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.
2
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1answer
36 views
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 ...
0
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0answers
39 views
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 ...
0
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0answers
15 views
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 ...
1
vote
0answers
24 views
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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vote
3answers
54 views
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.
$$
S=\int d \tau d \sigma ~\mathcal{L}=\frac{1}{4 \pi \alpha^{\prime}} \int d \tau \int_{0}^{\pi} d \sigma\left(\dot{X}...
3
votes
1answer
63 views
Why is there a Lagrangian? [duplicate]
In all discussions regarding the Lagrangian formulation it has always been said that $L = T - V $, only is a correct guess that when operated via through the Euler -Lagrange equation yields something ...
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0answers
31 views
Derivative with respect to a coordiante differential (geodesic equation)
If the arc length is chosen to be the action integral, that is $$ 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}...
2
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1answer
67 views
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
$$\frac{\partial{L}}{\partial{q}} - \frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{q}...
2
votes
1answer
56 views
Einstein-Palatini action in $d$-dimensions
The tetradic Einstein-Palatini action can be written as (see, for instance, arXiv:1804.09685)
$$S=\epsilon_{IJKL}\int_{\mathcal{M}}e^I\wedge e^J\wedge\Omega^{KL},$$
where $e^I$ are the frame ...
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2answers
802 views
Why does the 'metric Lagrangian' approach appear to fail in Newtonian mechanics?
A well known derivation of the free-space Lagrangian in Special Relativity goes as follows:
The action $\mathcal{S}$ is a functional of the path taken through
configuration space, $\mathbf{q}(\lambda)...
0
votes
2answers
57 views
Action principle and Functional derivative in CM
I want to extremize this well known action.
$$S[\phi]=\int \mathcal{L}(\phi(t),\dot{\phi}(t)) dt $$
The result is also well known. It turns out to be E-L equation.
The Action principle states that the ...
5
votes
1answer
65 views
Stability of Schwarzschild and Reissner-Nordstrom spacetimes
I am interested to know what is the best we can say about stability of Schwarzschild and Reissner-Nordstrom black holes.
I found some who study the behavior of perturbations that satisfy the ...
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0answers
130 views
Get the action from knowledge of all wave functions?
Say I know all the values of the wave functions at all times $\psi[\phi,t)$. Can I use this knowledge to find the action $S[\phi]$? i.e. to give the action as a function of the wave functions?
$\phi$ ...
1
vote
2answers
36 views
Convergence Property of Path-Integral
Let the action be $$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}.$$
Since the convergence is not clear we ...
1
vote
2answers
50 views
Grassmann-even action
I am currently studying supersymmetric quantum mechanics with the help of the book Mirror Symmetry by Kentaro Hori (and others). On page 155 where they introduce Grassmann variables they say that the ...
1
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0answers
29 views
How principle of least action? [duplicate]
I had learned the principle of least action.But I didn't get the motive behind taking the least action. Or why should the particle follow a path where it have a least action?
3
votes
2answers
57 views
Derivation of Hamilton-Jacobi equation
I am trying my own way of deriving the Hamilton Jacobi equation
$$\frac{\partial S}{\partial t} = -H \tag{1}$$
through direct variation. I think the difficulty of doing this is that the upper limit ...
0
votes
2answers
49 views
Taylor expansion in derivation of Noether-theorem
In my classical mechanics lecture we derived the Noether-theorem for a coordinate transformation given by:
$$ q_i(t) \rightarrow q^{'}_i(t)=q_i(t) + \delta q_i(t) = q_i(t) + \lambda I_i(q,\dot q,t).$$...
3
votes
1answer
296 views
Show two Lagrangians are equivalent
I need to show that these two Lagrangians are equivalent:
\begin{align}
L(\dot{x},\dot{y},x,y)&=\dot x^2+\dot y + x^2-y ,\\
\tilde{L}(\dot x, \dot y, x, y)&=\dot x^2+\dot y -2y^3.
\end{align}
...
2
votes
1answer
90 views
Proof of Noether's theorem: How to deal with transformations in time?
I was following the proof of Noether's theorem in Lemos - Analytical Mechanics, page 73.
He considers a full infinitesimal transformation:
$$t'=t+\epsilon X(q(t),t),$$
$$q'(t')=q(t)+\epsilon\Psi(q(t),...
0
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0answers
15 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
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1answer
44 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
23 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
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 ...
1
vote
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 $A$:
$$ S[A] = \frac{k}{4 \pi}\int_M \text{Tr}(A \wedge \text{d}A+ \frac{2}{3}A \...
