Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
14 views

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 ...
0
votes
1answer
35 views

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 ...
2
votes
1answer
21 views

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^...
0
votes
1answer
65 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 ...
1
vote
1answer
26 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 \...
2
votes
1answer
41 views

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,...
0
votes
1answer
32 views

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) = \...
-1
votes
0answers
23 views
0
votes
0answers
21 views

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}{...
4
votes
1answer
172 views

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 \...
1
vote
0answers
39 views

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. ...
0
votes
0answers
26 views

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 ...
1
vote
2answers
102 views

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 ...
5
votes
2answers
136 views

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 ...
0
votes
0answers
42 views

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 ...
0
votes
0answers
44 views

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(\...
2
votes
2answers
57 views

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 ...
1
vote
1answer
57 views

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 ...
8
votes
6answers
233 views

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 ...
-1
votes
1answer
74 views

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 $$...
2
votes
1answer
37 views

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 ...
1
vote
1answer
69 views

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 ...
1
vote
1answer
55 views

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 ...
1
vote
1answer
38 views

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 ...
5
votes
4answers
216 views

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 ...
0
votes
0answers
19 views

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 ...
1
vote
1answer
77 views

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: ...
0
votes
0answers
167 views

Supersymmetry and reparametrization invariance of superstring action

Section $4.3.5$ Superstring action and its symmetries, Page$229$ Superstring theory by Green, Schwarz and Witten. $S_1=-\frac1{2π} ∫d^2σ\ eh^{αβ}∂_αX^μ∂_βX_μ.\quad S_2=\frac i{2π}∫d^2σ\ e\barψ^μρ^α∇...
0
votes
1answer
55 views

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_{...
2
votes
1answer
44 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 ...
0
votes
1answer
53 views

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 ...
3
votes
0answers
57 views

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}{...
1
vote
1answer
93 views

What is the physical interpretation of the action integral, without the stationary action principle?

I'm still wondering about the physical interpretation of the action integral of some mechanical system (classical theory here, to simplify things): \begin{equation}\tag{1} A = \int_{t_1}^{t_2} L(q, \, ...
1
vote
1answer
100 views

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 ...
0
votes
1answer
42 views

Definition of integral functional [duplicate]

I'm reading the section of Marion and Thornton devoted to basics on the Calculus of Variations, and came across this definition for the functional: $$J = \int f(y(x), y'(x);x) dx$$ implying that $f$ ...
3
votes
1answer
156 views

Linearizing the Einstein-Hilbert action; shortcuts?

I am interested in linearizing actions that are constructed out of geometrical objects. By this I mean perturbing the metric (or vielbein) and keeping in the action terms which are quadratic in the ...
3
votes
3answers
113 views

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 = \...
2
votes
1answer
243 views

Stress-Energy Tensor of Electromagnetic Field with sources

I can find a lot of references which treat the derivation of Maxwell equations and the associated Energy-Stress Tensor from the action principle. But I cannot find any information on the Energy-Stress ...
2
votes
1answer
82 views

The action of Einstein Maxwell system for arbitrary dimensions

The question is as mentioned in the title. To write the action for the Einstein-Maxwell system in arbitrary dimension. Is it possible just to add them (The Lagrangian for gravity and for ...
1
vote
1answer
153 views

Action angle variables and Action

Action given by principle of least action ($S$) and action variable given by action angle variable theory ($J$) are same?
0
votes
1answer
63 views

Understanding Noether's second theorem

Wikipedia writes that "if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by $k$ arbitrary functions and their derivatives up to order $m$, ...
1
vote
1answer
49 views

Equation of motion for a massless scalar in 2 dimensions

I'm working through Polchinski's String Theory (Volume 1, Chapter 2, page 34), with $D$ scalar fields: The action is given by $$ S = \frac{1}{2 \pi \alpha'}\int d^{2}z \partial X^{\mu} \overline{\...
2
votes
1answer
78 views

On-shell and off-shell transformations in Noether's theorem

For any transformation of the fields, $$\varphi\to\varphi'=\varphi+\delta\varphi$$ the change in the Lagrangian can be written as $$\delta\mathcal L = \text{EoM} + \partial_\mu j^\mu\tag{1}$$where "...
2
votes
1answer
77 views

Supersymmetry transformation: why does the Lagrangian transform as total derivative?

There is something I don't understand at page 36 of these lecture notes (Author: Fiorenzo Bastianelli from the university of Bologna, title: Path integrals for fermions and supersymmetric quantum ...
1
vote
2answers
171 views

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-...
0
votes
1answer
52 views

Simulating trajectory of object in a non-uniform but static gravitational field

This is a physics question, but the motivation for it comes from game design. I want to simulate the motion of an object in 2D space with several point sources of gravity (actually stars). The point ...
1
vote
2answers
204 views

Why can we choose affine parameterization?

In general relativity when deriving the geodesic equation $$\ddot{x}^\mu + \Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta = 0\tag{1}$$ from the action $$S = \int d\tau \sqrt{|g_{\mu\nu} \dot{x}...
1
vote
0answers
33 views

Cauchy problem for Hamilton-Jacobi equation

In Arnol'd V.I, "Mathematical methods of classical mechanics" p.257, I was asked to find a solution for the Cauchy problem $$H=\frac{p^2}{2},\ \ \ S_0=\frac{q^2}{2}$$ of the Hamilton-Jacobi equation ...
1
vote
2answers
131 views

Definition of generalized 4-momentum

In nonrelativistic mechanics, given a lagrangian $L$, we define the action as $$S[q]=\int L(q(t),\dot q(t))dt\tag{1}$$ and we can prove (see, for example, this answer for eq. (2)) $$\begin{align} \...
2
votes
1answer
86 views

Doubt in Functional Derivative of Lagrangian

Lecture XXXIII: Lagrangian formulation of GR by Christopher M. Hirata NON-INTERACTING DUST Consider a system with a suite of particles {A} each of mass $\mu_{A}$ following some set of trajectories $...