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.
548
questions
0
votes
1answer
111 views
Is the Lagrangian of a non-relativistic particle just $\dot{x}$?
Let
$$
S= m \int_a^b \dot{x}dt
$$
Using the relation $L\to L^2/2$, (see Geodesic Equation from variation: Is the squared lagrangian equivalent?)
I obtain
$$
S=m\int_a^b\frac{1}{2}(\dot{x})^2dt
$$
...
1
vote
2answers
68 views
Given the action, derive the Lagrangian (Fields and non-fields)
Let us denote $L$ the Lagrangian, and $\mathcal{L}$ the Lagrangian density, and the action $S$.
It is common to find the action based on the Lagrangian. Here, however, I am interested in the reverse ...
2
votes
0answers
62 views
How does it make sense to talk about the size of a string if the string action is conformally invariant?
From what I understand the Polyakov action in string theory is essentially something like
$$S(\xi, g, G)=\kappa \int_{\Sigma} d \mu_{g} \operatorname{Tr}_{g} \xi^{*} G$$
where $\Sigma$ is a given ...
0
votes
0answers
80 views
On the Boundary term in $f(R,T)$ Gravity
In standard $f(R)$ gravity we consider the Lagrangian of the form $L=\frac{1}{16\pi G}f(R)\epsilon$, where $\epsilon$ is the spacetime volume form and similarly, we consider the boundary term to be of ...
1
vote
2answers
88 views
$\delta S=0$ only for $\frac{\partial\mathcal{L}}{\partial\phi}-\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}=0$?
Condition for the variation of action is:
$$0=\delta S$$
$$=\int d^4 x [\frac{\partial \mathcal{L}}{\partial \phi}\delta\phi-\partial_\mu(\frac{\partial\mathcal{L}}{\partial(\partial_\mu \phi)})\...
1
vote
1answer
55 views
Point particles as the limit of a short string
There's a common saying in the domain of the study of classical relativistic strings, that in the limit of a very short string, the action reduces to that of a point particle (there is for instance a ...
1
vote
1answer
75 views
Yang-Mills Action for Non-Trivial Bundle
Suppose we have a principal $G$ bundle $(P,M,π)$ where $M$ is a 4-dimenational manifold and $G$ a Lie group (and $\mathfrak{g}$ its Lie algebra).The Yang Mills action is a functional of the gauge ...
6
votes
1answer
76 views
Solving free particles with Fourier series
Here's a silly idea : take the action of a free particle,
$$S = \int_{t_1}^{t_2} \dot{x}^2 dt$$
Our configuration space is the space of $C^1$ functions over $[t_1, t_2]$, which is spanned by the ...
0
votes
0answers
27 views
Varying the action for gravitational field in Landau-Lifshitz book
I am attempting to prove a calculation found in the Landua-Lifshitz book, namely finding the action of the gravitational field, Sec $93$. The book attempts to write the action of the gravitational ...
0
votes
0answers
16 views
Pictogram extrinsic time variable
In Gravitation, it is said that the extrinsic time variable can be represented by a pictogram: Gravitation page 551 but I didn't find much data on it.
How is that pictogram called? Is it equivalent ...
1
vote
1answer
74 views
Gravitino Equation of motion in second-order formalism
In Freedman and Proeyen's text on supergravity they derive the equation of motion for the gravitino using the second order formalism. However, I'm not exactly clear as to how they use partial ...
0
votes
0answers
23 views
Unique Path in Principle of Least action [duplicate]
When in deriving Euler Lagrange equation how are we certain that there is only one unique path which would satisfy extremum of action?
Can't there exists more than one path which satisfies Euler ...
2
votes
1answer
44 views
Why is an action built from superfields guaranteed to be supersymmetric?
Given a superfield (in 0+1 spacetime + 2 superspace coordinates)
$$X(t,\theta_1,\theta_2) = x(t) + \theta_i \psi_i(t) + \theta_1 \theta_2 F_{12}(t)\tag{1}$$
and given the standard supercharges ...
2
votes
0answers
81 views
Deriving the stress-energy tensor from the Einstein-Hilbert action
I'm a mathematician who knows very little physics and is trying to learn relativity theory from a mathematical perspective.
Let $M$ be a compact, orientable manifold. In the vacuum, the Einstein-...
