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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Action functional of Born-Infeld model

I have a Born-Infeld action functional like this $$I[A,\phi]~=~\int b^2(\sqrt{1+(|\bigtriangledown\times A|^2)/b^2}-1)+|D_A\phi|^2 + b^2(1-\sqrt{(1-|\phi|^2)^2/b^2} ).$$ Have any books or notes talk ...
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What does it mean for an action to be defined “on-shell”?

Some actions like 11D supergravity are defined "on-shell". What does this mean exactly? Can you give me an example? Say for example the Klein-Gordon action. Can this be defined on-shell too?
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Behaviour of action with respect to time

I was wondering if it was possible to say something general on the behaviour of the action : $$ S[x(\tau)]=\int_0^T L(x,\frac{dx}{d\tau},t) dt $$ (where $x(\tau)$ defines a trajectory, with certain ...
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What is $\mathcal{L}_M$?

Usually we derive the Einstein field equation in vacuum starting from E-H action $$S= \int{\sqrt{-g}d^4x(\frac{c^4}{16\pi G})R}.$$ But in case we wanted to get ...
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Invariance under local diffeomorphisms

In the context of the Polyakov action, the action for a relativistic point particle $$ S_P = \frac{1}{2} \int \mathrm{d}\tau \, e(\tau) \left(\frac{1}{e^2(\tau)}\left(\frac{\mathrm{d} ...
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Why does overall action need to have an extremum?

Quoting from Landau's and Lifshitz' Mechanics : The integral ${\int\limits_{t_1}^{t_2}}L(q, \dot{q},t)\,dt$ for the entire path must have an extremum, but not necessarily a minimum. This, ...
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Function for which action is the minimum

On page 2 of "Mechanics" Landau & Lifshitz say that $q=q(t)$ is a function for which action is a minimum. Before this they say that at times $t_1$ and $t_2$, the system occupies coordinates ...
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Derivative of the action integral [closed]

I need to find the partial derivative of the action $S$ with respect to the generalized coordinate $q(t_f)$ and according to my textbook, it should equal the generalized momentum $p(t_f)$. I ...
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Covariant form of non-relativistic free particle

I have two questions about the action of free particle. $$S=\int dt~\frac{m}{2}~(\frac{d \vec{x} }{dt})^2 \tag{1}$$ The Covariant form is: (assume: $m=1$) $$S=\int d\tau ...
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Action max, min, or saddle?

It is well known that $\delta S = 0$ lays the foundation for variational mechanics. But I am confused as to whether or not this S is a minimum, a maximum, or a saddle point. Some books address this ...
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107 views

Derivation of Euler-Lagrange equations in Landau's and Lifshitz's “Mechanics”

There's an integral ${\int\limits_{t_1}^{t_2}}(\frac{\partial{L}}{\partial{q}}{\delta}q+\frac{\partial{L}}{\partial{v}}{\delta}v)dt=0$. [1.] $ {\delta}v={\frac{d{\delta}q}{dt}}$ [2.] I should get $ ...
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Lagrangians not related via a total time derivative lead to same Noether symmetries?

Having answered my initial two questions (v1), I now consider a third possibility. Consider two Lagrangians that both lead to equivalent equations of motion. Suppose that they are not related via a ...
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Maxwell and Bianchi equations

We frequently see: A certain action and then we are asked to solve for Bianchi identity and Maxwell equation. I have often solved for them but I never knew what is the difference between the two? In ...
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385 views

Physical meaning of the Lagrangian function [duplicate]

In Lagrangian mechanics, the function $L=T-V$, called Lagrangian, is introduced, where $T$ is the kinetic energy and $V$ the potential one. I was wondering: is there any reason for this quantity to ...
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Bulk action - Can a brane velocity be defined?

a) If a brane action in a bulk is defined, in that case, that a brane is modelwise moving through a bulk, how is this ratio defined? Is this a regular "velocity" in that meaning, that space is being ...
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338 views

Goldstein's derivation of the 'principle of least action'

I want make an punctual question ands it's about The derivation of the expression $$ \Delta\int_{t_1}^{t_2} Ldt=L(t_2)\Delta t_2-L(t_1)\Delta t_1 + \int_{t_1}^{t_2} \delta L dt. \tag{8.74}$$ You can ...
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55 views

Different actions equivalence in general relativity?

