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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Energy and momentum as partial derivatives of on-shell action in field theory

According to L&L, if we fix the initial position of a particle at a given time and consider the on-shell action as a function of the final coordinates and time, $S(q_1, \ldots, q_n, t)$, then... ...
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Coefficient matrix of quadratic Lagrangian

I've been studying path integrals from Weinbergs QToF vol 1. He says that when the $\mathcal{L_0}$ is quadratic in fields we can always write free term $I_0$ in the generalized quadratic form ...
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79 views

Proof that total derivative is the only function that can be added to Lagrangian without changing the eom

So I was reading this: Invariance of Lagrange on addition of total time derivative of a function of coordiantes and time and while the answers for the first question are good, nobody gave much ...
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Why are D'Alembert's Principle and the Principle of Least Action Related?

Why do we get the same differential equations from both principles? Surely there is a fundamental connection between them? When written out, the two seem to have nothing in common. $$\sum _i ( ...
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Why does reparameterisation invariance lead to gauge-fixing?

In Becker, Becker and Schwarz, the point particle action is given in terms of an auxiliary field $e(\tau)$ as: \begin{align} \tilde{S}_0 = \frac{1}{2}\int \,d\tau \left(e^{-1}\dot{X}^2 - m^2e\right) ...
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The meaning of action

The action $$S=\int L \;\mathrm{d}t$$ is an important physical quantity. But can it be understood more intuitively? The Hamiltonian corresponds to the energy, whereas the action has dimension of ...
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What is the physical meaning of the action in Lagrangian mechanics?

The action is defined as $S = \int_{t_1}^{t_2}L \, dt$ where $L$ is Lagrangian. I know that using Euler-Lagrange equation, all sorts of formula can be derived, but I remain unsure of the physical ...
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276 views

Energy-momentum tensor of Bosonic Ghost Action in String Theory

When quantizing bosonic string theory by means of the path integral, one inverts the Faddeev-Popov determinant by going to Grassmann variables, yielding: $$ S_{\mathrm{ghosts}} = ...
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Action and Action integral: Different kinds of variational principles

What are the difference between: the action $\int_{t_{1}}^{t_{2}}(L+H) dt$ that we use in the principle of least action, and the action integral $\int_{t_{1}}^{t_{2}}L dt$ that we use in ...
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Quantum entanglement and spooky action at a distance

When quantum entanglement is explained in "layman's terms", it seems (to me) that the first premise, that we have to accept on faith, is that a particle doesn't have a certain property (the particle ...
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Energy Tensor, covariant derivate, variation respect to the metric [duplicate]

I'm doing the variation of a Lagrangian respect to the metric, but I am having problem with a particular terminus. My action is: $$ S=\int d^4x \sqrt{-g}[ (\nabla_\mu A^\mu)^2]$$ My lagrangian is: ...
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Difference between Gravitational and Matter Scalar Fields

In the context of Scalar-Tensor theories of gravity (for example in Brans-Dicke) what is the difference between gravitational and matter scalar Fields? My doubt comes from "The scalar-tensor Theory ...
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Functional Derivative in the Linear Sigma Model

In the linear sigma model, the Lagrangian is given by $$ \mathcal{L} = \frac{1}{2}\sum_{i=1}^{N} \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) ...
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CM: Need to recover the Hamiltonian, knowing conserved quantities and information about the EOM, possibly via action-angle coordinates

Statement of the problem: I have a system with 2 degrees of freedom and I have found two independent conserved quantities, without knowledge of the Hamiltonian. I'm looking for a method to recover a ...
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Question about an integration by parts in Feynman's Quantum Mechanics [closed]

I have begun reading Feynman & Hibbs Quantum Mechanics and Path Integrals. Knowing little about variational calculus or Lagrangians I found the following integration by parts opaque. I think if I ...
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52 views

Derivation of the Noether current

(c.f Di Francesco et al, Conformal Field Theory, pp40-41) I am trying to derive eqn (2.142) or $\delta S = \int d^d x \partial_{\mu}j^{\mu}_a \omega_a$ in the book CFT by Di Francesco et al. I have ...
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Confusion regarding the principle of least action in Landau's “The Classical Theory of Fields”