2
votes
2answers
170 views
Does it make sense to say that the action is even or odd under time reversal?
The action of a system in mechanics is an integral over time defined as $$S[x(t)]=\int\limits_{t_1}^{t_2}L(x,\dot{x},t)dt.$$ Here, the time $t$ is integrated making the left hand side depend only on ...
0
votes
0answers
46 views
Why are these two variables being treated differently in the action?
I'm trying to understand the derivation provided in the section 2.4 of this paper. I have modified the notation and cut out the unimportant parts of the equations for clarity purposes, but for ...
0
votes
0answers
27 views
What's the partition function and action of thermofield double state (TFD)?
We know that TFD state is consisted of two CFT at each side, and their partition functions are $$ Z_{L/R}= {\rm tr}\,e^{-\beta H_{L/R}}=\int e^{-S_E}\,\,\,, $$ where $S_E$ is just Euclidean action of ...
1
vote
0answers
37 views
A simple question about equation of motion in polchinski's String theory?
In page 14 to get the equation of motion, it takes the variation of the action
$$
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
$$
...
1
vote
3answers
64 views
Motivation behind action when deriving ''Strings as Harmonic oscillators" in Zwiebach's book on String theory
Page 248 gives us this action and he simply says that we will assume it correct.
$$
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
73 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 ...
0
votes
0answers
33 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 $$L=\sqrt {g_{kn}\frac{dx^k}{ds}...
2
votes
1answer
69 views
Inconsistency? Lagrangian with its Euler–Lagrange equation as condition
Consider the action
$$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
66 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 ...
9
votes
2answers
811 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)...
1
vote
2answers
73 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
77 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 ...
1
vote
2answers
40 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
55 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
vote
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
74 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
51 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
302 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
94 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:
$$t'=t+\epsilon X(q(t),t),$$
$$q'(t')=q(t)+\epsilon\Psi(q(t),...
0
votes
0answers
15 views
Can all phase space conserving dynamics be described by a Lagrangian system? [duplicate]
Given a system described by a set of ODE's that can be shown to conserve phase space, does there necessarily exist a Lagrangian (or Action) formulation that describes my system?
I'm comfortable ...
0
votes
1answer
47 views
Free boson Equation motion from action
So in David tongs notes we have
$$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$$
now my ...
2
votes
1answer
28 views
Clarifications regarding the Maupertuis/Jacobi principle
I'm slightly confused regarding the Maupertuis' Principle. I have read the Wikipedia page but the confusion is even in that derivation. So, say we have a Lagrangian described by $\textbf{q}=(q^1,...q^...
0
votes
1answer
89 views
Confusing with the equation $(2.4)$ and $(2.5)$ of Landau and Lifshitz, Mechanics, Chapter 1, The principle of Least Action
I'm a 12th Grader and I'm interested in Lagrangian Mechanics and having a bit of knowledge about the Newtonian Mechanics. So, I found a book of Landau and Lifshitz's Mechanics and started reading from ...
1
vote
1answer
67 views
What is the trace in the Chern-Simons action
I have been looking at the Chern-Simons Lagrangian in $(2+1)$-dimensional spacetime $M$ in terms of a gauge field $A$:
$$ S[A] = \frac{k}{4 \pi}\int_M \text{Tr}(A \wedge \text{d}A+ \frac{2}{3}A \...
2
votes
1answer
53 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
34 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) = \...
0
votes
0answers
24 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: $$H(q_1,...,q_n,\frac{\partial F}{...
5
votes
1answer
279 views
Is Hilbert-Einstein action just the leading order of some kind of series?
Introducing the action for the gravitational field my GR professor stated that, in principle, one could write it as
$$S = k\int d^4x\sqrt{g}(\sum_n\sum_m a_{nm} R_n^m - 2\Lambda), \space \space \...
1
vote
0answers
41 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
29 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
188 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
153 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
53 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
59 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:
$$S = \int d^4x \sqrt{-g} e^{-2\phi}(R+4(\...
2
votes
2answers
83 views
How does Hamilton's Principle give us the path taken?
We defined the action as:
$$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$
where $q_i(t_1)$ and $q_i(t_2)$ are known and fixed.
Hamilton's principle states that the path that is ...
1
vote
1answer
58 views
How can Lagrangian method work whenever the Lagrangian is not convex?
Let $$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 ...
