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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Variation of the Ricci tensor “squared” and antisymmetrization of the derivatives
I'm dealing with some extension of GR, with action:
$S=\int d^4x\Big[\sqrt{-g} f(R,R_{\mu\nu}R^{\mu\nu})$
Varying this action gives:
$\delta S=\int d^4x\Big[\delta\sqrt{-g} f(R,R_{\mu\nu}R^{\mu\nu})...
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Difference between sign conventions in the action of $\mathcal{N}=4$ SYM
In the paper called Wilson Loops in N=4 Supersymmetric Yang-Mills Theory, the authors define the action for the $\mathcal{N}=4$ Supersymmetric Yang-Mills (SYM) theory including the following term:
$$...
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39 views
Deriving the equations of motion of a real vector field [duplicate]
I'm trying to calculate the euler-lagrange equations of the following Lagrangian density.
$$\mathcal{L} = -\frac{1}{2}(\partial_\alpha A_\beta)(\partial^\alpha A^\beta)+\frac{1}{2}(\partial_\alpha A^\...
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1answer
93 views
A Potential Euler-Lagrange Equation Alternate Derivation?
Can the Euler Lagrange Equation be derived with this overall strategy?
Step 1 – Define a geodesic in flat space to be $\frac{d}{dy} \frac{ds}{dx} = \frac{d}{dx} \frac{ds}{dy}$, where $ds$ represents ...
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1answer
55 views
Is there a superfluous statement in Schwartz's QFT book in deriving Euler-Lagrange equations?
Please help me with the following confusion.
Yesterday I was looking at the derivation of Euler-Lagrange equation in several QFT textbooks using stationarity of the action. At the last step one needs ...
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1answer
40 views
Determine canonical fields of action
I'm working on an exercise which asks me to determine the canonical fields, and their equations of motion, of this invariant action:
$$
S = \int d\tau \sqrt{g_{\tau\tau}}\left( \frac{\tilde m}{2} g^{\...
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1answer
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Simple calculation on coordinate transformation of Lagrangian (Qualls' CFT lecture note)
I have a question while reading "Lectures on conformal field theory" by Qualls (https://arxiv.org/abs/1511.04074). $^1$
Question. I cannot find that the transformation (1.12) makes the action ...
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1answer
39 views
Fermat's principle in classical mechanics?
I do know the principle of least action, but is it possible to formulate classical mechanics based on the principle of least time? That is, if we know the initial state $(x_i,p_i)$ of the particle and ...
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1answer
84 views
Can anybody identify these two equations for me? [closed]
I am trying to identify these two equations:
$$S=\int L\ \gamma \ d\tau \quad \tag{1}$$
$$\quad \frac{\partial L}{\partial \phi}-\partial_\mu\left[\frac{\partial L}{\partial(\partial_\mu \phi)}\...
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How to get the Nambu-Goto action from the Polyakov action by lagrangian multiplier? [duplicate]
I consulted some textbooks about string theory, but they only told me that the Nambu-Goto action and the Polyakov action is equivalent. So I searched relevant questions here and got some ways of ...
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Is there a constraint on the Lagrangian that prevents it from having multiple stationary points? [duplicate]
Particles only go through paths that make the Lagrangian stationary, but I haven't heard anyone explicitly say that the Lagrangian cannot have multiple stationary points. Is there such a constraint? ...
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1answer
66 views
What is the reasoning that leads one to postulate this second form for the relativistic particle action?
The action for the free relativistic particle with worldline $\gamma : I\subset \mathbb{R}\to M$ is
$$S[\gamma]=-m\int d\lambda\sqrt{-\dot{\gamma}^a(\lambda)\dot{\gamma}_a(\lambda)}\tag{1} $$
Now, ...
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1answer
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Clifford algebra formulation of the Nambu-goto action
Using the wedge product one can pair the generators the Clifford algebra $Cl_{1,3}(\mathbb{R})$ to produce 2-vectors (area elements).
The Nambu-Goto action is a statement on the evolution of ...
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2answers
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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
$$
S=m\int_a^b\frac{1}{2}(\dot{x})^2dt
$$
...
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2answers
75 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 ...
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63 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 ...
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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 ...
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$\delta S=0$ 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:
$$0=\delta S$$
$$=\int d^4 x [\frac{\partial \mathcal{L}}{\partial \phi}\delta\phi-\partial_\mu(\frac{\partial\mathcal{L}}{\partial(\partial_\mu \phi)})\...
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1answer
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Point particles as the limit of a short string
There's a common saying in the domain of the study of classical relativistic strings, that in the limit of a very short string, the action reduces to that of a point particle (there is for instance a ...