Edit: The previous title didn't really ask the same thing as the question (sorry about that), so I've changed it. To clarify, I understand that the action isn't always a minimum. My questions are in ...
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1answer
64 views

The Einstein-Hilbert Action On-Shell

If one consider the Maxwell action as $$S=-\int \mathrm{d^{4}}x\! \ \frac{1}{4}F_{ab}F^{ab} \,$$ one find the usual Maxwell equation $$\partial_{a}F^{ab}=0$$ Then one can simply arrive the following ...
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Deriving field equation in Yang Mills theory

Trying to show that $$D_\mu\vec{F^{\mu \nu}} = \partial_{\mu}\vec{F^{\mu \nu}} + g \vec{A_\mu} \times \vec{F^{\mu \nu}} = 4 \pi \vec{J^\nu},$$ or (correct me if I'm wrong) $$ \partial_{\mu} F^{\mu ...
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124 views

A Question on Hamilton's Principle

In some literatures, the Hamilton's principle for conservative systems is introduced by this equation: $$\delta \int_{t_1}^{t_2}(T-V) ~\mathrm{d}t~=~0$$ In some others, this principle is introduces ...
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Derivation of Noether's theorem - A problem with physical significance

My question is about the field theoretic version of Noether's theorem. I am deeply troubled by one of the hypotheses of the theorem. As it is the standard textbook for Lagrange mechanics, I'll follow ...
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2answers
107 views

Variation of Action with time coordinate variations

I was trying to derive equation (65) in the following review: http://relativity.livingreviews.org/open?pubNo=lrr-2004-4&page=articlesu23.html This slightly unusual then usual classical mechanics ...
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1answer
69 views

Getting the Lagrangian from the action in curved spacetime

Suppose I have this action: $$ S = \int \mathrm d^4 x\sqrt{-g}\times \text{something}$$ where $g$ is the determinant of the metric. Should I take the Lagrangian to be: $$ \mathcal L = \sqrt{-g} ...
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Source material desired for behavior of derivatives of action

I'm basically looking for concise commentary, and especially source material/ short discussion pertaining to the following, which I will (emphasizing loosely) state as follows: Suppose a given action ...
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1answer
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The Euler-Lagrange equation in special relativity

How can I derive the Euler-Lagrange equations valid in the field of special relativity? Specifically, consider a scalar field.
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Partial derivative of the classical action with respect to time [closed]

Does anyone know how to derive the general identity: $$\frac{\partial S}{\partial t}=-E$$ where $S$ is the classical action defined as $$S=\int_0^t\left[\frac{1}{2}m\dot x-V(x))\right]d\tau$$ and ...
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Deriving massless point particle action from Maxwell action?

Starting with the Maxwell action for a $U(1)$ vector gauge boson with a general metric and (I'm assuming) using a plane wave ansatz for the vector, is it possible to derive the action for a massless ...
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127 views

Stationary points of the action functional

In QFT the principle of stationary action states that we choose fields that will make the action stationary but what if the action has many stationary points? What's the significance of these other ...
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Is Einstein-Hilbert action the unique action whose variation gives Einstein's field equations?

I know that scaling the action with a non-zero multiplicative constant, or adding a total divergence term to the Lagrangian density do not change the Euler-Lagrange equations, cf. e.g. this ...
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7answers
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Why the Principle of Least Action?

I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action ...
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1answer
107 views

Functional field integral in condensed matter field theory (Altland)

This is the action for the 1+1 dimensional interacting electron system; $$S_{cl}[\theta , \phi]= \frac{1}{2\pi} \int dxd\tau \left(g^{-1}v(\partial_x \theta)^2 + gv(\partial_x \phi)^2 + ...
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Hamilton-Jacobi Equation

In the Hamilton-Jacobi equation, we take the partial time derivative of the action. But the action comes from integrating the Lagrangian over time, so time seems to just be a dummy variable here and ...
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Noether's current expression in Peskin and Schroeder

In the second chapter of Peskin and Schroeder, An Introduction to Quantum Field Theory, it is said that the action is invariant if the Lagrangian density changes by a four-divergence. But if we ...
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251 views

