The action is the integral of the Lagrangian over time, or the integral of the Lagrangian Density over both time and space.

learn more… | top users | synonyms

6
votes
2answers
390 views

Derivation of the Polyakov Action

As is usually done when first presenting string theory, the Nambu-Goto Action, $$ S_{\text{NG}}:=-T\int d\tau d\sigma \sqrt{-g} $$ ($g:=\det (g_{\alpha \beta})$ is the induced metric on the ...
1
vote
1answer
153 views

Expansion of action in general relativity?

I am reading a lot about GR lately (because of thesis), and one thing bothers me, and I'm not finding a direct answer to it. For instance in one article, the author says that they are expanding ...
3
votes
1answer
290 views

Euler's equations of rigid body motion from least action principle

I would like to derive Euler's equations of rigid body motion from least action principle. Suppose we are in free space so we have no gravity so Lagrangian is equal to kinetic energy. $$ L = T = ...
8
votes
2answers
340 views

How do I read the simple, but contradictory, Lagrangian ($\mathcal{L} = x + v$)?

I understand the lagrangian formulation of classical mechanics, to a degree. I can derive the Euler-Lagrange equations from the "least" action principle, and equivalently can determine the equations ...
1
vote
1answer
154 views

Is the Schwinger action principle important in renormalization?

Is the Schwinger action principle important in renormalization? I want to know if this principle could help us to see if a model is renormalizable of not. If you have any other comment or information ...
0
votes
2answers
190 views

What's an “Action” and what does the Lagrangian equation mean exactly?

How and why would a particle take the shortest path? $L=KE-PE$? What's the $KE-PE$ mean in English? I understand the 'mechanics' but not the idea itself. Please explain simply, I do know Calculus ...
6
votes
1answer
275 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}} = ...
3
votes
1answer
282 views

Principle of Least Action

Is the principle of least action actually a principle of least action or just one of stationary action? I think I read in Landau/Lifschitz that there are some examples where the action of an actual ...
9
votes
1answer
345 views

What makes a Lagrangian a Lagrangian?

I just wanted to know what the characteristic property of a Lagrangian is? How do you see without referring to Newtonian Mechanics that it has to be $L=T-V$? People constructed a Lagrangian in ...
6
votes
3answers
408 views

Principle of Least Action via Finite-Difference Method

I am reading Gelfand's Calculus of Variations & mathematically everything makes sense to me, it makes perfect sense to me to set up the mathematics of extremization of functionals & show that ...
1
vote
2answers
180 views

Is this a valid derivation of the Legendre transformation from the Euler-Lagrange condition

E-L condition: $$\frac{d p}{dt}=\frac{\partial L}{\partial q}$$ Where $p=\frac{\partial L}{\partial \dot{q}}$ Are the following steps valid: $$\frac{\partial q}{dt} dp=\partial L$$ $$\dot{q} \: ...
2
votes
1answer
387 views

Retrieving Maxwell's equations from the minimum action principle

I'm currently working at the start of Alexei Tsvelik's book Quantum Field Theory in Condensed Matter Physics. I'm kinda stumped on a few essential steps. Starting with the action: $$S = \int dt \int ...
2
votes
1answer
339 views

Why vary the action with respect to the inverse metric?

Whenever I have read texts which employ actions that contain metric tensors, such as the Nambu-Goto, Polyakov or Einstein-Hilbert action, the equations of motion are derived by varying with respect to ...
1
vote
1answer
196 views

Discretization of action in path integral

I am reading Peskin and Schroeder (path integrals) and it states that discretising the classical action gives: $$S~=~\int \left(\frac{m}{2}\dot{x}^{2}-V(x)\right) dt ~\rightarrow~ \sum ...
5
votes
1answer
157 views

What is the action for an electromagnetic field if including magnetic charge

Recently, I try to write an action of an electromagnetic field with magnetic charge and quantize it. But it seems not as easy as it seems to be. Does anyone know anything or think of anything like ...
1
vote
2answers
422 views

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 ...
6
votes
3answers
983 views

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 ...
6
votes
1answer
132 views

Does anybody know of any good sources that explain (generically) how we form Lagrangians/Actions/Superpotentials for different field content?

I regularly find that I'll understand where the field content in a particular physics paper comes from, but then a Lagrangian or action or superpotential is stated and I don't know how it's derived. ...
2
votes
1answer
137 views

Varying an action (cosmological perturbation theory)

I am stuck varying an action, trying to get an equation of motion. (Going from eq. 91 to eq. 92 in the image.) This is the action $$S~=~\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2).$$ ...
3
votes
2answers
372 views

Polyakov action: difference induced metric and dynamical metric

The Polyakov action is given by: $$ S_p ~=~ -\frac{T}{2}\int d^2\sigma \sqrt{-g}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu} ~=~ -\frac{T}{2}\int d^2\sigma ...
2
votes
2answers
249 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 ...
1
vote
2answers
329 views

Why lagrangian is negative number?

In the special relativistic action for a massive point particle, $$\int_{t_i}^{t_f}\mathcal {L}dt,$$ why is the Lagrangian $$\mathcal {L}=-E_o\gamma^{-1}$$ a negative number?
0
votes
0answers
80 views

path integrals: how/why can the phase be identified with the action?

In Peskin & Schroeder, chapter 9 introduces the functional methods. The idea, to recall, is simply to sum over all the possible paths: $U(x_a,x_b;T) = \sum_{\text{all paths}} e^{i . ...
9
votes
1answer
621 views

Do an action and its Euler-Lagrange equations have the same symmetries?

