For questions involving the Lagrangian formulation of a dynamical system. Namely, the application of an action principle to a suitably chosen Lagrangian or Lagrangian Density in order to obtain the equations of motion of the system.

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Mechanical similarity in Landau

I've read this very short paragraph from Landau & Lifshitz's Mechanics (Chap.2, Par.10) (that you can find here) about Mechanical similarity. I was looking for some more detailed explanations of ...
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3answers
2k views

Advantages of Lagrangian Mechanics over Newtonian Mechanics [closed]

Here, I'm going to pose a very serious list of doubts I have on Lagrangian Mechanics. Can we learn Lagrangian Mechanics without studying Newtonian Mechanics? Does Lagrangian help in solving problems ...
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2answers
2k views

Why is lagrangian density correct?

The textbooks I have available explain that due to the infinite degrees of freedom of a field, the relevant object in QFT is the lagrangian density. A lagrangian is then obtained for the field by ...
6
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2answers
945 views

Question about the apparent loophole in principle of least action

In Lagrangian formalism, given two points $(x_1,t_1)$ and $(x_2,t_2)$, we ask the question which paths $x(t)$ make the action $S=\displaystyle \int_{t_1}^{t_2}L\ \mathrm dt$ stationary and satisfy the ...
6
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2answers
812 views

A kind of Noether's theorem for the Hamiltonian formalism

How can I (conveniently?) show that an invariance of the Lagrangian and Hamiltonian (i.e. the kinetic as well as the potential energy are independently invariant) will lead to a conservation law using ...
6
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2answers
613 views

Can auxiliary fields be thought of as Lagrange multipliers?

In the BRST formalism of gauge theories, the Lautrup-Nakanishi field $B^a(x)$ appears as an auxiliary variable $$\mathcal{L}_\text{BRST}=-\frac{1}{4}F_{\mu\nu}^a F^{a\,\mu\nu}+\frac{1}{2}\xi B^a B^a + ...
6
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2answers
818 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 ...
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1answer
2k views

Invariance of Lagrangian in Noether's theorem

Often in textbooks Noether's theorem is stated with the assumption that the Lagrangian needs to be invariant $\delta L=0$. However, given a lagrangian $L$, we know that the Lagrangians $\alpha L$ (...
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2answers
296 views

QFT's that have no action

What does it mean to have a QFT that can not be encoded by an action. What is by far the most powerful approach of study in such a case. What is the best studied physical theory that falls into this ...
6
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3answers
958 views

The number of independent variables in the Lagrangian and Hamiltonian methods in Classical Mechanics

It's told in Landau - Classical Mechanics, that in the Hamiltonian method, generalized coordinates $q_j$ and generalized momenta $p_j$ are independent variables of a mechanical system. Anyway, in the ...
6
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1answer
232 views

Why isn't $F = \frac{\partial \mathcal{L}}{\partial q}$?

If momentum is, $$p = \frac{\partial \mathcal{L}}{\partial \dot{q}}$$ and force is, $$ F = \frac{dp}{dt}$$ and by Euler-Langrange equations, $$ \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{...
6
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2answers
998 views

Is the Dirac Lagrangian Hermitian?

I'm wondering of the Dirac Lagrangian density $$\mathcal{L} =\overline{\psi}(-i\gamma^\mu \partial_\mu +m)\psi $$ is an hermitian operator, since upon complex conjugating one gets $$\mathcal{L}^\...
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5answers
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What is the Lagrangian for a relativistic charge that includes the self-force?

The usual Lagrangian for a relativistically moving charge, as found in most text books, doesn't take into account the self force from it radiating EM energy. So what is the Lagrangian for a ...
6
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2answers
204 views

Why are there only 3 Additive Integrals of Motion?

1. I was reading Landau & Lifschitz's book on Mechanics, and came across this sentence on p.19: "There are no other additive integrals of the motion. Thus every closed system has seven such ...
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2answers
208 views

In QFT how do you write down the most general interactions?

