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Questions tagged [action]

The action is the integral of the Lagrangian over time, or the integral of the Lagrangian Density over both time and space.

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Does it make sense to say that the action is even or odd under time reversal?

The action of a system in mechanics is an integral over time defined as $$S[x(t)]=\int\limits_{t_1}^{t_2}L(x,\dot{x},t)dt.$$ Here, the time $t$ is integrated making the left hand side depend only on ...
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Why are these two variables being treated differently in the action?

I'm trying to understand the derivation provided in the section 2.4 of this paper. I have modified the notation and cut out the unimportant parts of the equations for clarity purposes, but for ...
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What's the partition function and action of thermofield double state (TFD)?

We know that TFD state is consisted of two CFT at each side, and their partition functions are $$ Z_{L/R}= {\rm tr}\,e^{-\beta H_{L/R}}=\int e^{-S_E}\,\,\,, $$ where $S_E$ is just Euclidean action of ...
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A simple question about equation of motion in polchinski's String theory?

In page 14 to get the equation of motion, it takes the variation of the action $$ S_P[X,\gamma]=-\frac{1}{4\pi\alpha'}\int_Md\tau d\sigma(-\gamma)^{1/2}\gamma^{ab}\partial_a X^\mu\partial_b X_\mu $$ ...
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Motivation behind action when deriving ''Strings as Harmonic oscillators" in Zwiebach's book on String theory

Page 248 gives us this action and he simply says that we will assume it correct. $$ S=\int d \tau d \sigma ~\mathcal{L}=\frac{1}{4 \pi \alpha^{\prime}} \int d \tau \int_{0}^{\pi} d \sigma\left(\dot{X}...
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Why is there a Lagrangian? [duplicate]

In all discussions regarding the Lagrangian formulation it has always been said that $L = T - V $, only is a correct guess that when operated via through the Euler -Lagrange equation yields something ...
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Derivative with respect to a coordiante differential (geodesic equation)

If the arc length is chosen to be the action integral, that is $$ S=\int \sqrt {g_{kn}\frac{dx^k}{ds} \frac{dx^n}{ds}} dx \tag{11.13} $$ Then Lagrangian is given by $$L=\sqrt {g_{kn}\frac{dx^k}{ds}...
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Inconsistency? Lagrangian with its Euler–Lagrange equation as condition

Consider the action $$A_{1} = \int{L(q, \dot{q})}{dt}\tag{1}$$ and the corresponding Euler–Lagrange equation $$\frac{\partial{L}}{\partial{q}} - \frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{q}...
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Einstein-Palatini action in $d$-dimensions

The tetradic Einstein-Palatini action can be written as (see, for instance, arXiv:1804.09685) $$S=\epsilon_{IJKL}\int_{\mathcal{M}}e^I\wedge e^J\wedge\Omega^{KL},$$ where $e^I$ are the frame ...
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Why does the 'metric Lagrangian' approach appear to fail in Newtonian mechanics?

A well known derivation of the free-space Lagrangian in Special Relativity goes as follows: The action $\mathcal{S}$ is a functional of the path taken through configuration space, $\mathbf{q}(\lambda)...
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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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Stability of Schwarzschild and Reissner-Nordstrom spacetimes

I am interested to know what is the best we can say about stability of Schwarzschild and Reissner-Nordstrom black holes. I found some who study the behavior of perturbations that satisfy the ...
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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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Convergence Property of Path-Integral

Let the action be $$S= \int \bigg\{ \frac{1}{2} \big(\frac{dX}{dt}\big)^2 - V(X) \bigg\} d\tau$$ and the corresponding Path-Integral $$Z= \int DX(t) e^{iS}.$$ Since the convergence is not clear we ...
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2answers
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Grassmann-even action

I am currently studying supersymmetric quantum mechanics with the help of the book Mirror Symmetry by Kentaro Hori (and others). On page 155 where they introduce Grassmann variables they say that the ...
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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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Derivation of Hamilton-Jacobi equation

I am trying my own way of deriving the Hamilton Jacobi equation $$\frac{\partial S}{\partial t} = -H \tag{1}$$ through direct variation. I think the difficulty of doing this is that the upper limit ...
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2answers
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Taylor expansion in derivation of Noether-theorem

In my classical mechanics lecture we derived the Noether-theorem for a coordinate transformation given by: $$ q_i(t) \rightarrow q^{'}_i(t)=q_i(t) + \delta q_i(t) = q_i(t) + \lambda I_i(q,\dot q,t).$$...
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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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1answer
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Proof of Noether's theorem: How to deal with transformations in time?

