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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How would someone discover the Einstein-Hilbert Action?

Usually in textbooks or on online resources, when you are learning General Relativity, propose the following: $$ \mathcal S[g_{\mu\nu}] =\frac{1}{2 \kappa}\int_{\mathcal M} {\mathrm d^4 x \; R\sqrt{-g}...
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What is the generally expected or more useful form of supersymmetry, on-shell or off-shell?

If the on-shell numbers of degrees of freedom of bosons and fermions match we have on-shell supersymmetry, if the off-shell numbers match we have off-shell supersymmetry. When LHC people say they are ...
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Question about integration limits in the special relativistic action

We can read in this article that the action of a particle in special relativity, is It seems like nitpicking maybe, but shouldn't the two coordinate time limits be changed to proper time limits, or ...
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Lagrangian and Action for a plane pendulum under the dampling force exerted by air resistance [duplicate]

Consider a plane pendulum which is composed of an ideal string and a sphere of mass m and length $l$. As a consequence of the presence of air, it exerts a force proportional to the speed characterized ...
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How does nature know Hamilton's principle? [duplicate]

I have gone through some of the questions asked here re Hamilton's principle, but could not readily find an answer to the following: Hamilton's principle states that paths particles follow extremizes ...
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Is the Nambu-Goto action defined only for the torus?

For simplicity, I will use the Nambu-Goto action, but the following question would probably be the same for the Polyakov action. According to David Tong's lecture notes on string theory, the Nambu-...
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Is there a Lagrangian $L$ (equivalently an action functional $S$) which yields the Navier-Stokes equation?

The Navier-Stokes equation or the Euler equation are usually derived as the conservation laws. However, I wonder if there exists a Lagrangian $L$ or equivalently, an action functional $S[\...
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Magnitude of the variations $\delta q_i$ in the principle of stationary action

To determine the equation of motion using the principle of stationary action, one has to consider the variation of the action due to variations $\delta q_i$ in all the generalized coordinates $q_i$. ...
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How do I calculate the functional derivative of the EM action on the curved spacetime with respect to the metric?

I am having trouble with computing the functional derivative with respect to the metric of the EM on a curved spacetime: \begin{equation} S:=\frac{1}{16\pi^2 G}\int R \sqrt{-g}\text{ }d^4x-\frac{1}{4}\...
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Constraining the metric on the boundary of space-time to get rid of unwanted equation of motion

I've posted this question on maths.stackexchange discussing the action I'm working with, which is (in Dirichlet boundary conditions) just the effective action of a minimally coupled free massive ...
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Confusion with the variational operator $\delta$ and finding variations

I have recently started studying String Theory and this notion of variations has come up. Suppose that we have a Lagrangian $L$ such that the action of this Lagrangian is just $$S=\int dt L.$$ The ...
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Effective action as a generating functional and its derivative expansion

On page 381 of Peskin and Schroeder, equation (11.90) reads $$ \frac{\delta^2 \Gamma}{\delta \phi_{cl}(x)\delta \phi_{cl}(y)} = iD^{-1}(x,y).\tag{11.90}$$ I am having a bit of trouble interpreting ...
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How are rigid solids approached in the context of Lagrangian formalism?

Maybe it's my own fault, but neither in my classical mechanics class nor within any book I've read on the subject I have found an extensive use of analytical mechanics to discuss the motion of solids. ...
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Symmetry of Scalar Action Associated with (Conformal) Killing Tensor

Short version: Consider the action for a scalar field coupled to the Ricci scalar in $d$ spacetime dimensions: $$S = -\frac{1}{2}\int d^dx \, \left(\nabla_\mu \phi \nabla^\mu \phi + \xi R \phi^2\right)...
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Anti-Symmetry of Dirac Operator

In his paper Fermion Path Integrals And Topological Phases, Witten states “Whenever one has a theory of fermions, the quadratic part of the fermion action is always antisymmetric by virtue of fermi ...
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Why does Action-Reaction pair cancel out in a system?

When I first studied Newton's third law, I heard that the action-reaction pair of forces do not cancel out as they are applicable on two different bodies. And that makes sense. However, people say in ...
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Is action ever stationary in quantum theory, or is it always minimal? [duplicate]

The question says it all. Is there an example from quantum theory of quantum field theory where action is only stationary, but not minimal? (There are such examples in classical physics - but do they ...
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Is it possible to built a variational principle for this first-order system?