Assume a certain action $S$ with certain symmetries, from which according to the Lagrangian formalism, the equations of motion (EOM) of the system are the corresponding Euler-Lagrange equations. Can ...
2
votes
3answers
262 views

Is the path of stationary action unique? What are the physical implications of $L_{\dot{x}}=L_x$

Below, for any function $Q$ the notation $Q_x$ means $\frac{\partial Q}{\partial x}$, and $Q_{xx}$ means $\frac{\partial^2 Q}{\partial x^2}$. In physics, the trajectory of a particle is given by the ...
8
votes
3answers
1k views

Entropy and the principle of least action

Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other ?
4
votes
3answers
535 views

Action for a point particle in a curved spacetime

Is this action for a point particle in a curved spacetime correct? $$\mathcal S =-Mc \int ds = -Mc \int_{\xi_0}^{\xi_1}\sqrt{g_{\mu\nu}(x)\frac{dx^\mu(\xi)}{d\xi} \frac{dx^\nu(\xi)}{d\xi}} \ \ d\xi$$
0
votes
0answers
59 views

Help identifying an expression for the action

I found the following expression for the action of a (free, I think) relativistic particle in my notes but I can't remember from what it came from: $$ S = \int_{0}^{N} \left [ ...
1
vote
1answer
281 views

What's the motivation behind the action principle? [closed]

What's the motivation behind the action principle? Why does the action principle lead to Newtonian law? If Newton's law of motion is more fundamental so why doesn't one derive Lagrangians and ...
1
vote
2answers
292 views

What is the relativistic action of a massive particle?

all Lorentz observers watching a particle move will compute the same value for the quantity $$ds^2 = -(c \, dt)^2 + dx^2 + dy^2 + dz^2,$$ $$ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu},$$ and ''ds/c'' is then ...
4
votes
3answers
933 views

What is the meaning of the word “Principle” in Physics?

What is the meaning of the word principle in Physics? For example in the "action principle". Is it an action law, an action equation, or an unproved assumption? (I have an idea what an action is). ...
3
votes
2answers
160 views

Could we get rid of explicit fields derivatives in Quantum Field Theories?

For instance, if we choose the following scalar field Lagrangian, which is (I hope) Lorentz-invariant, where $l$ is a a length scale, and with a $(-1,1,1,1)$ metric: $$ \mathfrak{L}(x) \sim ...
2
votes
1answer
278 views

Find the action from given equations of motion

Is there a systematic procedure to generally obtain an appropriate action that corresponds to any given equations of motion (if I know that it exists)?
4
votes
0answers
219 views

Dirac action and conventions

I have a (possibly) fundamental question, which is driving me crazy. Notation When considering the Dirac action (say reading Peskin's book), one have $\int ...
4
votes
2answers
539 views

How to apply Noether's theorem

Say I have a point transformation: $$x' ~=~ (1 +\epsilon)x,$$ $$t' ~=~ (1 +\epsilon)^2t,$$ and Lagrangian $$ L ~=~ \frac{1}{2}m\dot{x}^2 - \frac{\alpha}{x^2}.$$ How do I go out about showing ...
13
votes
6answers
2k views

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 ...
4
votes
2answers
411 views

Gauge fixing and equations of motion

Consider an action that is gauge invariant. Do we obtain the same information from the following: Find the equations of motion, and then fix the gauge? Fix the gauge in the action, and then find the ...
3
votes
3answers
280 views

Is the Lagrangian “math” or “science”?

I've seen in class that we can get from Lagrangian to derive equations of motion (I know its used elsewhere in physics, but I haven't seen it yet). It's not clear to me whether the Lagrangian itself ...
18
votes
4answers
798 views

Is the principle of least action a boundary value or initial condition problem?

Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating: In analytic (Lagrangian) mechanics, the derivation of the ...
10
votes
2answers
832 views

Deriving the action and the Lagrangian for a free point particle in Special Relativity

My question relates to Landau & Lifshitz, Classical Theory of Field, Chapter 2: Relativistic Mechanics, Paragraph 8: The principle of least action. As stated there, to determine the action ...
1
vote
1answer
216 views

What do I call the inverse of a propagator?

Let's suppose I have a theory described by a Lagrangian as follows: $ \mathcal{L} = A_\mu \underbrace{\left( \partial^2 g^{\mu\nu} - \partial^\mu \partial^\nu + m^2 g^{\mu \nu} \right)}_{K^{\mu \nu}} ...
2
votes
2answers
513 views

What is the significance of action?

What is the physical interpretation of $$ \int_{t_1}^{t_2} (T -V) dt $$ where, $T$ is Kinetic Energy and $V$ is potential energy. How does it give trajectory?
12
votes
1answer
300 views

Is there some connection between the Virial theorem and a least action principle?

Both involve some 'averaging' over energies (kinetic and potential) and make some prediction about their mean values. As far as the least action principles, one could think of them as saying that the ...
7
votes
2answers
1k views

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 ...
4
votes
3answers
720 views

$\hbar$, the angular momentum and the action

Is there anything interesting to say about the fact that $\hbar$, the angular momentum and the action have the same units or is it a pure coincidence?
5
votes
2answers
471 views

Is it circular reasoning to derive Newton's laws from action minimization?

Usually, a typical example of the use of the action principle that I've read a lot is the derivation of Newton's equation (generalized to coordinate $q(t)$). However, in the classical mechanics ...
14
votes
2answers
2k views

Deriving the Lagrangian for a free particle

I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading "Mechanics" by Landau and Lifshitz recently and hit a few roadblocks right away. Proving that a free particle ...
7
votes
4answers
1k views

Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?

All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before. Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
4
votes
3answers
399 views

Calculating lagrangian density from first principle

In most of the field theory text they will start with lagrangian density for spin 1 and spin 1/2 particles. But i could find any text where this lagrangian density is derived from first principle.
4
votes
2answers
360 views

How do I show that there exists variational/action principle for a given classical system?

We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's ...