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 variables in canonical transformations

Let's suppose we have a Hamiltonian $H(p_k, q_k)$ and we want to transform it via a canonical transformation to one Hamiltonian which doesn't depend on the new coordinates $w_k$, but only in the ...
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Actions that are not integrals

So far every action I've seen in physics has been an integral of a Lagrangian, be it a point particle: $$S = \int dt\ L$$ or fields (relativistic or not): $$S = \int d^4x\ \mathcal{L}$$ and so ...
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Principle of Least Action Question

Let's say we have a particle with no forces on it. The path that this classical particle takes is the one that minimizes the integral $$\frac{1}{2}m\int_{t_i}^{t_f}v^2dt.$$ So if we graph this for ...
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Why are the Nambu-Goto action and Polyakov action equivalent at quantum level?

It's a well known elementary fact that the Nambu-Goto action $$S_{NG} = T \int d \tau d \sigma \sqrt{ (\partial_{\tau} X^{\mu})^2 (\partial_{\sigma} X^{\mu})^2 - (\partial_{\sigma} X^{\mu} \partial_{\...
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What is the intuitive concept of the action of a relativistic point particle? [duplicate]

The action of a relativistic point particle is its negative rest energy along its worldline, the parameter being its own proper time. $$ S = - mc^2 \int d\tau $$ (see Wikipedia) Action is energy ...
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Why the action of a relativistic point particle is considered to be negative? [duplicate]

The action of a relativistic point particle is its negative rest energy along its worldline, the parameter being its own proper time. $$ S = - mc^2 \int d\tau $$ (see Wikipedia) Is there a ...
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Nambu-Goto and Polyakov Actions

This might be a little bit of a technical question, so bear with me. Ok, so from string theory we know that the action for a relativistic string is found from the worldsheet when we embed the string ...
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Are the partial derivatives of Lagrangian in the varied action functional derivatives?

In particle mechanics Lagrangian $L$ depends upon position, velocity (and may be explicitly on time), whereas in field theory the Lagrangian density ${\cal L}$ similarly (or analogously) depends upon ...
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Vary action with respect to velocity

Variation of the action $S$ corresponding to a Lagrangian e.g. $L(x(t),\dot{x}(t))$ gives the Euler-Lagrange equations: $$ \frac{\delta S}{\delta x(t)} = 0 \\ \int du \left ( \frac{\delta L}{\delta x(...
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Why the Lagrangian $L$ is KE - PE? Why not KE + PE!

With Lagrangian, is there any way to intuitively grasp why total energy equals the difference between the kinetic and potential energy? Seems counter-intuitive - whereas Hamiltonian calculation (sum ...
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Action of a massive free point-particle in relativistic mechanics

I was reading about the formulation of mechanics in special relativity and found that the action for a massive free point-particle as $$ S = -mc\int_a^b ds $$ So, I did a few observations, ie. $$ S =...
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Problem obtaining string equations from Polyakov action [closed]

I am trying to obtain the string equations of motion from the Polyakov action in the conformal gauge, i.e.: $$ S=T\int{d\tau d\sigma (\dot{x}^2-x^{'2})}\equiv\int{d\tau d\sigma \mathcal{L}} $$ where ...
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Going to the Einstein frame in f(R) theories

First of all thank you for your time! I have a question that I can't solve. In every review that I read, I find that when you want to go to the Einstein frame in a $f(R)$ theory what you have to do ...
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Feynman's Path Integral Approach: The Complex Exponentiated Action [duplicate]

I'm working on a project covering Feynman's Path Integral Approach. I'm having trouble intuitively grasping what motivates the introduction of the expression $e^\frac{iS}{\hbar}$, where S is the ...
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How are Lagrangians in QFT constructed?