This past year I took a QFT class and I now feel comfortable solving scattering problems, but I am still a bit perplexed by how physicists write down a Lagrangian in the first place. In particular, ...
6
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1answer
639 views

Lagrangian and Hamiltonian EOM with dissipative force

I am trying to write the Lagrangian and Hamiltonian for the forced Harmonic oscillator before quantizing it to get to the quantum picture. For EOM $$m\ddot{q}+\beta\dot{q}+kq=f(t),$$ I write the ...
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2answers
181 views

Pass to globally conserved currents from locally conserved currents in curved spacetime

Let us begin with a Lagrangian of the form $$\mathscr L= \frac 12 \sqrt{-g}g^{\mu\nu}\partial_\mu\phi(x)\partial_\nu\phi(x)+\mathscr L_g,$$ where $$\mathscr L_g=\frac 1{16\pi k}\sqrt{-g}R.$$ ...
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2answers
430 views

When can we add a total time derivative of $f(q, \dot{q}, t)$ to a Lagrangian?

The other day, I was listening to this lecture on the Lagrangian for a charged particle in an electromagnetic field, and at one point in the video, the lecturer mentions that we can add any total time ...
6
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1answer
175 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 ...
6
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3answers
601 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 ...
6
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2answers
978 views

Two carts connected by spring on frictionless track

I have the following homework problem: Consider two carts of equal mass m on a horizontal, frictionless track. The carts are connected by a single spring of force constant k, but are otherwise ...
6
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1answer
505 views

Noether's Theorem: Lie algebra, Lie groups

I've had a brief look through similar threads on this topic to see if my question has already been answered, but I didn't find quite what I was looking for, perhaps it is because I'm finding it hard ...
6
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1answer
973 views

Pendulum with a rotating point of support from Landau-Lifschitz

I found this problem in Landau-Lifschitz vol.1 (Mechanics) A simple pendulum of mass $m$, length $l$ whose point of support moves uniformly on a vertical circle with constant frequency $\gamma$. $$...
6
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1answer
317 views

Counterterm Lagrangian and Renormalisation?

I am going through the notes on QFT by M. Srednicki (online: http://web.physics.ucsb.edu/~mark/qft.html), and I am having a hard time to understand the "renormalised" Lagrangian. Consider a Klein-...
6
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1answer
176 views

Uses for Action from Lagrangian Mechanics

In my course on Lagrangian/Hamiltonian mechanics I noticed that we dealt with finding the stationary point of the change in action $ \delta S $ and we were never really doing anything with $ S $ ...
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2answers
391 views

Mass particle trajectory on a sphere [closed]

So, I am trying to simulate mass particle motion on the outer surface of sphere using cartesian coordinates. Let's conclude just a gravity and frictionless movement. Sphere $x^2 + y^2 + z^2 = 1$, ...
6
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1answer
105 views

Use partial or covariant derivatives when deriving equations of a field theory?

I feel like this question has been asked before but I can't find it. would the Euler Lagrange equation for, say, the standard model Lagrangian be $$\frac{\partial L}{\partial \phi}=\partial_\mu \frac{\...
6
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2answers
180 views

How general are Noether's theorem in classical mechanics?

I'm going through the derivations of Noether's theorems and I have several criticisms as to how they are presented in popular sources (note that I'm only referring to classical mechanics here and not ...
6
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1answer
133 views

Why does the Palatini formalism of GR work? [duplicate]

We can get the Einstein field equations of GR from the Einstein-Hilbert action via two distinct methods: First, by taking the metric as the only degree of freedom, and imposing right away that the ...
6
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1answer
93 views

Feynman propagator for arbitrary values of the gauge parameter $\zeta$

For the choice $\zeta = 1$ the Lagrangian can be brought into a particularly simple form upon integration by parts in the action integral. Equation$$\mathcal{L}' = -{1\over4}F_{\mu\nu}F^{\mu\nu} - {1\...
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3answers
2k views

Constraint force on a rod

I really hope someone will take a quick look at the following, I would just love to better understand it... This exercise is from Arnold's "Mathematical Methods of Classical Mechanics", p. 97 in the ...
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4answers
1k views

Lagrangian for relativistic massless point particle

For relativistic massive particle, the action is $$S ~=~ -m_0 \int ds ~=~ -m_0 \int d\lambda ~\sqrt{ g_{\mu\nu} \dot{x}^{\mu}\dot{x}^{\nu}} ~=~ \int d\lambda \ L,$$ where $ds$ is the proper time of ...
6
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1answer
454 views

Dirac Lagrangian density in curved spacetime

I'm trying to derive this form of the Dirac Lagrangian density in curved space-time: $$ \mathcal{L}~=~\det\left(e\right)\bar{\Psi}\Bigg (\frac{i}{2}\gamma^{a}\partial_{a}-m+\gamma^{a}\gamma^{5}B_{a}\...
6
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1answer
385 views

Determinant for a coupled fluctuation Lagrangian

Lets consider a bosonic physical system in variables $t, x$ and $y(x)$ ($x$ dependent) with a classical Lagrangian $L$. To first order in fluctuations $x \to x+\xi_1$ and $y \to y+\xi_2$ the ...
6
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2answers
636 views

What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?