I started reading up on the different actions that appear in GR. I have come across two: When looking at geodesics I found: $$S=\int \sqrt{-g_{\mu \nu} \frac{dx^\mu}{d\lambda} ...
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qualitative explanation of Principle of Least Action (vertical movement)

Consider the following situation I want to understand what the PLA means here from an intuitive and qualitative point of view. I understand the mathematical approach. Combining ...
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Conserving Action (Lagrangian Mechanics) [closed]

I posted a question similar to this yesterday. The reason was because I was trying to figure out this problem and the question, while not pertaining directly to the problem, I thought could help me ...
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Deriving Expression for variation of Action (Lagrangian Mechanics)

I am studying Lagrangian mechanics and I have come across something that I do not understand. Basically the text I am reading skipped steps and I do not know how to get from point A to point B. I ...
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175 views

Why are the Euler-Lagrange equations invariant if we add a surface term to the action?

In the lecture on Noether's theorem and the Lagrange formulation of classical field theories, my professor wrote A symmetry is a field variation that maps solutions to solutions, which is true if ...
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Deriving velocity after elastic and inelastic collision via the Principle of Least Action

I am reading on the Principle of Least Action from a historical perspective. I am also trying to make sense of it from a contemporary point of view -- though my training in contemporary physics is ...
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Why should physical theories always have a Lagrangian formalism? [duplicate]

I've often heard that every physical theory has some kind of Lagrangian formalism, or a formalism in terms of a principle of stationary action. The Standard Model has one, General Relativity has one, ...
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Equivalence of functional and partial derivatives

I am trying to derive Newton's second law from the principle of least action, that is, setting the functional derivative $\frac{\delta S}{\delta x(t)}$ equal to 0. $$S = \int dt' \left[ \frac{m}{2} ...
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Is there any reason for principle of least action to be true? [duplicate]

My question is not rigidly related to physics. The principle of least actions says that for any dynamical system there exists a function parameterized by $q$'s and $\dot{q}$'s such that the line ...
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What is a macro world example of “action”? [duplicate]

What is a real world example of action? This question originally came to me when learning about Planck's constant $h$. Planck's constant $h$ is measured in Joule * Seconds. I know Joule * Seconds can ...
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Why don't all free particles lose their kinetic energy?

I'm currently studying Action. I've been reading about how a particle has particular probabilities of ending at an infinite number of events. Say I have a free particle that isn't experiencing any ...
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On the connection between forces and the principle of stationary action

Feynman tries to account for the relation between the principle of stationary action, which is a statement about the whole path of a particle, and Newton's second law, which is a statement about the ...
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Finding the action of a discretized Lagrangian

I am trying to find the action associated with the Lagrangian density $$ \mathcal{L} = \frac{1}{2}\left( \frac{\partial\phi}{\partial x} \right)^2 + \frac{1}{2}m^2\phi^2. \tag{1} $$ I am supposed to ...
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56 views

Physical significance of omitting a purely time dependent term from a Lagrangian

For a simple pendulum whose point of support moves on a vertical circle of radius $a$ with constant frequency $\gamma$, you can write the Lagrangian down. The potential energy can be written as ...
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3-cylinder surface element (Poisson's “A Relativist's Toolkit”)

From Poisson's "A Relativist's Toolkit": he introduces the non-dynamical term $$ S_0=\frac{1}{8\pi}\int_{\partial\Omega}\epsilon K\sqrt{\lvert h\rvert}d^3x $$ in the GR action, where $h$ is the ...
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The Nambu-Goto action how do we know the Hamilton's principle applies?

I am reading 'A first course in string theory' by Barton Zwiebach (2ed) on page 112 he comes up (after a small derivation) the action formula: $$S=-\frac{T_0}{c} \int d\tau d \sigma \sqrt{-\gamma}.$$ ...
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Help me calculate the Euclidean action of a gravitating system!

