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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142 views

Understanding the effective low-energy Lagrangian for hadrons

My course in Higgs Physics is discussing a two-nucleon low-energy effective theory of hadron interaction. With $\psi=(p,n)$, the pion is defined as $\vec{\pi}= i \bar{\psi}\vec{\tau} \gamma_5 ...
10
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2answers
466 views

For a particle to have physical mass, is it always necessary to have a mass term in the lagrangian?

Since the self-energy adds to the bare mass defined in the Lagrangian, is it possible to create a physical particle mass from the self-energy alone, with no mass terms occuring in the Lagrangian? On ...
2
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0answers
225 views

Hoop rolling inside a circular hole

A hoop of radius $b$ and mass $m$ rolls without slipping within a stationary circular hole of radius $a > b$ and is subject to gravity. Use the generalized coordinates the rotation angle $\phi$ of ...
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1answer
140 views

Finding the magnetic vector potential by calculus of variations

Given the functional $$F[A]=\int_{\mathbb{R}^3}\{\frac{1}{2\mu(x)}|\nabla\times\vec{A}|^2-\vec{J}\cdot\vec{A}\}d^3x$$ with $\vec{A}$ is a candidate vector potential for the field ...
2
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1answer
62 views

Motion with damping involving both $-v^2$ and $-v$ terms

I'm trying to solve for the motion of a particle, accounting for both viscous and drag forces. (There is no potential) The total resistance by medium is modeled by: $\vec{F}=-(\mu_1v + ...
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1answer
118 views

The Euler-Poincare equation

Can anyone tell me very basically how the Euler-Poincare equation generalises the Euler-Lagrange equation? Further does anyone know if there is an "easy" relationship between the two, i.e. can anyone ...
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0answers
68 views

Total Vs Partial in Lagrange density?

I have a question regarding the red term below. This is the integration by parts during the derivation of the Euler-Lagrange equation for continuous systems. Why is this not the time derivative ...
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0answers
89 views

Hamiltonian Operator for nonrenormalizable Effective Field Theories?

Assuming we have a Effective Field Theory, for example a Real Scalar Field Theory, defined through a Lagrangian density of the form $\mathcal{L}_{eff} = \frac{1}{2}\partial_\mu\phi \partial^\mu\phi - ...
3
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2answers
411 views

Classical Field Theory - Continuum limit in forming the Lagrangian density and the elasticity modulus

I have been looking at taking the continuum limit for a linear elastic rod of length $l$ modeled by a series of masses each of mass $m$ connected via massless springs of spring constant $k$. The ...
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0answers
38 views

Equations of motion with replacing the Lagrangian by irrep diagrams generating functional

I have read that equations of motion of ghosts is equal to $$ \tag 1 \frac{\delta \Gamma}{\delta \bar{c}^{a}(x)} = -\partial^{\mu}_{x}\frac{\delta \Gamma}{\delta K^{\mu , a}(x)}, $$ where $\Gamma = W ...
3
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2answers
157 views

What is the “associated scalar equation” of equations of motion?

In an essay I am reading on celestial mechanics the equations of motion for a 2 body problem is given as: $$\mathbf{r}''=\nabla(\frac{\mu}{r})=-\frac{\mu \mathbf{r}}{r^3}$$ Fine. Then it says the ...
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0answers
205 views

Intuition behind the principle of virtual work

To derive Lagrange's Equations we need the principle of virtual work first. This principle states that whenever a system of $K$ particles is constrained to a submanifold $\mathcal{M}\subset ...
2
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1answer
711 views

Proof that total derivative is the only function that can be added to Lagrangian without changing the eom

So I was reading this: Invariance of Lagrange on addition of total time derivative of a function of coordiantes and time and while the answers for the first question are good, nobody gave much ...
4
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2answers
294 views

Why do the $1/2$ factor appear in the Majorana mass Lagrangian?

In case of Dirac neutrino there is no $1/2$ factor in the mass Lagrangian but for Majorana type neutrino there is a half factor in the mass Lagrangian.
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1answer
75 views

Coefficient matrix of quadratic Lagrangian

I've been studying path integrals from Weinbergs QToF vol 1. He says that when the $\mathcal{L_0}$ is quadratic in fields we can always write free term $I_0$ in the generalized quadratic form ...
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2answers
674 views

How can I derive the Hamiltonian of simple harmonic oscillator from this Lagrangian?

I'm working through Leonard Susskind's Theoretical Minimum: Classical Mechanics and I can't seem to understand how the Hamiltonian of a simple harmonic oscillator is derived from the following ...
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0answers
101 views

Equations of motion for controlled/driven classical systems? Does D'Alembert's principle apply?

I'm puzzled about how to derive the equations of motion for certain classical systems where some entity is controlling some of the DOFs. For example, consider a double-pendulum, with lengths $l_1$ ...
5
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1answer
764 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 ( ...
5
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2answers
410 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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1answer
216 views

Finding equilibrium of mechanical system

A system is described as follows: Consider a system consisting of two rotating bars of length $l$ and with uniform mass density and each with total mass $m$. The bars are attached to a common ...
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1answer
83 views

Effective field theory for fermion gas

Reading about fermion gas in a paper they used the following Lagrangian, which describes an effective field theory for nonrelativistic fermions (I neglect the four point interaction term). $$ L = ...
2
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1answer
446 views

“A Crash Course in Lagrangian Dynamics”. Is it still available online?