I have a problem with one of my study questions for an oral exam: The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov ...
6
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1answer
596 views

Lagrangian density for a Piano String

So I'm trying to do this problem where I'm given the Lagrangian density for a piano string which can vibrate both transversely and longitudinally. $\eta(x,t)$ is the transverse displacement and $\xi(x,...
6
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0answers
132 views

Why don't we have logarithms or exponentials of the fields in the Lagrangians?

All tbe Lagrangian densities I have seen have always been polynomials of the fields. Is this a coincidence or is there a reason forbid, say, Lagrangians with logarithms or exponentials of the fields?
6
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1answer
368 views

How do you determine the Lagrangian? [duplicate]

I have always been puzzled by how do you arrive at Lagrangians? That is, how do you know that the functional you need to get Newton's equations is $$L = T-V(x)~?$$ Do you derive the Lagrangian ...
5
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2answers
553 views

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 ...
5
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2answers
564 views

How can you solve this “paradox”? Central potential

A mass of point performs an effectively 1-dimensional motion in the radial coordinate. If we use the conservation of angular momentum, the centrifugal potential should be added to the original one. ...
5
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3answers
914 views

Meaning of kinetic part in the Lagrangian density?

What is the physical meaning of the kinetic term in the classical scalar field Lagrangian $$\mathcal{L}_{kin}~=~\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)~?$$ It gives how does the field change ...
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5answers
670 views

What distinguishes time from space in Quantum Field Theory?

Consider the following expression for a general QFT action: $$ S ~=~ \int_0^t\mathrm dt~L ~=~\int_0^t\mathrm dt\int_\mathbb {R^3}\mathrm d^3x~\mathcal L ~=~\int\mathrm d^4x~\mathcal L.$$ Here we ...
5
votes
2answers
505 views

Euler Lagrange equation in different frames

Suppose I have an inertial frame with coordinate $\{q\}$. Now I define another reference frame with coordinate $\{q'(q,\dot q,t)\}$. I obtain the equation of motion in $\{q'\}$ in two different ways: ...
5
votes
2answers
796 views

Stress-energy tensor for a fermionic Lagrangian in curved spacetime - which one appears in the EFE?

So, suppose I have an action of the type: $$ S =\int \text{d}^4 x\sqrt{-g}( \frac{i}{2} (\bar{\psi} \gamma_\mu \nabla^\mu\psi - \nabla^\mu\bar{\psi} \gamma_\mu \psi) +\alpha \bar{\psi} \gamma_\mu \...
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3answers
1k views

Hanging chain in a planet's gravitational field

The curve for a chain hanging between two poles in a uniform gravitational field is known as the catenary. Is there known an expression for the curve of a hanging chain on a planet of mass $M$ which ...
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2answers
1k views

How the Lagrangian of classical system can be derived from basic assumptions?

It is well known that the Lagrangian of a classical free particle equal to kinetic energy. This statement can be derived from some basic assumptions about the symmetries of the space-time. Is there ...
5
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3answers
86 views

The $\dot{q}$ term in the Euler-Lagrange equation

The Euler-Lagrange equation is about the functional $$ \int_{t_1}^{t_2} L(q, \dot{q}, t ) dt . $$ From a mathematical point of view, a simpler functional might be $$ \int_{t_1}^{t_2} L(q, t ) ...
5
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1answer
235 views

How will SR EM Lagrangian change if we find a magnetic charge?

When we introduce electromagnetic field in Special Relativity, we add a term of $$-\frac e c A_idx^i$$ into Lagrangian. When we then derive equations of motion, we get the magnetic field that is ...
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1answer
1k views

Elementary derivation of the motion equations for an inverted pendulum on a cart

Consider a cart of mass $M$ constrained to move on the horizontal axis. A massless rod is attached to the midpoint of the cart, having a mass $m$ on its endpoint. See wikipedia for a picture and for a ...
5
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2answers
213 views

Lagrangian $L' = L + \frac{df}{dt}$ gives the same equations of motion

It is well known that when a Lagrangian $L$ is incremented by the total time derivative of a function $f$ that does not depend on the time derivatives of the generalized coordinates, the same ...