I recently read Gibbons and Hawking's paper Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752. I am interested in repeating their calculations. It is fairly ...
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50 views

Reversing time for a closed system of particles

For a closed system of particles, the lagrangian in classical mechanics is $$L=\sum \frac{1}{2}mv_a^2 - U(\mathbf{r_1},\mathbf{r_2}, \cdots)$$ For an arbitrary position function $x(t)$, to see the ...
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What exactly is the Action? (Learning lagrangian)

I have been trying to wrap my head around lagrangian mechanics but I find some parts confusing. For example, what exactly is action and why is it defined by the Kinetic energy minus the potential ...
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Classical trajectories that are not a minimum of the action [duplicate]

Are there physically realizable dynamical systems where the true trajectory is not a minumum action trajectory? Formally, Lagrangian mechanics only requires that the trajectory be an extremum (or ...
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Derivation of an ordinary, Lagrangian/Hamiltonian and action formulation

I am confused as to how the different formulations in physics are derived. In many fields of physics, we usually begin with an ordinary formulation (e.g Newton's Laws in classical mechanics), and ...
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Hamilton's Principle - achieving Hamilton equations

Consider the action function: $$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$ where $\mathcal{L}$ is the Lagrangian of the system. The Hamiltonian is defined by the following ...
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Does the principle of stationary action always work? [duplicate]

Give some Lagrangian we use the principle of stationary action to find the desired euqations of motion for something (e.g. a field). A lot of modern physics seems to be based on the principle of ...
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How does one express a Lagrangian and Action in the language of forms?

In Lipschitzs Classical Mechanics a Lagrangian is defined as: $L(q,q',t)$ for some trajectory $q(t)$ of a particle And the action is defined as: $S:=\int^a_b L(q,q',t) dt$ How does one ...
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Does Conformal Invariance of the Polyakov Action in Conformal Gauge imply Conformal Invariance of the Pre-gauge-fixed Polyakov Action?

In bosonic string theory the Polyakov action can be put in into conformal gauge. It is then possible to show that the resulting gauge fixed action is conformally invariant. Actually it's shown that ...
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Proving independence of the lagrangian on position of a free particle using the euler-lagrange equation

I asked a similar question some time back but am trying to work this from another angle. In deriving the lagrangian of a free particle, we use the homogeneity of space to conclude that the lagrangian ...
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138 views

Equation of motion of an auxiliary field

I'm a newbie in the field of QFT and SUSY, so I'm warning you: this might be a stupid question. I'm working with auxiliary fields to describe supersymmetric models and I understand that upon ...
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Deriving lagrangian of a free particle - How do you arrive at Lagrangian independency conclusions

I guess this question has been asked before, but I'm looking at a slightly different aspect. I'm reading Landau's book on classical mechanics. In deriving the lagrangian for a free particle, I ...
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Decoupling of generalized coordinates in lagrangian

Say you have a lagrangian $L$ for a system of 2 degrees of freedom. The action, S is: $S[y,z] = \int_{t_1}^{t_2} L(t,y,y',z,z')\,dt \tag{1}$ If $y$ and $z$ are associated with two parts of the ...
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If I flex my arm, where is the “equal and opposite” reaction?

In my sight, nothing happens at all. Is the opposite reaction pressure applied to my bones? It certainly seems so; however, since I flex my arms in a curve, shouldn't the opposite reaction direction ...
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Information contained in Lagrangians and actions [duplicate]

I've been looking into analytical mechanics with the intention of finding out more about Lagrangians and actions. As far as I currently understand it, the Lagrangian is formed with positions and ...
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Formulating a symplectic integrator for a non-local Hamiltonian

I recently asked two questions, Q. [1] and Q. [2], regarding reformulating non-local Lagrangians as Hamiltonians. In these questions, the Hamiltonian is formulated as an integral because of it's ...
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An inconsistency in Hamiltonian formulation for non-local Lagrangian: what am I doing wrong?

This question is based on a previous question I asked, Q. [1] In this question, I proposed an example of a non-local Lagrangian (functional), I'm revisiting it here: $$\mathbb{L}=\frac{1}{2}\int^t_0 ...
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Legendre transform for non-local Lagrangians, or Hamiltonian of non-local Lagrangian and their properties

This is sort of a multi-part question, mostly dealing with how to treat non-local Hamiltonians and how the corresponding properties of Hamiltonians work in a non-local framework. I proposed an example ...