Various particle equations (like the K-G equation, the Dirac equation, the Proca equation etc.) in QFT are derived by applying the Euler-Lagrange equations to the Lagrangian density. But how are these ...
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Is there any general theorem which specifies conditions where the critical solution of an action is unique (for given boundary conditions)? [duplicate]

Consider a classical mechanical system with generalized coordinates $q_i$, $i \in \{1,\dots\,n\}$. And Lagrangian $L$. Given a path $\gamma$ (with coordinates $\gamma_i$) and two times $t_1$ and $t_2$ ...
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Total derivatives in GR

Without gravity we can easily switch between terms in a Lagrangian, such as $\partial\phi\partial\bar{\phi}$ and $\phi\Box\bar{\phi}$, since total derivative vanishes. But in GR we have additional $e\...
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Lagrangian in polar coordinates [closed]

$$L=\frac{1}{2}mv^2=\frac{1}{2}m(\dot{x}^2+\dot{y}^2)$$ $$L=\frac{1}{2}mv^2=\frac{1}{2}m(\dot{r}^2+r^2\dot{φ}^2)$$ I dont get this part. $$\frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{φ}}}\...
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Supersymmetrizing bosonic actions at higher orders

Given only the bosonic terms of a supersymmetric action, using a knowledge of the (local) supersymmetry transformations, is there a systematic way of reconstructing the fermionic terms? More generally,...
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Field equations of a given action

Provided an action: $$S[A_\nu] = \int\left(\frac{1}{4\mu_0}(A_{\gamma,\mu}-A_{\mu,\gamma})(A_{\zeta,\alpha}-A_{\alpha,\zeta})\eta^{\gamma\zeta}\eta^{\mu\alpha}+\frac{1}{2}\nu^2A_\mu A_\gamma -\beta A_\...
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Physical motivation for Lagrangian formalism

This is more of a request for clarification of understanding and intuition rather than a question, but I hope people can help me with it. I have learned calculus of variations and have subsequently ...
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Higher than Lagrangian/action?

When you begin learning physics, you start with equations of motion applied to various physics systems. In classical mechanics course you learn, that exists Lagrangian/action of a system, which gives ...
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Why is a theory Lorentz invariant if the Lagrangian is Lorentz invariant?

For if I started by trying to make the Hamiltonian Lorentz invariant, I would have failed. Indeed, the Hamiltonian is part of a covariant tensor. But how do I know that the Lagrangian is not a part of ...
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Must there exist a Lagrangian for any 2nd order ordinary derivative equation?

We know if there exist a Lagrangian of some ODE, then it must exist many equivalent Lagrangian. My question: Then must there exist a Lagrangian for any 2nd order ODE? If not, do we have some ...
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Is there an action for every physical law?

Given an action, I can get the differential equation governing the evolution of the system by applying the principle of least action. Does it work the other way around? Given any differential ...
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Understanding Derivation of Euler-Lagrange

I am trying to understand the derivation of the Euler-Lagrange equation. I drew a graph below. So, according to the graph, $$ \int_{t_1}^{t_2} L(x+\delta{x},\dot{x}+\delta\dot{x}\,t) dt - \int_{...
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In the Principle of Least Action, how does a particle know where it will be in the future?

In his book on Classical Mechanics, Prof. Feynman asserts that it just does. But if this is really what happens (& if the Principle of Least Action is more fundamental than Newton's Laws), then ...
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Computing the value of an Action given some boundary conditions

Having being dealing with Actions for a while I have come across a question in which I am required to calculate the value for $S$ an action in the form of a function for some given boundary conditions....
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Meaning of dt/dx when deriving the law of reflection

One way to derive the law of reflection, you can use the principle of least action to minimize the action path of motion of light. They key concept while doing this is to take the derivative of the ...
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Principle of Stationary Action and Euler-Lagrange Equation

Principle of Stationary Action: Given a mechanical system, there exists an action $S$ such that it is extremitized, or $\delta S=0$, for the actual motion of the system. $$S = \int_{t_1}^{...
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How to calculate Lagrangian density function in classical field theory

In Lagrangian mechanics observing the possible degrees of freedom we first write down our Lagrangian. Then we use E-L equation to determine equation of motion and using sufficient boundary condition ...
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Coordinates from action-angle variables

I'm interested into getting the original coordinates, $q(t)$ and $p(t)$, from the action, $J=\oint p dq$, and angle, $w(t)=\frac{dH}{dJ}t+\beta$, variables for a 1-D, one particle system. I know that ...
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Max & inflection point in the principle of least action [duplicate]

Short question: What is the physics interpretation of max & inflection points in the principle of least action? Long question: If $$L(q_1,q_2;t)=K-V$$ then let $$S = \int^{t_1}_{t_2} L(q_1,...
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Polyakov From Nambu-Goto Directly, for Strings?

