian that depends on second-order derivatives:
$$L = L(q, \dot{q}, \ddot{q},t).\tag{1}$$
If we're working on the variational problem for this Lagrangian, then I know that we'...
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Optics: Derivation of $\vec\nabla{n} = \frac{d(n\hat{u})}{ds}$
I have been given this formula from optics here, with no background:
$$\vec\nabla{n} = \frac{d(n\hat{u})}{ds}$$
Where $n$ is the refractive index and $\hat{u}$ is a unit vector tangent to the path $...
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Variation of a term in the Lagrangian
I don't understand why
$$\frac{\delta}{\delta\phi}\left(\frac12\partial^\mu\phi\partial_\mu\phi\right)~=~\partial^\mu\partial_\mu\phi.\tag{1}$$
If we use integration by parts, there should be a minus ...
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Is there some connection between the Virial theorem and a least action principle?
Both involve some 'averaging' over energies (kinetic and potential) and make some prediction about their mean values. As far as the least action principles, one could think of them as saying that the ...
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Geodesics: Straightest or Shortest? When and Why?
In classical General Relativity (meaning not modified) one can think of geodesics in two ways.
One way is to say that a geodesic is the curve which is the straightest (in analogy with the flat case) ...
user78618
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Theory invariance after substitution of theory's field equations back into theory's action functional?
Suppose I have a theory $A$ concerning the evolution of a set of fields $T_1, \dots, T_n$. Let the action functional for this theory be $S[T_1, \dots, T_n]$. Suppose in the action, in addition to ...
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Euler's equations of rigid body motion from least action principle
I would like to derive Euler's equations of rigid body motion from least action principle.
Suppose we are in free space so we have no gravity so Lagrangian is equal to kinetic energy.
$$
L = T = \...
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Explicit Variation of Gibbons-Hawking-York Boundary Term
Are there any references that present the explicit variation of the Hilbert-Einstein action plus the Hawking-Gibbons-York boundary term, and demonstrate the cancellation of the normal derivatives of ...
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Variational Derivation of Schrodinger Equation
In reading Weinstock's Calculus of Variations, on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles.
Unfortunately I don't ...
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Why is it so coincident that Palatini variation of Einstein-Hilbert action will obtain an equation that connection is Levi-Civita connection?
There are two ways to do the variation of Einstein-Hilbert action.
First one is Einstein formalism which takes only metric independent. After variation of action, we get the Einstein field equation.
...
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When is numerical value of Lagrangian evaluated on-shell a full differential?
I noticed recently that for many field equations, Lagrangian evaluated on-shell (i.e. using equations of motions) is a full derivative- a divergence or something, or in other words a boundary term. ...
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Must the action be a Lorentz scalar?
Page 580, Chapter 12 in Jackson's 3rd edition text carries the statement:
From the first postulate of special relativity the action integral must be a Lorentz scalar because the equations of motion ...
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How is Fermat's least time principle proven?
How is Fermat's least time principle proven? Or it is what usually is observed and is basis for the theories?
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