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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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Is it possible for the Action $S$ to *not* have a stationary point?

So the path of an object in configuration space is given by Hamilton's principle, which states that the path which the particle travels on is the one on which the action is stationary: $$\delta S = \...
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Curve for fastest time down a ramp [duplicate]

I came across a physics experiment video showing three balls released from a point A, going down three different kinds of ramps leading to a point B (https://www.youtube.com/watch?v=61S0KW7e-rc) ...
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Example in which light takes the path of maximum optical length [duplicate]

According to the modern version of Fermat's principle,"A light ray in going from point A to point B must traverse an optical path length that is stationary with respect to variations of that path.".Is ...
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How to deal with explicit time dependence of the Lagrangian?

Clearly, if the Lagrangian in explicitly time dependent, the Euler-Lagrange equations being satisfied does not extremise the action. I am unclear as to how to deal with systems with an explicitly time-...
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Deriving the geodesic equation using a Lagrange multiplier to fix affine parametrisation

The geodesic equation can be derived using the action $$S_0 ~=~ \int d\tau \sqrt{-\dot{x}_\mu\cdot \dot{x}^\mu}\tag{1}$$ (I am using the (-+++) convention and $\dot{x} = \frac{dx}{d\tau}$). To ...
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Polyakov Lagrangian and Lagrange multipliers

I'm reading Polchinski's Introduction to String Theory (volume I) and something got me quite puzzled in the beginning (At the top of page 19 to be precise). This part is about the open string and the ...
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Doubt in notation of Variation with respect to a function

I cannot find this notation used anywhere on the internet or on SE (maybe I am searching using wrong tags). Hence, I am asking this question here. I don't know whether this even qualifies as a valid ...
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When can we say $x$ and $p$ are “independent variable”, in order to find the Vlasov equation?

I have a question about "independent variable" in physics, and especially variable in Lagrangian or Density Function. I read several questions about it in this forum and although I have the feeling I ...
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Must the varied paths in the action be physically possible?

For simplicity without loss of generalization, consider a free particle. When using the Principle of Least Action, I imagine all variations of the true path between $t_1, t_2$ regardless of whether ...
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Relativistic action is a constant?

Say that you want to find the equations of motion of a free relativistic massive point particle by minimizing the action $$S=-m\int\mathrm{d}\tau\,\sqrt{\eta_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}\...
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What does it mean for a force to 'produce' virtual displacement?

Book: Variational Principles of Mechanics by Lanczos (page 80) Statement: "Two systems of forces which produce virtual displacements are dynamically equivalent." I don't understand the part about ...
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Calculus of Variations. Finding the extremals of a perturbed Lagrangian [closed]

Im trying to solve the following problem: Approximate with an error of $O(\epsilon ^3)$ the extremals of the Lagrangian $$L(y,y',x) = y^2 + (y')^2 - 2y \sin(x) + \epsilon y^3$$ with $y(0)=1$ ...
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Schwinger's variation of the action of point particle with *both* time and position as independent variables

In Chapter 8, pages 86-87, equations (8.5)-(8.11) of Julian Schwinger et al., Classical Electrodynamics, the equations of motion for the following action principle of a point particle in an external ...
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Independence of generalized coordinates and generalized velocities [duplicate]

I think this might be a very basic doubt, but in the Lagrangian method of classical mechanics, we assume the generalized coordinate $q_{i}$ and the corresponding velocity $\dot q_{i}$ are independent. ...
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Is action for free particle really minimal?

On my mechanics classes I have a problem: show, that the action for free non-relativistic particle $$S=\int\limits_{t_i}^{t_f}\frac{m\dot{x}^2}{2}dt\tag{1}$$ is really the least (but not maximal). ...
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When and why is $\frac{d}{dt}\delta q^{i}=\delta \frac{dq^{i}}{dt}$ true? [duplicate]

Apparently my question is different from Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}δq=δ\frac{dq}{dt}$. I hadn't noticed because the answer given in the comments to this question was ...
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Functional Poincaré's lemma and the inverse Lagrangian problem

I have only encountered the inverse Lagrangian problem in mathematics books that treat Lagrangian field theory using jet bundles and homological algebra, and while I am studying this approach, I still ...
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What is the principle of least action? [duplicate]

I want to understand the principal of least action intuitively, away from any mathematical proof.
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Why are action principles so powerful and widely applicable?