The following derivation, for a classical relativistic point particle, of the 'Polyakov' form of the action from the 'Nambu-Goto' form of the action, without any tricks - no equations of motion or ...
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Simplify calculation of geodesics from action principle

I don't understand a step with the calculation of geodesics equations from action principle on this link : demo geodesics equations My issue is the following step : $$\int \bigg(\dfrac{dx^{\mu}}{d\...
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General Relativity as a Special Relativistic Field Theory

In this question, I want to consider only the classical case. I have seen the statement that general relativity can be considered as a spin-2 field living on a Minkowski background. In that case, you ...
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Are cumulus clouds analogous to airships flying downwards?

First of all, cumulus clouds are amazing. Big puffy white clouds floating on the air. Some of them produce updrafts of over 100 Km per hour. Now, if an airship had its engine pointed towards the ...
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Are there (interesting) Poincare-invariant QFTs with non-invariant Lagrangian densities?

In all QFTs I know, the Lagrangian density is completely invariant under the Poincare group, $$ \mathcal L \to \mathcal L. $$ On the other hand, the action would be invariant even if the Lagrangian ...
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Action with self-dual field strength

It is said that writing down an action in presence of a self-dual field strength is subtle and not known till date. The familiar example people give is that of type IIB super-gravity which has a self-...
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Equivalence between principle of least action and minimum potential energy

Are the principle of least action and the principle of minimum potential energy equivalent? How does one show that? Also, are Newton's laws of motion equivalent to the principle of least action? How ...
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Tidal Bulge on Earth due to Line of Action [duplicate]

I am in an intro to Statics course, and we briefly went over why tidal bulges occurred due to vectors in the line of action between the earth and the moon. I am confused because I do not understand ...
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Principle of Most Action? [duplicate]

In Landau-Lifshitz - Vol 1. Mechanics, right after the introduction of the principle of leas action, there is the following comment: It should be mentioned that this formulation ($S = \int\limits_{...
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Conceptual problem with action considered as function of endpoints

I am having some trouble with understanding why it makes sense to consider action in classical mechanics as function of endpoints $q_{initial}, \ q_{final}$ and endtimes $t_{initial}, \ t_{final}$. ...
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Functional Derivative of action

Consider the action of free Klein-Gordon theory $S[\phi]=\frac{1}{2}\displaystyle\int d^4y(\partial_\mu\phi(y)\partial^\mu\phi(y)-m^2\phi^2(y))$ Integrating by parts in the first term gives me $S[\...
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When the equations of motion are not unique (eg. when they are given by eigenvectors), which will the free particle adhere to?

For this question I think it will be easier to express the usual equation describing the motion of a "free particle,"--viz. $g_{ij}\dot{x}^i\dot{x}^j$--in matrix form as follows: $$g_{ij}\dot{x}^i\...
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How does satisfying the Euler-Lagrange equation put a Classical Path on-shell?

I am thinking of what the Euler-Lagrange equation, $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = 0 $$ specifically represents in satisfying the ...
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Principle of least action: $\frac{d S_{cl}}{dt_b} = \frac{\partial S_{cl}}{\partial t_b} + \frac{\partial S_{cl}}{\partial x_b}\dot{x}_b$

Question I cannot see how I can obtain the yellow highlighted section on the RHS from that of the LHS. The following equation can be found in both my lecture notes(*1) (page 9, equation 2.7) and is ...
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Are the generalized coordinates in Lagrangian mechanics really independent?

In Goldstein's Classical Mechanics, Chapter 2.3: Derivation of Lagrange's Equations From Hamilton's Principle part of the derivation involves each of the generalized coordinates being independent. $$ ...
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How to describe time-shifts in Noether's theorem in Hamiltonian formalism

As was described in, for example, this post, one can formulate Noether's Theorem also in Hamiltonian Mechanics. Symmetries are then represented by vector fields generated by observables whose Poisson ...
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With radian as a unit, should action and angular momentum have the different units?

If one accepts radian as a fundamental unit, does it make sense that action and angular momentum have units differing in radian to the power of one? The same question applies for energy and torque. ...