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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Something fishy with canonical momentum fixed at boundary in classical action

There's something fishy that I don't get clearly with the action principle of classical mechanics, and the endpoints that need to be fixed (boundary conditions). Please, take note that I'm not ...
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Worldsheet action in the presence of background fields in complex coordinates

We will start with the worldsheet action under massless background fields - the graviton $G_{\mu\nu}$ and Kalb-Ramond field $B_{\mu\nu}$ (we choose to exclude the dilaton $\Phi$ that also appears in ...
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Deeper explanation for Principle of Stationary Action [duplicate]

The Principle of Stationary Action (sometimes called Principle of Least Action and other names) is successfully applied in a wide variety of fields in physics. It can often can be be used to derive ...
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Why the choice of Configuration Space in Hamilton's Principle is $(q, \dot{q}, t)$? [closed]

In most physics books I've read, such as Goldstein's Classical Mechanics, the explanation of Hamilton's principle took into consideration the equation (1) known as Action: $$\displaystyle I = \int_{...
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Is there a proof that a physical system with a *stationary* action principle cannot always be modelled by a *least* action principle?

I'm aware that with Lagrangian mechanics, the path of the system is one that makes the action stationary. I've also read that its possible to find a choice of Lagrangian such that minimization is ...
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Minimality of the action [duplicate]

How is it proved that the extremal of the action obtained with Hamilton's principle in classical mechanics or classical field theory is in fact a minimum of the action and not just a stationary point? ...
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Howe-Tucker to Nambu-Goto Action

Aim to find from the Howe-Tucker action: $$S_{\text{HT}}=-\frac{1}{2}\int d^d\sigma\sqrt{-\gamma}(\gamma^{ab}\partial_a X^{\mu}\partial_b X^{\nu}\eta_{\mu\nu}-m^2(d-2))$$ (which is a Polyakov-like ...
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Directly integrating the Lagrangian for a simple harmonic oscillator

I've just started studying Lagrangian mechanics and am wrestling with the concept of "action". In the case of a simple harmonic oscillator where $x(t)$ is the position of the mass, I ...
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Lagrangian for Kerr-Newman black holes

I am trying to write down the action that is extremized by Kerr-Newman solutions in General Relativity. Specifically, I am interested in parametrizing the Lagrangian by the mass $M$, angular momentum $...
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Does a quantum field theory have an effective single-particle action in the single-particle subspace?

In non-interacting quantum field theories, the particle number is conserved so we can restrict to a given subspace of fixed particle number. On the single-particle subspace, the state will evolve ...
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Conformal transformation to Einstein frame for a Non-minimally coupled Ricci and Maxwell term

I am currently working on a modified gravity theory which has non-minimal coupling between Ricci scalar and Maxwell term. The precise action is $$\int d^4x\sqrt{-g} \left(R + \alpha R^2 + (1 + \beta R)...
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Is there a Lorentz invariant action for a free multi-particle system?

I want to write down a Lorentz-invariant action of free multi-particle systems. I know that a Lorentz-invariant action for each particle might be expressed as $$ S[\vec{r}]=\int dt L(\vec{r}(t),\dot{\...
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Landau/Lifshitz action as a function of coordinates [duplicate]

In Landau/Lifshitz' "Mechanics", $\S43$, 3ed, the authors consider the action of a mechanical system as a function of its final time $t$ and its final position $q$. They consider paths ...
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Variation of the action in a non-flat spacetime

In Padmanabhan's Quantum Field Theory, The Why, What and How, in chapter $3$, section $3.1.6$, in the paragraph between equations ($3.75$) and ($3.76$), he states regarding the Belinfante tensor that: ...
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QFT by Schwartz Problem 3.1 Solution

I am having trouble while solving in the Problem 3.1 of the QFT book by Schwartz. Problem Find the generalization of the Euler-Lagrange equations for general higher-order Lagrangians of the form $\...
darkphysics's user avatar
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Does it follow from Least Action Principle that particles do not go back in time, or do we stipulate this?

Consider the action integral, $S[\gamma] := \int L(\gamma(t),\dot{\gamma}(t),t)dt$. We can always re-write it in terms of an arbitrary curve parameter $\tau$ which need not coincide with time $t$: $$S[...
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Interpreting non-normalized covariance matrix eigenvalues as physical Actions

Summary: Eigenvalues of a "non-normalized" covariance matrix of time-series measurements from a linear system have units of Action (energy * time). Can we interpret this to obtain ...
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Action of a Scalar Field in Path Integral Formulation Peskin & Schroeder (Pag. 285)

I'm really confused on the discretization stuff on this chapter of P&S. My question is related to the computation of the Action in scalar field theory done in page 285. When they compute the ...
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Infinitesimal transformations of fields

We usually want to study the infinitesimal transformations on the action which depends on the field $\phi(x)$ and its derivatives. Such transformations may in general be written as, \begin{equation} x'...
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Does classical electrodynamics have a Lagrangian that gives both the Lorentz force and Maxwell equations?

