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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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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5answers
896 views

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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3answers
529 views

Do we need inertial frames in Lagrangian mechanics?

Do Euler-Lagrange equations hold only for inertial systems? If yes, where is the point in the variational derivation from Hamilton's principle where we made that restriction? My question arose ...
6
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1answer
123 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 ...
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1answer
323 views

Peskin & Schroeder Chapter 3.1 EoM Lorentz Invariant under Lorentz Invariant Lagrangian

From Peskin & Schroeder QFT page 35: The Lagrangian formulation of field theory makes it especially easy to discuss Lorentz invariance. And equation of motion is automatically Lorentz ...
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3answers
434 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
162 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 ...
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2answers
187 views

Semiclassical limit of Quantum Mechanics

I find myself often puzzled with the different definitions one gives to "semiclassical limits" in the context of quantum mechanics, in other words limits that eventually turn quantum mechanics into ...
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1answer
87 views

Curvilinear Coordinates and basis vectors

In these notes, $\frac{\partial \vec{r}} {\partial q_i}$ is stated to form a basis set for the vector space. How does this happen? Also, how does one justify this equation from Goldstein's ...
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1answer
162 views

Sign in front of QFT kinetic terms

I'd like to know if the sign in front of a kinetic term in QFT important. For the scalar field we conventionally write (in the $ + --- $ metric), \begin{equation} {\cal L} _{ kin} = \frac{1}{2} ...
6
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2answers
1k views

Can a force in an explicitly time dependent classical system be conservative?

If I consider equations of motion derived from the pinciple of least action for an explicilty time dependend Lagrangian $$\delta S[L[q(\text{t}),q'(\text{t}),{\bf t}]]=0,$$ under what ...
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1answer
2k views

Invariance of Lagrange on addition of total time derivative of a function of coordiantes and time

My question is in reference to Landau's Vol. 1 Classical Mechanics. On Page 6, the starting paragraph of Article no. 4, these lines are given: If an inertial frame $К$ is moving with an ...
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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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2answers
119 views

Does the lagrangian contain all the information about the representations of the fields in QFT?

Given the Lagrangian density of a theory, are the representations on which the various fields transform uniquely determined? For example, given the Lagrangian for a real scalar field $$ \mathscr{L} = ...
6
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1answer
244 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 ...
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2answers
350 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 ...
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1answer
130 views

Pressure and Density Using a General Lagrangian

Given a lagrangian of a form: \begin{equation}\mathcal{L}=f(\phi,\partial_{\mu}\phi\partial^{\mu}\phi)\end{equation} where $f$ is a function, I need to derive pressure and density in a FLRW universe ...
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1answer
146 views

Caldeira-Leggett Dissipation: frequency shift due to bath coupling

I am trying to understand the Caldeira-Leggett model. It considers the Lagrangian $$L = \frac{1}{2} \left(\dot{Q}^2 - \left(\Omega^2-\Delta \Omega^2\right)Q^2\right) - Q \sum_{i} f_iq_i + ...
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0answers
207 views

Equation of motion for cyclic model of the universe

I recently started to study about cyclic universe. I came across this article [1]. My question is about the action that used for describing the cyclic model: $$S = \int d^{4}x\sqrt{-g}(\frac{1}{16\pi ...
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5answers
503 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
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3answers
369 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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2answers
404 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. ...
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1answer
960 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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1answer
204 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 ...
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2answers
476 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
232 views

Why should it be allowed to set the einbein to unity?

The Polyakov action for a massive free point particle with worldline $\gamma$ is given by $$ S[\gamma] = \frac{1}{2}\int_\gamma e \biggl(\frac{1}{e^2}\dot{x}^2 - m^2\biggr)\mathrm{d}\tau $$ where ...
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2answers
185 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 ...
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1answer
190 views

Any good resources for Lagrangian and Hamiltonian Dynamics?