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1answer
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Yang-Mills Action for Non-Trivial Bundle
Suppose we have a principal $G$ bundle $(P,M,π)$ where $M$ is a 4-dimenational manifold and $G$ a Lie group (and $\mathfrak{g}$ its Lie algebra).The Yang Mills action is a functional of the gauge ...
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Solving free particles with Fourier series
Here's a silly idea : take the action of a free particle,
$$S = \int_{t_1}^{t_2} \dot{x}^2 dt$$
Our configuration space is the space of $C^1$ functions over $[t_1, t_2]$, which is spanned by the ...
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Varying the action for gravitational field in Landau-Lifshitz book
I am attempting to prove a calculation found in the Landua-Lifshitz book, namely finding the action of the gravitational field, Sec $93$. The book attempts to write the action of the gravitational ...
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Pictogram extrinsic time variable
In Gravitation, it is said that the extrinsic time variable can be represented by a pictogram: Gravitation page 551 but I didn't find much data on it.
How is that pictogram called? Is it equivalent ...
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1answer
80 views
Gravitino Equation of motion in second-order formalism
In Freedman and Proeyen's text on supergravity they derive the equation of motion for the gravitino using the second order formalism. However, I'm not exactly clear as to how they use partial ...
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25 views
Unique Path in Principle of Least action [duplicate]
When in deriving Euler Lagrange equation how are we certain that there is only one unique path which would satisfy extremum of action?
Can't there exists more than one path which satisfies Euler ...
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1answer
46 views
Why is an action built from superfields guaranteed to be supersymmetric?
Given a superfield (in 0+1 spacetime + 2 superspace coordinates)
$$X(t,\theta_1,\theta_2) = x(t) + \theta_i \psi_i(t) + \theta_1 \theta_2 F_{12}(t)\tag{1}$$
and given the standard supercharges ...
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Deriving the stress-energy tensor from the Einstein-Hilbert action
I'm a mathematician who knows very little physics and is trying to learn relativity theory from a mathematical perspective.
Let $M$ be a compact, orientable manifold. In the vacuum, the Einstein-...
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2answers
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Does it make sense to say that the action is even or odd under time reversal?
The action of a system in mechanics is an integral over time defined as $$S[x(t)]=\int\limits_{t_1}^{t_2}L(x,\dot{x},t)dt.$$ Here, the time $t$ is integrated making the left hand side depend only on ...
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Why are these two variables being treated differently in the action?
I'm trying to understand the derivation provided in the section 2.4 of this paper. I have modified the notation and cut out the unimportant parts of the equations for clarity purposes, but for ...
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What's the partition function and action of thermofield double state (TFD)?
We know that TFD state is consisted of two CFT at each side, and their partition functions are $$ Z_{L/R}= {\rm tr}\,e^{-\beta H_{L/R}}=\int e^{-S_E}\,\,\,, $$ where $S_E$ is just Euclidean action of ...
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A simple question about equation of motion in polchinski's String theory?
In page 14 to get the equation of motion, it takes the variation of the action
$$
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
$$
...
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Motivation behind action when deriving ''Strings as Harmonic oscillators" in Zwiebach's book on String theory
Page 248 gives us this action and he simply says that we will assume it correct.
$$
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}...
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1answer
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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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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 $$L=\sqrt {g_{kn}\frac{dx^k}{ds}...
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1answer
70 views
Inconsistency? Lagrangian with its Euler–Lagrange equation as condition
Consider the action
$$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}...
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1answer
94 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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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)...
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2answers
85 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 ...
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1answer
86 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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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 ...
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2answers
57 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 ...
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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?
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2answers
112 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 ...
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2answers
63 views
Taylor expansion in derivation of Noether-theorem
In my classical mechanics lecture we derived the Noether-theorem for a coordinate transformation given by:
$$ q_i(t) \rightarrow q^{'}_i(t)=q_i(t) + \delta q_i(t) = q_i(t) + \lambda I_i(q,\dot q,t).$$...
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1answer
327 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}
...
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1answer
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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:
$$t'=t+\epsilon X(q(t),t),$$
$$q'(t')=q(t)+\epsilon\Psi(q(t),...
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16 views
Can all phase space conserving dynamics be described by a Lagrangian system? [duplicate]
Given a system described by a set of ODE's that can be shown to conserve phase space, does there necessarily exist a Lagrangian (or Action) formulation that describes my system?
I'm comfortable ...
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1answer
48 views
Free boson Equation motion from action
So in David tongs notes we have
$$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
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1answer
30 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
100 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 ...