2
votes
1answer
45 views
Classical field theory with fields on different base spaces
Keeping things at a "basic level", a field is a function from a base manifold (of dimension D) to some other space. Usually the base manifold is the spacetime but may be something different (a lattice,...
0
votes
1answer
32 views
Necessity and sufficiency of Euler-Lagrange equations in making an integral stationary
Suppose we want to make an integral $S$ of the form $$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$$ stationary with the constraint $y_1\left(x_1\right) = \...
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0answers
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Why is the generating function in the Hamilton–Jacobi equation equal to the action? [duplicate]
The aim in Hamilton Jacobi formalism is to find a canonical transformation that generates a new Hamiltonian $H'$ which is equal to $0$.
Therefor we find the equation: $$H(q_1,...,q_n,\frac{\partial F}{...
4
votes
1answer
223 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
$$S = k\int d^4x\sqrt{g}(\sum_n\sum_m a_{nm} R_n^m - 2\Lambda), \space \space \...
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0answers
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A doubt in Modified gravity
I am new to dark matter and modified gravity so excuse and inform me if I am wrong.
If changes are made in the Friedman equations then there wouldn't always be an underlying action action principle. ...
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0answers
28 views
How general is the Lagrangian formulation? [duplicate]
Haven't seriously tackled this problem myself because it's been awhile since I've done any hard mathematics and I'm a bit rusty. However, you needn't spare the math in your answers.
I've been ...
0
votes
2answers
134 views
Euler-Lagrange equations from a complex Lagrangian
I'm looking for generalizations of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” and “maximum” don't have obvious meaning for a ...
5
votes
2answers
142 views
Is Action Always “Locally” Least?
In general, I know it's true that the Principle of Least Action is more properly called the Principle of "Stationary" Action. However, there are results which seem to suggest that for sufficiently ...
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0answers
49 views
Intuition behind the use of the Principle of Stationary Action in Classical Field Theory [duplicate]
Whilst studying Field Theory and after checking numerous sources it appears that people always just state the action without providing some sort of motivation/intuition as to why we should/can use the ...
0
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0answers
54 views
Equations of motion from action variation
I was reading about dilaton gravity in 2D, and I was trying to reproduce the equations of motion of a related theory. If I consider the following action:
$$S = \int d^4x \sqrt{-g} e^{-2\phi}(R+4(\...
2
votes
2answers
67 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 ...
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votes
6answers
260 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 ...
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votes
1answer
111 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 $$...
2
votes
1answer
44 views
Euler-Lagrange equations for electrodynamics: matter vs. gauge field change
I know how to derive Maxwell's equations from a Lagrangian density via the Euler-Lagrange equations, but I'm bothered by one little detail.
Variation of the Lagrangian seems only to be done with ...
1
vote
1answer
72 views
Time component of momentum four-vector
In Landau-Lifshitz, Classical theory of fields (second chapter), the four-momentum is defined by the equation
$$-\frac{\partial S}{\partial x^i}=p_i\tag{9.12},$$
where $S$ is the action integral. The ...
1
vote
1answer
59 views
Why only 2 derivatives in classical mechanics? [duplicate]
The title conveys my true question, but for the sake of clarity I will now rephrase it in a more mathematical flavor using the Hamiltonian formalism of classical mechanics and the terminology of ...
1
vote
1answer
44 views
EoM for scalar field in Brans-Dicke Theory
The action is given by
$$ 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]$$
I am trying to vary with respect to ...
4
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4answers
223 views
What is the physical content of the principle of least action?
Say the world is governed by the Principle of Least Action (or Hamiltonian mechanics) and let's not worry about quantum mechanics too much.
Independently of any Lagrangian or Hamiltonian, does that ...
0
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0answers
20 views
Branes without Fayet-Iliopoulos term
Often D-brane effective actions are given as (let's take D3-branes)
$$
S=-T\int d^4x \sqrt{-det(g_{mn}+2\pi\alpha'F_{mn})}.
$$
After expansion in $2\pi\alpha'$ there is a constant Fayet-Iliopoulos ...
1
vote
1answer
99 views
Obtaining Brans-Dicke theory scalar (wave) equation
I have trouble with obtaining d'Alambert equation for scalar field in Brans-Dicke gravity (http://www.scholarpedia.org/article/Jordan-Brans-Dicke_Theory).
B-D gravity langrangian density is given by:
...
0
votes
1answer
61 views
Variation of action for massive point particle (pp)
So I'm pretty sure I'm missing something obvious, but for the life of me I cannot replicate the step between 1.2.2 and 1.2.3 in Polchinski Vol 1. Basically, I'm trying to find the variation of:
$$S_{...
2
votes
1answer
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 ...
0
votes
1answer
55 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
70 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}{...
