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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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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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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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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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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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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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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 ...
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Geodesic curve definition [duplicate]

Do we have a choice in defining the covariant derivative by the use of a set of coefficient functions(Christoffel gammas)? If so, could we then say that these coefficient functions need not to ...
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How does Fermat's principle of least time come from this statement? [duplicate]

In Wikipedia Fermat's Principle is stated as: A ray of light prefers the path such that there are other paths, arbitrarily nearby on either side, along which the ray would take almost exactly the ...
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Geodesic equations from action with auxiliary field

A textbook says that the geodesic equations (for both massive and massless) can be derived from the following action: $$ S = -\frac{1}{2} \int d\tau \:\eta \: (\eta^{-2} \dot{x}^\mu \dot{x}^\nu g_{\...
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If there is no destination does light actually get emitted? [closed]

If the action of least principle says light needs a destination what happens when we shine a laser into the dark region of the microwave background radiation? Alternatively, if we put a light source ...
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Least action in differential steps

It says here that the entire integral is minimum if and only if the actions for each step is minimum but here is a contradiction. Suppose there are two fixed points in space, time: A and C. it takes ...
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Why does light go in the path where time is minimum and not maximum?

On deriving Snell's Law from Fermat's Principle there is a part where $\frac{ds}{dt}=0$ where $s$ is the distance gone by light. But the principle states that light takes the path where it takes ...
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Feynman Lecture Principle of Least Action: Glossed over Taylor expansion?

His initial one dimensional derivation of Newton's Second Law using the Principle of Least Action, I believe is fairly concise and easy to read. However, I did get hung up on his use of the Taylor ...
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Lagrangian of free particle - classical case

I have a question, more related to a mathematical aspect of physics, which seems I am not understanding very well. So, by applying Galilean transformation between two reference frames, which move at ...
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2answers
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Why the Lagrangian of a free particle cannot depend on the position or time, explicitly?

On p. 5 in $\S$3 pf the book of Mechanics by Landau & Lifshitz, it is claimed that [...] for a free particle, the homogeneity of space and time implies that Lagrangian cannot depend on ...
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Overconstrained equilibrium binary mixture phase seggregation

Assume a binary mixture, with the component $A$ in concentration $c$ and the component $B$ in concentration $1-c$. The total energy $E(c)$ is thus given by a function of one variable $c$ (if we do not ...
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Maxwell Lagrangian where $F$ and its derivatives are the variables (i.e., without replacing $F={\rm d}A$) [duplicate]

The way the electromagnetic Lagrangian is usually constructed is by noticing that the EM fields are always constrained to satisfy ${\rm d}F=0$ (half of Maxwell's equations). We can immediately solve ...
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3answers
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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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2answers
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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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2answers
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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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1answer
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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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1answer
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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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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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1answer
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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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1answer
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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? [duplicate]

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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1answer
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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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2answers
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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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1answer
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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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1answer
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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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1answer
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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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2answers
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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 ...