I'm taking a course on Lagrangian and Hamiltonian Dynamics, and I would like to find a good book/resource with lots of practice questions and answers on either or both topics. So far at my university ...
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2answers
367 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 ...
5
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1answer
270 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
923 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 ...
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5answers
484 views

Noether Theorem and Energy conservation in classical mechanics

I have a problem deriving the conversation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
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2answers
665 views

The Faddeev-Popov Lagrangian

This is a non-abelian continuation of this QED question. The Lagrangian for a non-abelian gauge theory with gauge group $G$, and with fermion fields and ghost fields included is given by $$ ...
5
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2answers
509 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 ...
5
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1answer
313 views

formal framework for talking about 'minimal couplings'

usually on physical theories one would have Lagrangians or Hamiltonians with multiple fields; say, a vector $A_{\mu}$ and a scalar $\phi$ and one would postulate ad hoc a coupling between the fields ...
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1answer
208 views

Why are D'Alembert's Principle and the Principle of Least Action Related?

Why do we get the same differential equations from both principles? Surely there is a fundamental connection between them? When written out, the two seem to have nothing in common. $$\sum _i ( ...
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2answers
158 views

Questions about the degree of freedom in General Relatity

I'm confused about the number of degrees of freedom in General Relatity. There are two ways to count it. However, they are contradictory. For simplicity, we consider vacuum solution. First, ...
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2answers
184 views

Functional Derivative in the Linear Sigma Model

In the linear sigma model, the Lagrangian is given by $$ \mathcal{L} = \frac{1}{2}\sum_{i=1}^{N} \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) ...
5
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1answer
220 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 $\gamma$. ...
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2answers
774 views

The Lagrangian in Scalar Field Theory

This is perhaps a naive question, but why do we write down the Lagrangian $$\mathcal{L}=\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi - \frac{1}{2}m^2\phi^2$$ as the simplest ...
5
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1answer
97 views

Proca Lagrangian manipulation

How can I show that the Lagrangian density $$\mathcal{L} = -\frac{1}{2}\partial_\alpha \varphi_\beta \partial^\alpha \varphi^\beta + \frac{1}{2} \partial_\alpha \varphi^\alpha \partial_\beta ...
5
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1answer
82 views

How does one prove that the current of a spontaneously broken symmetry generates a particle?

I am having a hard time arguing that, after spontaneous breaking of a continuous symmetry of a field Lagrangian, local fluctuations around the vacuum can be interpreted as particles (without referring ...
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5answers
245 views

Euler-Lagrange equation for continuous systems

I'm having a little trouble with wrapping my head around a part of a method which is fairly 'new' in some fashions to me. I imagine it should be fairly obvious, but I am not seeing something at the ...
5
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2answers
117 views

Defining quantum effective action (Legendre transformation), existence of inverse (field - source)?

Given a Quantum field theory, for a scalar field $\phi$ with generic Action $S[\phi]$, we have the generating functional $$Z[J] = e^{iW[J]} = \frac{\int \mathcal{D}\phi e^{i(S[\phi]+\int d^4x ...
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1answer
258 views

On a trick to derive Noether current

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$. The trick to get the Noether current consists in making the ...
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1answer
138 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 = ...
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1answer
406 views

What is Maupertuis' principle good for?

The strength of Hamilton's principle is obvious to me and I see the advantage. Now, for conservative systems we also have Maupertuis' principle that says: $$ \delta \int p dq =0$$ and I am not sure ...
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1answer
70 views

Least action principle — numerical simulation strangeness

I'm trying to get some experience with the least action principle, and for this I chose a simple 1-dimensional problem of a particle moving in some field. The least action principle would then look ...
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2answers
316 views

Kugo and Ojima's Canonical Formulation of Yang-Mills using BRST

I am trying to study the canonical formulation of Yang-Mills theories so that I have direct access to the $n$-particle of the theory (i.e. the Hilbert Space). To that end, I am following Kugo and ...
5
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1answer
87 views

Variational form of Euler's incompressible fluid equations?

I am trying to derive Euler's incompressible fluid equations in terms of a variational stationary principle. Given Euler's flow equations: $$\frac{\partial v}{\partial t} = -\nabla p$$ $$\nabla\cdot ...