Imagine there is a mechanical system described in unitary units by the equation: $$\dot{x} = -\text{sgn}(x)\sqrt{|x|},\quad x(0)=1 \tag{Eq. 1}$$ such it has a finite duration solution: $$x(t) = \frac{...
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How to deal with actions with interfering extrema?

How can one deal with actions with multiple extrema which difference is comparable to $\hbar$ (the extrema of the action are not widely separated) in the path integral method?
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How to Derive the 0 Temperature Action of Bose-Hubbard System?

I am currently reading chapter 8 of the textbook A Modern Approach to Critical Phenomena. This chapter deals with the Bose Hubbard Model and is filled with many equations that aid the analysis of this ...
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Why is the action integral of relativity particles $S = -mc\int ds$?

In my classical mechanic course material, it states that (In context of relativity) The path of a particle is called its "world line". Each world line can be noted mathematically using the ...
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What is the difference between variational principle, principle of stationary action and Hamilton's principle?

In advanced mechanics, we learn about the variational principle, the principle of stationary action, and the Hamilton's principle. I feel that the difference between them is not very clearly organized ...
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2 votes
1 answer
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Question about the Weiss variational of gravitational action and related equations of motion

I was reading The Weiss Variation of the Gravitational Action by Feng and Matzner, where the authors take the variations of the gravitational action with respect to the bulk metric $g$, the induced ...
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6 votes
1 answer
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Computing the DBI action on $S^5$

I've been looking at this paper (arXiv: 1103.4079). On page 7, from the metric of the giant gravtiton on $AdS_5 \times S^5$, $$ds^2 = -\cosh^2\rho \, dt^2 + d\rho^2 + \sinh^2 \rho \, d\tilde \Omega_3^...
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1 vote
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Numerical Solution of least action principle

I am trying to numerically find the path of least action between two points (ignoring the time step normalizing factor): $$S=\sum_i (x_{i+1}-x_{i})^2/2-V(x_i)$$ I don't have the potential in explicit ...
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Equivalent of Lagrangians after integration by parts

Given these three Lagrangians of a spin-1 quantum field $A_\mu$ $$ \begin{align} \mathscr{L}_1 &= \partial_\mu A^\nu \partial^\mu A_\nu \tag{1}\\ \mathscr{L}_2 &= \partial_\mu A^\mu \partial_\...
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2 votes
2 answers
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Hamiltonian from Lagrangian defined as an integral

I need to derive the Hamiltonian from the Lagrangian defined in the following way: $$L[x, \dot{x}] = \int_{t_0}^{t_1} f(x(t), t) \sqrt{1 + \dot{x}^2} \mathrm{d}t.\tag{1}$$ The usual method is to ...
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1 answer
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D'Alembert Principle and Euler-Lagrange. Virtual displacement

I have a little trouble with d'Alembert Principle and with virtual displacement. Imagine that with the d'Alembert Principle: $$ \sum_i \boldsymbol{\mathrm{F_i}} \; \cdot \delta \boldsymbol{\mathrm{...
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4 votes
3 answers
512 views

Does the Dirac Hamiltonian uniquely define the Dirac Lagrangian?

Example Hamiltonian: the linearised graphene Hamiltonian In condensed matter, we typically write down Hamiltonians instead of Lagrangians. An example is given by the Hamiltonian for graphene. When we ...
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How to show that the Action has units Energy·time?

The Lagrangian, which has units of Energy, is defined as that which when summed over time gives the Action, the action being more fundamental. But how does summing over units of Energy across time ...
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The meaning of symmetry in field theories (probably a notational problem)

I'm quite confused about the meaning of "symmetry" in the context of field theories. After reading many posts like 1-, 2-, 3-, 4- and 5-, I'm even overwhelmed. My first approach would be the ...
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What if the minimising path for the least action isn't unique? [duplicate]

Using the Lagrangian method, we find the path which minimises the action. But what if multiple paths each have the same minimum action? How do we resolve such a situation?
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What is self-action in quantum theory?

I read that the gravitational field in any quantum theory will be self-acting. What does it mean? How can a field interact with itself?
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1 answer
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Functional derivative for the action $S$

From Lancaster and Blundell's Quantum Field Theory for the Gifted Amateur, p. 15: Example 1.3 The Lagrangian $L$ can be written as a function of both position and velocity. Quite generally, one can ...
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1 vote
1 answer
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Demonstration of Noether's Theorem [closed]

So, as many, many people before me, I'm trying to get some insight on Noether's Theorem and its demonstration. As I'm in the process of self-teaching here, there are several things I'm "missing&...
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How can a boundary have an equation of motion? (Quadratic gravity with boundaries)

This question is the following of my previous one on the appearance of quadratic gravity from a simple scalar field treated non-perturbatively. In the following, I will not specify the $\epsilon$ term ...
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1 vote
2 answers
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Noether identities and the relativistic point particle

I am trying to better understand Noether identities, i.e. relations between equations of motion in the presence of gauge symmetries for the example of the relativistic point particle. Formally, a ...
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How can the scaling dimension of a scalar field be $1$ in a $1d$ CFT?

