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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Is this field redefinition for free scalar field theory non-local?

The action of free scalar field theory is as follows: $$ S=\int d^4 x \frac{\dot{\phi}^2}{2}-\frac{\phi(m^2-\nabla^2)\phi}{2}. $$ I have been thinking to redefine field as ...
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2answers
178 views

Quantum Anomalies and Quantum Symmetries

In Quantum Field Theories (QFT) there is a well known phenomenon of anomalies, where a classical symmetry is broken in the quantum theory due to a so called anomaly. This symmetry breaking can be ...
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2answers
282 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} = ...
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1answer
89 views

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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216 views

How do derivative couplings affect canonical quantization?

Consider a Lagrangian for a scalar field $\phi$ with an interaction term $$\mathcal{L}_{int} = (\partial^2 \phi)^2 \phi.$$ Here I'm suppressing all indices for brevity. Now, this is just a ...
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153 views

How do you build a Lagrangian in particle/nuclear physics? (A specific example)

I know that the terms in the Lagrangian needs to be scalars (with respect to Lorentz symmetry etc.). Also I know that [see C. G. Tully (EPP) p. 85] in general, for $\psi$ in the fundamental ...
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1answer
160 views

Is it possible to derive the brane action in pure supergravity?

The branes that source the RR fields of supergravity are described by the DBI action plus a CS term. I know this only from superstring considerations. Is there a way to find this result without ...
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151 views

Lagrangian formalism and Contact Bundles

In his Applied Differential Geometry book, William Burke says the following after telling that the action should be the integral of a function $L$: A line integral makes geometric sense only if ...
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1answer
262 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 + ...
8
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1answer
257 views

Orbit through L4 and L5

I was reading the Wikipedia article on Lagrangian points and doing the requisite wiki walk through the various quasi-satellites of Earth when a question occurred to me: Could there be a stable or ...
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2answers
270 views

Why so many arguments for the transformation equations of generalized coordinates?

For a system of $N$ particles with $k$ holonomic constraints, their Cartesian coordinates are expressed in terms of generalized coordinates as $$\mathbf{r}_1 = \mathbf{r}_1(q_1, q_2,..., q_{3N-k}, ...
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730 views

Finding 3-Sphere Christoffel connection coefficients using variational calculus, Sean Carrol problem

I have A 3-Sphere with coordinates $x^{\mu} = (\psi,\theta,\phi)$ and the following metric: \begin{equation} ds^2 = d\psi^2 + \text{sin}^2\psi(d\theta^2 + \text{sin}^2\theta d\phi^2) \end{equation} ...
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4answers
1k views

Why can't we ascribe a (possibly velocity dependent) potential to a dissipative force?

Sorry if this is a silly question but I cant get my head around it.
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766 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 ...
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1answer
145 views

Free Field theory to Interacting Field theory

Free field theory: Why is it said that different Fourier modes in case of a free field (say, real Klein-Gordon field) are independent of each other? Interacting field theory: How exactly does the ...
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1answer
714 views

what this Lagrangian stands for?

i saw this Lagrangian in notes i have printed: $$ L(x,dx/dt) = (m^2(dx/dt)^4)/12 + m(dx/dt)^2*V(x) -V^2(x) $$ what is it? is it physical? it seems like it doesn't have the right units of energy, ...
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3answers
680 views

Noether theorem with semigroup of symmetry instead of group

Suppose You have semigroup instead of typical group construction in Noether theorem. Is this interesting? In fact there is no time-reversal symmetry in the nature, right? At least not in the same ...
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1answer
402 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
230 views

What canonical momenta are the “right” ones?

I'm doing some classical field theory exercises with the Lagrangian $$\mathscr{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}$$ where $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. To find the ...
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1answer
183 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. ...
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5answers
321 views

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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2answers
349 views

How do we know if a formulation of classical mechanics is correct?

For example, the Lagrangian formulation. I may be missing something, i.e. not having done it in enough detail, but here is my issue: from the definition of the lagrangian ($\mathcal{L}$) and from ...
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138 views

Group of symmetries of Lagrange's equations

Consider the following statements, for a classical system whose configuration space has dimension $d$: Lagrange equations admit a smaller group of "symmetries" (coordinate change under which ...
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1answer
54 views

Are there Trojan family or Hilda family satellites locked in Earth's orbit?

Jupiter has many Trojan asteroids located at Lagrangian points L4 and L5 and Hilda asteroids dispersed between points L3, L4, and L5. Does the Earth have similar satellites? If so, how many?
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1k views

When is the principle of virtual work valid?

The principle of virtual work says that forces of constraint don't do net work under virtual displacements that are consistent with constraints. Goldstein says something I don't understand. He says ...
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447 views

What corresponds to this Lagrangian density?

Is there a physical example of a field that would have the following Lagrangian density $$ L= \sqrt{1+\phi_x^2 +\phi_y^2+\phi_z^2} $$ where the subscripts denote partial derivatives and $\phi$ is a ...
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281 views

Naive questions on the concept of effective Lagrangian and equations of motion?

Let us consider a LC circuit containing an electric dipole moment, the quantum system (electric field $E$ coupled with a dipole moment) can be described by the path integral $$Z=\int DEDxe^{i\int ...
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699 views

Constraints of massive relativistic point particle in Hamiltonian mechanics

I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint: ...
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3answers
438 views

Why not formulate Quantum Mechanics using Lagrangians? [duplicate]

As the title implies, why is it that the most common formalisms we use in quantum mechanics prefer to describe systems in the terms of a Hamiltionian instead of a Lagrangian? Is there some ...
7
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1answer
588 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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1answer
778 views

Lagrangian of 2D square lattice of point masses connected by springs

Zee's QFT book mentions the Lagrangian of a square 2D horizontal lattice of point masses, connected by springs, and considering only vertical displacements $q_{i}$, as $ L = \frac{1}{2} ...
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3answers
386 views

Physical meaning of the Lagrangian function [duplicate]

In Lagrangian mechanics, the function $L=T-V$, called Lagrangian, is introduced, where $T$ is the kinetic energy and $V$ the potential one. I was wondering: is there any reason for this quantity to ...
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2answers
326 views

Deriving Newton's first law from the principle of least action

Newton's first law states that if the net force on an object is zero, then this object moves with constant velocity. I'm interested in the derivation of this law from the principle of least action. ...
7
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1answer
320 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} ...
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1answer
242 views

Heuristic Motivation for Lagrangian Formalism

Does anyone know a good heuristic motivation for the Lagrangian Formalism? I think most physicist just accept at one point that it works and thats that. I think I understand the historic origin. ...
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2answers
2k 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
134 views

Lie algebra of axial charges

Starting from the lagrangian (linear sigma model without symmetry breaking, here $N$ is the nucleon doublet and $\tau_a$ are pauli matrices) $L=\bar Ni\gamma^\mu \partial_\mu N+ \frac{1}{2} ...
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2answers
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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1answer
257 views

To construct an action from a given two-point function

This is really a basic question whose answer I guess may have to do with the way we construct Feynman rules and diagrams. The question is: Suppose I have been given a two-point function (found in some ...
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0answers
242 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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730 views

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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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 ...
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2answers
926 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
746 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 ...
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2answers
578 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 + ...
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2answers
777 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
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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
282 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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3answers
922 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 ...
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1answer
231 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 ...