There is a Lagrangian for a particle of mass $m$ and charge $q$ $$\mathcal{L}_1 = \mathcal{L}_k(m, \vec{v}) - q\phi + q\vec{v}\cdot\vec{A}$$ where $\mathcal{L}_k(m, \vec{v})$ is either $\frac{1}{2}m\...
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Derivation by Zee of a relativistic point particle action in a EM field in curved space

In Einstein Gravity in a Nutshell by Zee, in section IV.1 page 241, he tries to write down the action for electromagnetism and gravity in an intuitive and patchwork way, Starting from the relativistic ...
mathemania's user avatar
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Doubts about Noether's theorem derivation

Assume you have an action: $S[q] = \int L(q, \dot q, t)$ (i.e $q$ is a function of time). (1) Then you do a transformation on $q(t)$ such as $\sigma(q(t), a)$ where $a$ is infinetisemal and this ...
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7 votes
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Action as the integral of a differential form

I'm wondering if the action functional can be written as the integral of a differential form and if so how it relates to the Lagrangian. I tried considering in 1D the Lagrangian $L(q,v,t)$ as a ...
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Relationship between adding total time derivative to a Lagrangian and symmetry transformation

Given the Lagrangian $L(x(t),\dot{x}(t),t)$ one can perform an infinitesimal transformation \begin{align*} x &\mapsto x' = x + \delta x \\ t &\mapsto t' = t + \delta t \end{align*} ...
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On the physical meaning of functionals and the interpretation of their output numbers

I am studying about functionals, and while looking for some examples of functionals in physics, I have run into this handout . Here are two questions of mine. 1- This handout starts as follows (the ...
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Lagrangian without explicit time dependence, analysing the changes in the action

I'm studying the Mirror Symmetry book, available here. I'm reading chapter 10 on quantum mechanics. The following is from page 170. Although this is a basic Lagrangian/Hamiltonian type issue, so the ...
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Derivation of Hamiltonian by constraining $L(q, v, t)$ with $v = \dot{q}$

I am trying to reconstruct a derivation that I encountered a while ago somewhere on the internet, in order to build some intuition both for $H$ and $L$ in classical mechanics, and for the operation of ...
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Non-abelian Chern-Simons forms and Wess-Zumino action

Is there any closed analytic relationship between the Wess-Zumino action and the (non-abelian) Chern-Simons forms for arbitrary spacetime dimensions that appear in brane theories?
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Why does the Minkowski matrix appear in the free particle action?

It is usual to write the "kinetic" part of the SR action as the Minkowski space-time interval, here $(-,+,+,+)$, times $mc$ $$ S_{kin} = -\int_{\tau_1}^{\tau_2}mc\sqrt{-\eta_{\mu\nu}\dot{x}^{...
K. Pull's user avatar
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"Polyakov trick" for general relativity

In introductory String Theory textbooks, one typically starts with the Nambu-Goto action \begin{equation} S_{NG} \sim \int d^2 \sigma \sqrt{\dot{X}^2 - X'^2} \end{equation} and then arrive later at ...
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2 votes
1 answer
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Confirming an action is invariant under a supersymmetric transformation

I am studying chapter 9 of the book Mirror Symmetry, available here. My question is relating to page 156/157 where Supersymmetry is being introduced for the first time in QFT in 0-dimensions. We are ...
Gleeson's user avatar
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Finding equation of motion from total energy of $E$ and $B$ fields

I would like to find out how to obtain the equation of motion from the total energy of the $B$ and $E$ fields $$E_\text{tot} = \frac{1}{2\mu_0}\int\mathbf{B}(\mathbf{r},t)^2\ d^3r + \frac{\epsilon_0}...
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Hamiltonian analysis of relational $N$-Particle Dynamics

I am following "A Shape Dynamics Tutorial, Flavio Mercati" (https://arxiv.org/abs/1409.0105), and have problems understanding the hamiltonian formulation of $N$-particle dynamics as sketched ...
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GR Action and Ashtekar Connection

Palatini action $S_{Pal}$ is (assuming Cosmological constant $\Lambda=0)$: $$ S_{Pal}[e,\omega]=\int e\wedge e\wedge R[\omega].$$ Motion equations (varying this action) gives us Einstein Equation and ...
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2 answers
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What is the most general transformation between Lagrangians which give the same equation of motion?

