3
votes
3answers
114 views

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 ...
1
vote
0answers
66 views

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 ...
6
votes
1answer
93 views

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 ...
2
votes
1answer
45 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} ...
1
vote
1answer
88 views

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 ...
7
votes
4answers
219 views

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 ...
4
votes
1answer
89 views

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 = ...
0
votes
1answer
109 views

Can't understand the principle of least action [closed]

I tried many hours to understand the principle of least action, and those hours become days... and I still didn't understand that principle/ and how it relates to Newtonian mechanics? Could someone ...
2
votes
1answer
105 views

Curved spacetime point particle Lagrangian density

This is probably trivially related to the question: Action for a point particle in a curved spacetime , but am a bit unsure how to write it as a Lagrangian density. In curved spacetime the action is ...
4
votes
1answer
181 views

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 ...
3
votes
1answer
114 views

Why can we assume independent variables when using Lagrange multipliers in nonholonomic systems?

I'm studying from Goldstein's Classical Mechanics. In section 2.4, he discusses nonholonomic systems. We assume that the constraints can be put in the form $f_\alpha(q, \dot{q}, t) =0$, $\alpha = 1 ...
3
votes
1answer
94 views

Hamilton-Jacobi formalism and on-shell actions

My question is essentially how to extract the canonical momentum out of an on-shell action. The Hamilton-Jacobi formalism tells us that Hamilton's principal function is the on-shell action, which ...
0
votes
0answers
28 views

How can a transversality condition be invoked to reduce the Euler-Lagrange equation?

I asked this question regarding the Euler-Lagrange equation at MSE and have gotten no response. I will ask it here too. I think I might have more luck here since the E-L equation is at the core of ...
5
votes
1answer
381 views

How to find the Stress-Energy tensor?

I am a bit at loss about how to proceed to find the stress-energy tensor given some distribution of matter. The Wikipedia page gives some examples, and some (inequivalent) definitions for it: Using ...
1
vote
1answer
74 views

Finding the EOM for a charged relativistic particle

For an exercise sheet of a course in general relativity I'm asked to derive the equations of motion for a charged particle in an EM-field given by a potential $A^\mu$. I am give the action: $$S = ...
6
votes
2answers
313 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 ...
3
votes
1answer
264 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
308 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 ...
0
votes
2answers
153 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 ...
3
votes
1answer
213 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
293 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
372 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
156 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
308 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 ...
1
vote
2answers
334 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
1answer
119 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
122 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 $$S~=~\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2).$$ ...
2
votes
2answers
212 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
306 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?
8
votes
1answer
529 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
238 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 ...
1
vote
1answer
251 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
256 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 ...
2
votes
1answer
269 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
2answers
494 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 ...
11
votes
5answers
1k 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
360 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
267 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 ...
16
votes
4answers
710 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 ...
9
votes
2answers
731 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 ...
2
votes
2answers
440 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
260 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 ...
6
votes
2answers
934 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 ...
5
votes
2answers
431 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
364 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
327 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 ...
3
votes
1answer
215 views

Differentiation of the action functional

In the QFT book by Itzykson and Zuber, the variation of the action functional $I=\int_{t_1}^{t_2}dtL$ is written as: $$\delta I=\int_{t_1}^{t_2}dt\frac{\delta I}{\delta q(t)}\delta q(t)$$ How is ...
12
votes
1answer
346 views

Lagrangian for Euler Equations in general relativity

The stress energy tensor for relativistic dust $$ T_{\mu\nu} = \rho v_\mu v_\nu $$ follows from the action $$ S_M = -\int \rho c \sqrt{v_\mu v^\mu} \sqrt{ -g } d^4 x = -\int c \sqrt{p_\mu ...