Calculating the (on-shell) action of a free particle

I am having difficulty with the first problem from Feynman and Hibbs' book. For a free particle $L = (m/2)\dot{x}^2$. Show that the (on-shell) action $S_{cl}$ corresponding to the classical ...
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Lagrangian for relativistic massless particle

For relativistic massive particle, the action is $$S ~=~ -m_0 \int ds ~=~ -m_0 \int d\lambda ~(\dot x ^\mu \dot x_\mu)^{\frac{1}{2}} ~=~ \int d\lambda \ L,$$ where $ds$ is the proper time of the ...
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Why does Principle for least action hold for classical fields [duplicate]

Let $\mathscr L (\phi(\mathbf x), \partial \phi(\mathbf x))$ denote the Lagrangian density of field $\phi(\mathbf x)$. Then then actual value of the field $\phi(\mathbf x)$ can be computed from the ...
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1answer
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Field equations in extended EH-GHY action. Is Schwarzschild a solution?

When taking the EH action, $$S_{EH} = \frac{1}{16\pi G}\int_M d^4x \sqrt{-g}R$$ and making a small variation in the metric while ignoring boundary terms, we obtain $$\delta S_{EH} = \frac{1}{16\pi ...
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Classical electrodynamics formulated in terms of forces

The Newton's law of universal gravitation is described in terms of a force, which is produced by an action at a distance. It also can be described using the concept of a field, and that would be an ...
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Action for $p-p'$ strings (equation 13.5.21 in Polchinski's textbook)

This action reads $$S=-\frac{1}{4g_{D9}^2}\int d^{10}x F_{MN} F^{MN}-\frac{1}{4g_{D5}^2}\int d^{6}x F'_{MN} F'^{MN}- \int d^6 x \left[ D_{\mu} \chi^{\dagger} D^{\mu} \chi + ...
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1answer
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Why we can set variations for the metric and its derivatives to zero at infinity?

This question is the continuation of the following one. I still don't understand why $(1)$ may be set to zero. This refers to the zero value variations of metric and its derivatives on the infinitely ...
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Intuition for actions written as integrals over spacetime

Right now I'm simply looking for an intuitive explaination of actions that integrate over a 4-volume element, $d^4x$ rather than a parameter say $\lambda$. More specifically I'm well versed in action ...
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1answer
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Can I applicate the law of action and reaction on energy?

It is clear that if you push on some object, there is reaction of the same force. But is it the same energy? Thanks a lot.
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1answer
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Normal to the Hypersurfaces

I am trying to understand the derivation of the Hilbert-Einstein action. However it requires a knowledge about hyper-surfaces for the boundaries of the integrals and also about the normal to the ...
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How do I obtain the Lagrangian in standard for using action? [closed]

I have action as shown below $$S=\int \mathrm{d}t \int \mathrm{d}x^3 \bar\psi\left(i\partial_t\psi +\frac1{2m}\bar\nabla^2\psi-V(x)\psi\right)$$ How do I manipulate it to obtain the Lagrangian ...
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Must the action be a Lorentz scalar?

Page 580, Chapter 12 in Jackson's 3rd edition text carries the statement: From the first postulate of special relativity the action integral must be a Lorentz scalar because the equations of ...
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Does the action and Lagrangian have identical symmetries and conserved quantities?

From the book Introduction to Classical Mechanics With Problems and Solutions by David Morin, page 236 states: Noether's Theorem: For each symmetry of the Lagrangian, there is a conserved ...
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How the boundary term in the variation of the action vanishes

In David Tong's QFT lecture notes (Quantum Field Theory: University of Cambridge Part III Mathematical Tripos, Lecture notes 2007, p.8), he states that We can determine the equations of motion by ...
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How Hamilton's Principle was found?

Hamilton's principle states that the actual path a particle follows from points $p_1$ and $p_2$ in the configuration space between times $t_1$ and $t_2$ is such that the integral $$S = ...
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1answer
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Does action really have to be Lorentz-invariant in SR?

From Landau & Lifshitz The Classical Theory Of Fields it is said: To determine the action integral for a free material particle (a particle not under the influence of any external force), we ...
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106 views

Differential Operators in Polyakov Action

What do the differential operators in the Polyakov action mean? How does one derive the Polyakov action and treat the differential operators?