I've been trying to wrap my head around Lagrangian mechanics and Lagrangians in general, and I've found it difficult. After some thinking, I believe that the issue I have is with action principles. ...
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Principle of least action for dissipative systems [duplicate]

I am trying to solve a problem with involves friction, and am having trouble solving it with $F=ma$. I am considering to use Lagrangian mechanics, but I am puzzled. There doesn't seem to be a ...
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For the Lagrangian stationary action formula does the eta function for a specific path vary the distance from the true path? [closed]

This question can apply to any variation calculus problem although it has come up in my case for the stationary action principle so I will stick to the application I am using it for. The action is ...
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Euler-Lagrange equations using $\vec{E}$ and $\vec{B}$ instead of $A^\mu$ [duplicate]

We all know that the lagrangian for the free electromagnetic field is given by $$ \mathscr{L} = -\frac{1}{4}F^{\mu \nu}F_{\mu \nu} $$ where $ F^{\mu \nu} = \partial^\mu A^\nu -\partial^\nu A^\mu $ is ...
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Variational wave functions in many-body physics

One of the very famous variational wave functions is Gutzwiller wave function (GWF) which explained Mott-insulator transition back in 60s/70s. It is analoguous to the idea of Projector Monte Carlo ...
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Extremum of the action functional [duplicate]

Is there an example where a classical particle follows a path of maximal action rather than that of minimum action?
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Shape of a compressed wristband

What is the curve of the top part of this wristband after I squish the two ends closer? It's a curve of fixed length with given start $(x_1, 0)$ and end $(x_2, 0)$, and zero slope at these points, ...
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Is Feynman wrong about the principle of virtual work to find forces in a dielectric?

In The Feynman Lectures on Physics, Vol II, 10–5 Fields and forces with dielectrics describes a method for finding the force between two charged conductors in a dielectric. I accept the first part of ...
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Schrödinger's variational method

In Schrödinger's Quantisation as an Eigenvalue Problem he solves the Hydrogen atom through a precursor of Schrödinger's Equation, derived from the Hamilton-Jacobi equation through a variational method ...
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Equations of motion from Polyakov action, before choosing the conformal gauge

My question is the following: It is usual in the standard textbooks to firstly choose a gauge (usually the conformal gauge) and then extract the equations of motion from the Polyakov action by ...
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Palatini action: variation of spin connection: show that torsion vanishes

Consider the tetrad-Palatini action: $$S[e,\omega] = \int e \wedge e \wedge F[\omega]^\star,$$ where $\star$ denotes the Hodge dual, i.e. $F_{IJ}^\star = \frac{1}{2} \varepsilon_{IJKL} F^{KL}$. The ...
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Can the Euler-Lagrange equation be used to derive the stationary action formula? [duplicate]

From what I understand I can use the Euler-Lagrange equation to find the function ( Let us call L. ) where L can be the function as stated in the action formula. But how difficult is it to actually ...
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Can it be shown that the principle of least action applies to problems with no analytical solutions?

After thinking about the earlier version of my question, I realize that what I’m really asking about is whether it can be shown that the principle of least action applies to systems without an ...
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Euler-Lagrange equation in General Relativity

In Relativity the Lagrangian of a free particle is \begin{align} \mathcal L=\sqrt{g_{ab}\frac{dx^a}{d\tau}\frac{dx^b}{d\tau}}\end{align} Since $\mathcal L$ is parameterization invariant we can always ...
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General relativity - scalar gravitational field, variation principle

I have a basic question about the variation principal when applied to a scalar gravitational field in general relativity. Consider the action $$S_M = \int d^4 x\sqrt{|g|}g^{uv}\partial_u \phi\...
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Why does Fermat's principle (optics) not apply to all paths?