This question is made up from 5 (including the main titular question) very closely related questions, so I didn't bother to ask them as different/followup questions one after another. On trying to ...
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In which cases does the action obey $\frac{\partial S}{\partial t}=-E$? [duplicate]

I'm reading https://web.physics.utah.edu/~starykh/phys7640/Lectures/FeynmansDerivation.pdf and the article states that there are cases where the action obeys $\frac{\partial S}{\partial t}=-E$. Is ...
Jeb Song's user avatar
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2 answers
167 views

Are all actions time reparameterization invariant?

Let's concentrate on point particle mechanics on a one dimensional manifold for simplicity. The action is $$S [q,\dot{q}]=\int dt L(q,\dot{q},t).$$ Time reparameterization would involve $t \to t'=f(t)$...
Sanjana's user avatar
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1 answer
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Change of action after a transformation with space-time dependent parameters

I've been following David Tong's lecture on introduction to quantum field theory. In his lecture notes page number 19 (and his video class on Youtube), he talks about global transformation that ...
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How to construct the final action for a supergravity theory $N = 1$ in 4D?

I am trying to understand the basics of supergravity theory $N = 1$ in 4D, which combines supersymmetry and general relativity. I know that the theory involves a spin 2 field whose quantum is the ...
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Notion of $4$-vectors in Lorentz symmetry violating case

$4$-vectors can be defined in various ways. One very common way among them is by saying if $A^{\mu}$ is some $4$-vector, then it will transform via the Lorentz transformation rule and will obey the ...
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Derive Navier-Stokes equation from the action principle (Euler-Lagrange's equation) [duplicate]

Navier-Stokes equations are the most general equations in fluid dynamics: We ususlly derive it by the consevation laws and $F=ma$. But how to derive it from the Action principle or equivalently Euler-...
user353731's user avatar
5 votes
1 answer
699 views

Confusion in derivation of Euler-Lagrange equations

Here's a screenshot of derivation of Euler-Lagrange from feynman lecture https://www.feynmanlectures.caltech.edu/II_19.html My doubt is in the last paragraph. I get that $\eta = 0$ at both ends, but ...
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How do we construct an action on a superspace lattice?

I am interested in the formulation of supersymmetric theories on a discrete spacetime, such as a lattice. I know that there are some difficulties in preserving supersymmetry on a lattice, such as the ...
Olandelie's user avatar
1 vote
1 answer
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Help deriving Maxwell's equations from the Lagrangian [duplicate]

Starting with the Lagrangian density $$\mathcal{L} = -\frac{1}{2}(\partial_\mu \mathcal{A}_\nu)(\partial^\mu \mathcal{A}^\nu)+\frac{1}{2}(\partial_\mu \mathcal{A}^\mu)^2,$$ I don't understand how to ...
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How would you interpret the relativistic action, and how does the mass alter the action exactly? [duplicate]

How would you apply the lagrangian in special relativity? How do you interpret the expression? The action is given by: $$S=-mc^2 \int d\tau$$ where $m$ is the rest mass. If I understand correctly the ...
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Noether’s second theorem: about the action principle

Noether's second theorem is supposed to show that the invariance of the Lagrangian by the Lie group (infinite in dimension) of certain theories necessarily implies that the field equations proper to ...
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1 answer
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Divergence of gauge kinetic coupling at the AdS boundary

This is the Einstein-Maxwell-Dilaton Gravity action: \begin{eqnarray*} S_{EM} = -\frac{1}{16 \pi G_5} \int \mathrm{d^5}x \sqrt{-g} \ [R - \frac{f(\phi)}{4}F_{MN}F^{MN} -\frac{1}{2}D_{M}\phi D^{M}\...
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2 answers
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Feynman's Derivation for Principle of Least Action

In the Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1964), Vol. II, Chap. 19, Feynman demonstrates how the principle of stationary action for one particle implies Newton's second law (or ...
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How did Landau & Lifshitz (Mechanics) get Equation 2.5?

I understood everything in Landau & Lifshitz's mechanics book until Equation 2.4,but I'm not sure what he means when he says "effecting the variation" and gets Equation 2.5. Can anyone ...
PhysicsNoob101's user avatar
1 vote
1 answer
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Lagrange momentum for position change

After the tremendous help from @hft on my previous question, after thinking, new question popped up. I want to compare how things behave when we do: $\frac{\partial S}{\partial t_2}$ and $\frac{\...
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