I was following the proof of Noether's theorem in Lemos - Analytical Mechanics, page 73. He considers a full infinitesimal transformation: $$t'=t+\epsilon X(q(t),t),$$ $$q'(t')=q(t)+\epsilon\Psi(q(t),...
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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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Free boson Equation motion from action

So in David tongs notes we have $$S=\frac{m}{8\pi}\int d^2x\partial_i\varphi\partial^i\varphi$$ and he finds that the equation of motion is $$[\partial_{t}^2-v^2\partial_{x}^2]\varphi=0$$ now my ...
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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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73 views

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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1answer
43 views

What is the trace in the Chern-Simons action

I have been looking at the Chern-Simons Lagrangian in $(2+1)$-dimensional spacetime $M$ in terms of a gauge field $A$: $$ S[A] = \frac{k}{4 \pi}\int_M \text{Tr}(A \wedge \text{d}A+ \frac{2}{3}A \...
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1answer
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Classical field theory with fields on different base spaces

Keeping things at a "basic level", a field is a function from a base manifold (of dimension D) to some other space. Usually the base manifold is the spacetime but may be something different (a lattice,...
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1answer
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Necessity and sufficiency of Euler-Lagrange equations in making an integral stationary

Suppose we want to make an integral $S$ of the form $$S = \int_{x_1}^{x_2} f\left[y_1(x), \dots, y_n(x), y'_1(x), \dots, y'_n(x), x\right]dx$$ stationary with the constraint $y_1\left(x_1\right) = \...
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Why is the generating function in the Hamilton–Jacobi equation equal to the action? [duplicate]

The aim in Hamilton Jacobi formalism is to find a canonical transformation that generates a new Hamiltonian $H'$ which is equal to $0$. Therefor we find the equation: $$H(q_1,...,q_n,\frac{\partial F}{...
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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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A doubt in Modified gravity

I am new to dark matter and modified gravity so excuse and inform me if I am wrong. If changes are made in the Friedman equations then there wouldn't always be an underlying action action principle. ...
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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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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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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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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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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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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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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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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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Einstein-Hilbert action and Lagrangian density for vacuum Ricci scalar

From the action, $$\int L\,\mathrm dt=\int R \sqrt{|g|}\,\mathrm d^4x,$$ why is the Lagrangian density for the gravitational field replaced by the Ricci scalar, which yield field equations in vacuum $$...
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1answer
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Euler-Lagrange equations for electrodynamics: matter vs. gauge field change

I know how to derive Maxwell's equations from a Lagrangian density via the Euler-Lagrange equations, but I'm bothered by one little detail. Variation of the Lagrangian seems only to be done with ...
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1answer
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Time component of momentum four-vector

In Landau-Lifshitz, Classical theory of fields (second chapter), the four-momentum is defined by the equation $$-\frac{\partial S}{\partial x^i}=p_i\tag{9.12},$$ where $S$ is the action integral. The ...
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1answer
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Why only 2 derivatives in classical mechanics? [duplicate]

The title conveys my true question, but for the sake of clarity I will now rephrase it in a more mathematical flavor using the Hamiltonian formalism of classical mechanics and the terminology of ...
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1answer
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EoM for scalar field in Brans-Dicke Theory

The action is given by $$ S^{(BD)} = \int d^4 x \sqrt{|g|} \left[ \phi R - \frac{\omega}{\phi} g^{\mu \nu} \, \nabla_\mu \phi \nabla_\nu \phi - V(\phi) \right]$$ I am trying to vary with respect to ...
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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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Branes without Fayet-Iliopoulos term

Often D-brane effective actions are given as (let's take D3-branes) $$ S=-T\int d^4x \sqrt{-det(g_{mn}+2\pi\alpha'F_{mn})}. $$ After expansion in $2\pi\alpha'$ there is a constant Fayet-Iliopoulos ...
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Obtaining Brans-Dicke theory scalar (wave) equation

I have trouble with obtaining d'Alambert equation for scalar field in Brans-Dicke gravity (http://www.scholarpedia.org/article/Jordan-Brans-Dicke_Theory). B-D gravity langrangian density is given by: ...
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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
47 views

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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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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Chern-Simons Gravity term in 3D and equations of motion

In the book "Quantum Gravity in 2+1 dimensions" by Steven Carlip he writes down a possible modification to the Einstein-Hilbert Action in 3d (eq. 1.16 to eq. 1.18) \begin{equation} I_{GCS}=-\frac{1}{...