Feynman's statement of Fermat's Principle regarding optics is the following, "a ray going in a certain particular path has the property that if we make a small change (say a one percent shift) in ...
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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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From non-relativistic to relativistic action

There is a derivation of relativistic action that treats space and time symetricaly which is just playing arround with the square of kinetic energy in the non-relativistic action and plugging in speed ...
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Does the integral in the action formula regarding the principle of stationary action represent an area or a length?

I am referring to the Feynman Lectures. The second volume has the "Principle of Least Action" as one of his lectures. (See after the 2nd paragraph below figure 19-6.) Although he does not explicitly ...
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Matrix Euler’s rigid-body equation

Define the action $$S[g]=\displaystyle\frac{1}{2}\int^1_0 Tr(I(g^{-1}\dot g)~g^{-1}\dot g)~dt.$$ $I:SO(N)\to SO(N)$ denotes the endomorphism $\omega \to I(\omega)$ with $I(\omega)_{ij}=\omega_{ij}/...
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Why is the formula for stationary action expressed as kinetic minus potential energy instead of potential minus kinetic energy?

I am sure this is a duplicate but I could not spot it exactly. And I am sure folks have covered this topic online here in great detail. I am referring to the Lagrangian here in the "Action" formula ...
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Why isn't it important, after which coordinates the Variation of the action integral is done?

I often read,that if the lagrangian $L=p\dot{q}-H$ of a pair of coordinates in phase space $(q,p)$ and $P\dot{Q}- K $, for some new pair of coordinates $(Q,P)$ only differ by a total time derivative $...
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Is the Euler-Lagrange equation a special case of the principle of least action?

Is the Euler-Lagrange equation a special case of the principle of least action? I have some confusion after reading a few dozen stackexchange articles of the "principle of least action". I follow the ...
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System of forces reducible to single force

I'm self-studying Lanczos book The Variational Principles of Mechanics and in the chapter on the principle of virtual work there's a problem Show that any given system of forces acting on a rigid ...
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How is the problem of finding an object's motion an functional extrema problem? [duplicate]

I'm learning about Lagrangian mechanics and I've learned about the Lagrangian. I've read a proof of the fact that if $$J = \int_a^b F(x,f(x),f'(x) dx .$$ Then the function $f(x)$ that extremizes ...
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Variational calculus and KKR method for band structure calculation

I’ve been studying the KKR method from the original Kohn and Kostoker’s paper (https://journals.aps.org/pr/abstract/10.1103/PhysRev.94.1111). On the text, they use variational calculus for dealing ...
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Does the principle of least action assume the kinetic energy law as an axiom? [closed]

I should probably wait till I finish the complete derivation before asking this. but maybe it will help me to know ahead of time. I like to always know what the axiom's are before I start a topic and ...
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Variation of gamma and Christoffel

I am trying to experiment with Lagrangian densities and came across a term similar to $$\gamma^i\Gamma^j_{ik}A^k$$ where the $\gamma$ are the gamma matrices, $\Gamma$ are the Christoffel Symbols and $...
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Variational principle, functional gradient

Given the energy functional $$E[\Psi] = \frac{\langle \Psi \vert H \vert \Psi \rangle}{\langle \Psi \vert \Psi \rangle},$$ its functional gradient is $$\frac{\delta E[\Psi]}{\delta \langle \Psi \vert}...
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Euler-Lagrange equations for free rigid body [closed]

How do I derive Euler's equations of motion for a free rigid body using a Lagrangian formulation? The required equations are, in vector form, $J\dot{\omega} = -\omega \times J\omega$ where $J$ is ...
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Variational Principle to find the Mean Field Bose-Hubbard Ground State

I'm trying to find the ground state energy of the Bose-Hubbard in the context of a mean-field approximation. Assuming small fluctuations, the mean field Hamiltonian can be decoupled into the sum of ...
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How to find the Lagrangian of this system?

I am trying to find the Lagrangian $L$ of a system I am studying. The equations of motion is: $$\left\{ \begin{array}{c l} r \ddot{\phi} + 2\dot{r} \dot{\phi}+k(r) \cdot r \dot{r} \dot{\phi} = ...