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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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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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 ...
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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^...
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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 ...
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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 $A$:
$$ S[A] = \frac{k}{4 \pi}\int_M \text{Tr}(A \wedge \text{d}A+ \frac{2}{3}A \...
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1answer
41 views
Classical field theory with fields on different base spaces
Keeping things at a "basic level", a field is a function from a base manifold (of dimension D) to some other space. Usually the base manifold is the spacetime but may be something different (a lattice,...
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1answer
32 views
Necessity and sufficiency of Euler-Lagrange equations in making an integral stationary
Suppose we want to make an integral $S$ of the form $$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$$ stationary with the constraint $y_1\left(x_1\right) = \...
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In field theory，why from the invariance of density of Lagrangian one could get the invariance of equation of motion？ [duplicate]
Somebody who can give me a concrete derivation process from Principle of least action？
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21 views
Why is the generating function in the Hamilton–Jacobi equation equal to the action? [duplicate]
The aim in Hamilton Jacobi formalism is to find a canonical transformation that generates a new Hamiltonian $H'$ which is equal to $0$.
Therefor we find the equation: $$H(q_1,...,q_n,\frac{\partial F}{...
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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
$$S = k\int d^4x\sqrt{g}(\sum_n\sum_m a_{nm} R_n^m - 2\Lambda), \space \space \...
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A doubt in Modified gravity
I am new to dark matter and modified gravity so excuse and inform me if I am wrong.
If changes are made in the Friedman equations then there wouldn't always be an underlying action action principle. ...
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26 views
How general is the Lagrangian formulation? [duplicate]
Haven't seriously tackled this problem myself because it's been awhile since I've done any hard mathematics and I'm a bit rusty. However, you needn't spare the math in your answers.
I've been ...
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2answers
102 views
Euler-Lagrange equations from a complex Lagrangian
I'm looking for generalizations of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” and “maximum” don't have obvious meaning for a ...
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2answers
136 views
Is Action Always “Locally” Least?
In general, I know it's true that the Principle of Least Action is more properly called the Principle of "Stationary" Action. However, there are results which seem to suggest that for sufficiently ...
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Intuition behind the use of the Principle of Stationary Action in Classical Field Theory [duplicate]
Whilst studying Field Theory and after checking numerous sources it appears that people always just state the action without providing some sort of motivation/intuition as to why we should/can use the ...
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44 views
Equations of motion from action variation
I was reading about dilaton gravity in 2D, and I was trying to reproduce the equations of motion of a related theory. If I consider the following action:
$$S = \int d^4x \sqrt{-g} e^{-2\phi}(R+4(\...
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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 ...
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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 ...
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6answers
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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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1answer
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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 $$...
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1answer
37 views
Euler-Lagrange equations for electrodynamics: matter vs. gauge field change
I know how to derive Maxwell's equations from a Lagrangian density via the Euler-Lagrange equations, but I'm bothered by one little detail.
Variation of the Lagrangian seems only to be done with ...
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1answer
69 views
Time component of momentum four-vector
In Landau-Lifshitz, Classical theory of fields (second chapter), the four-momentum is defined by the equation
$$-\frac{\partial S}{\partial x^i}=p_i\tag{9.12},$$
where $S$ is the action integral. The ...
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1answer
55 views
Why only 2 derivatives in classical mechanics? [duplicate]
The title conveys my true question, but for the sake of clarity I will now rephrase it in a more mathematical flavor using the Hamiltonian formalism of classical mechanics and the terminology of ...
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1answer
38 views
EoM for scalar field in Brans-Dicke Theory
The action is given by
$$ 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]$$
I am trying to vary with respect to ...
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4answers
216 views
What is the physical content of the principle of least action?
Say the world is governed by the Principle of Least Action (or Hamiltonian mechanics) and let's not worry about quantum mechanics too much.
Independently of any Lagrangian or Hamiltonian, does that ...
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0answers
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Branes without Fayet-Iliopoulos term
Often D-brane effective actions are given as (let's take D3-branes)
$$
S=-T\int d^4x \sqrt{-det(g_{mn}+2\pi\alpha'F_{mn})}.
$$
After expansion in $2\pi\alpha'$ there is a constant Fayet-Iliopoulos ...
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1answer
77 views
Obtaining Brans Dicke theory scalar (wave) equation
I have trouble with obtaining d'Alambert equation for scalar field in Brans-Dicke gravity (http://www.scholarpedia.org/article/Jordan-Brans-Dicke_Theory).
B-D gravity langrangian density is given by:
...
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0answers
167 views
Supersymmetry and reparametrization invariance of superstring action
Section $4.3.5$ Superstring action and its symmetries, Page$229$ Superstring theory by Green, Schwarz and Witten.
$S_1=-\frac1{2π} ∫d^2σ\ eh^{αβ}∂_αX^μ∂_βX_μ.\quad
S_2=\frac i{2π}∫d^2σ\ e\barψ^μρ^α∇...
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1answer
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Variation of action for massive point particle (pp)
So I'm pretty sure I'm missing something obvious, but for the life of me I cannot replicate the step between 1.2.2 and 1.2.3 in Polchinski Vol 1. Basically, I'm trying to find the variation of:
$$S_{...
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1answer
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 ...
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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 ...
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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}{...
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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, \, ...
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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 ...
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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 $f$ ...
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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 ...
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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:
$$\delta S = \...
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1answer
243 views
Stress-Energy Tensor of Electromagnetic Field with sources
I can find a lot of references which treat the derivation of Maxwell equations and the associated Energy-Stress Tensor from the action principle. But I cannot find any information on the Energy-Stress ...
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1answer
82 views
The action of Einstein Maxwell system for arbitrary dimensions
The question is as mentioned in the title. To write the action for the Einstein-Maxwell system in arbitrary dimension.
Is it possible just to add them (The Lagrangian for gravity and for ...
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1answer
153 views
Action angle variables and Action
Action given by principle of least action ($S$) and action variable given by action angle variable theory ($J$) are same?
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1answer
63 views
Understanding Noether's second theorem
Wikipedia writes that
"if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by $k$ arbitrary functions and their derivatives up to order $m$, ...
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1answer
49 views
Equation of motion for a massless scalar in 2 dimensions
I'm working through Polchinski's String Theory (Volume 1, Chapter 2, page 34), with $D$ scalar fields:
The action is given by
$$ S = \frac{1}{2 \pi \alpha'}\int d^{2}z \partial X^{\mu} \overline{\...
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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 "...
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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 ...
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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-...
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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 ...
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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
$$S = \int d\tau \sqrt{|g_{\mu\nu} \dot{x}...
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0answers
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Cauchy problem for Hamilton-Jacobi equation
In Arnol'd V.I, "Mathematical methods of classical mechanics" p.257, I was asked to find a solution for the Cauchy problem
$$H=\frac{p^2}{2},\ \ \ S_0=\frac{q^2}{2}$$
of the Hamilton-Jacobi equation
...
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2answers
131 views
Definition of generalized 4-momentum
In nonrelativistic mechanics, given a lagrangian $L$, we define the action as
$$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}
\...
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1answer
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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 $...