In a one-dimensional scalar field theory, the kinetic term of the action takes the following form: $$S_\text{kin} = \int_\mathbb{R} dt\, \frac{1}{2} \dot{\varphi}(t) \dot{\varphi}(t)\,, \tag{1}$$ with ...
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Is Weyl gauge together with Coulomb gauge a possible choice?

I am working in the Weyl gauge with an action in euclidean signature (so the Lagrangian is a Hamiltonian): \begin{equation} S=-\frac{1}{96\pi^2}\int_{\mathbb{R}^4} d^4 x\,\text{Tr}\left( E_i \cdot ...
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3 votes
1 answer
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How can I show that the action of a SHO is a saddle-point solution if $t_{f} - t_{0} > T/2$?

In this post and in this post, QMechanic claims that a simple harmonic oscillator with Dirichlet boundary conditions has saddle-point solutions if $t_{f} - t_{0} > T/2$ where $T$ is the ...
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6 votes
2 answers
210 views

In a theory with spinor fields, what general condition on the action ensures that the stress-energy tensor can be made symmetric?

In general relativity, the stress-energy tensor is normally defined by $$ T^{ab}\equiv \frac{2}{\sqrt{|g|}}\,\frac{\delta S_m}{\delta g_{ab}}, \tag{1} $$ where $S_m$ is the "matter" action (...
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2 votes
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Boundary terms of action in AdS

Let's consider the action of a free scalar field in a space-time with metric $g_{\mu \nu}$ $$ S = -\frac{\eta}{2} \int d^{d+1}x \, \sqrt{g} \{g^{AB} \partial_{A} \phi \partial_{B} \phi + \frac{1}{2} m^...
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Help solving one dimensional third order Euler-Lagrange condition

I have the following Lagrangian: $$ \mathcal{L}(t, x, \dot{x}) = \frac{df}{dt} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} \dot{x} + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} b(...
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What is the reason behind the stationarity of action? [duplicate]

I am reading Goldstein right now to understand the least action principle. I understood that the action needs to be stationary under small variation and this specifies the equation of motion, but do ...
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3 votes
1 answer
128 views

Wald's approach to deriving the Einstein field equations and the Levi-Civita connection through Palatini's action

I'm reading Appendix E of Wald's General Relativity book and I'm a bit confused in how he derives the Einstein field equations and the Levi-Civita connection through Palatini's action. The Palatini ...
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Missing minus sign when taking derivative

I'm trying to understand to get the following formula (first formula on pg 33) in Altland Simons second edition: $$\Delta S \simeq \int d^m x (1 + \partial_{x_\mu} \, (\omega_a \, \partial_{\omega_a} \...
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1 answer
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Einstein-Hilbert action and antisymmetry of derivative of $g_{\mu \nu}$ in Christoffel symbols

In the context of the Einstein-Hilbert action $S_{EH}$, we have to compute $\delta R_{\mu\nu}$ and at a moment, we have a term $\delta \Gamma^{\alpha}_{\alpha \mu}$ to compute. I'm wondering why do we ...
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How would a time-dependent $G$ affect the derivation of the Einstein field equations?

Let's say that the gravitational constant changes with time. $~G~\to G(t)$. ​​​​ (It is essentially an isotropic and homogeneous scalar field) If we were to re-derive the Einstein field equations ...
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When does a Lagrangian exist for arbitrary equations of motion? [duplicate]

Let's say I have some equations of motion for an arbitrary system, i.e. some implicitly or explicitly defined ODE involving $q = (q_1, q_2, q_3, \dots)$ and $\dot q = (\dot q_1, \dot q_2, \dot q_3, \...
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2 votes
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Calculating the path integral for cubic interactions (perturbatively)

I'm trying to apply the Coleman-Weinberg mechanism to the weakly interacting, $g \ll 1$, $\mathbb{Z}_2$-symmetric $\phi^6$-theory in $d = 3 - \epsilon$ dimensions (in Euclidean signature) \begin{...
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