In an Amazon review of "Schaum's Outline of Lagrangian Dynamics" I found this: I recommend that you type "Lagrangian Dynamics" into Google and look at some of the excellent sets of lecture notes ...
2
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1answer
67 views

Deriving conserved currents by promoting parameter

I currently reading Tong's text on String Theory. In Chapter 4.1.1 he alludes to a technique to derive conserved currents Recall that we can usually derive conserved currents by promoting the ...
2
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1answer
205 views

Lagrangian and grassmann numbers

Why sometimes we remember that "classical" lagrangians of fermions are constructed from grassmann numbers, while sometimes don't? For example, for Majorana's field in terms of 2-component spinors ...
16
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3answers
5k views

What exactly is a virtual displacement in classical mechanics?

I'm reading Goldstein's Classical Mechanics and he says the following: A virtual (infinitesimal) displacement of a system refers to a change in the configuration of the system as the result of any ...
2
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1answer
151 views

Mass term in the Lagrangian

I have read that the mass term appearing in the electroweak Lagrangian stops it (the Lagrangian) from becoming gauge invariance. Can someone explain where and why this term is creating the problem?
3
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0answers
92 views

Parity invariance of Einstein-Hilbert Lagrangian

How can we show that the Einstein-Hilbert action is Parity invariant? $$ S_{EH}=\int \sqrt{-g}R d^4x $$
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0answers
38 views

Does the order of variables matter for a quantum Lagrangian in the path integral formula for quantum mechanics? [duplicate]

For a single particle or field, I can't see how the path-integral formulation depends on the order of terms in the Lagrangian. It seems that you integrate the classical Lagrangian to get the action on ...
2
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2answers
271 views

Pendulum with changing length over time. What's wrong?

I tried to find the equation of this pendulum, but I think I did something wrong. I know I have to get the Bessel's equation but I can't see it. It's a simple 2-D pendulum, without any dissipation. ...
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0answers
80 views

Constructing conserved current given the lagrangian

Consider the following Lagrangian for a massive vector field $A_{\mu}$ in Euclidean space time: $$\mathcal L = \frac{1}{4} F^{\alpha \beta}F_{\alpha \beta} + \frac{1}{2}m^2 A^{\alpha}A_{\alpha}$$ ...
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0answers
129 views

Lagrangian Systems

Given a manifold $M$, Arnold's "Mathematical Methods of Classical Mechanics" defines a Lagrangian system as a pair $(M,L)$ where $L$ is some smooth function on the tangent bundle $TM$. The function ...
7
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2answers
287 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 ...
3
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0answers
90 views

Is the “Force” of Gravity Simply Hamilton's Principle on a Curved Spacetime?

It's my understanding that General Relativity abstracts away the concept of gravity as a force, and instead describes it as a feature of spacetime by which massive objects cause curvature. Then it ...
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0answers
64 views

Conditions for hyperregularity

(Definitions from Marsden & Ratiu, ``Introduction to Mechanics and Symmetry''): A Lagrangian is regular if the Hessian $\partial^2 L/(\partial \dot{q}^i \partial \dot{q}^j)$ is weakly non ...
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0answers
116 views

Lagrangian for FRW metric

For the metric $$ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2),$$ $$L= \sqrt{-g_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt}}$$ How does this become $$L= \sqrt{1-a^2 (\frac{dx}{dt})^2}~? $$ I guess ...
4
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1answer
430 views

Sufficient conditions for the energy to be not conserved?

I'm almost embarrased to ask this question because I thought I was by now very confident with classical mechanics. Someone has stated that given a mechanical system with a Lagrangian $L$ s.t. ...
4
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1answer
116 views

Reference Request: Fluid dynamics/Elasticity via Lagrangians

Would there be a book that does what Landau does in Fluid Mechanics and Theory of Elasticity using Lagrangian's/Action-principles, analogous to the presentation in Landau's mechanics? I have only ...
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1answer
180 views

Subtlety in derivation of Noether's theorem by Di Francesco

In the book 'Conformal Field Theory' by Di Francesco et al, a derivation of Noether's theorem is demonstrated by imposing that, what I believe is said to be a more elegant approach, the parameter ...
4
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1answer
392 views

Invariance of canonical Hamiltonian equation when adding the total time derivative of a function of $q_i$ and $t$ to the Lagrangian

The following is exercise 8.2 in 3rd edition (and exercise 8.19 in 2nd edition) of Goldstein's Classical Mechanics. Adding the total time derivative of a function of $q_i$ and t to the Lagrangian ...
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3answers
465 views

Eigenvalues of the Lagrangian?

It is often stated that the Lagrangian formalism and the Hamiltonian formalism are equivalent. We often hear people talk about eigenvalues of Hamiltonians but I have never ever heard a word about ...
3
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1answer
177 views

Lagrangian description of Brownian motion?

I'm interested in the existence of a Lagrangian field theory description of Bronwnian motion, does such a thing exist? Given a particle of some spin $\sigma$, which has a Lagrangian associated with ...
0
votes
1answer
112 views

How can a generalised force be dependent on an angle i.e. not a vector?

I'm currently working through an example question in Patrick Hamill's 'A Student's Guide to Hamiltonians and Lagrangians'. The question I'm having conceptual difficulty with is: A particle is ...
1
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1answer
331 views

Where can some worked problems in classical mechanics (and more specifically the Lagrangian and Hamiltonian formalisms) be found? [duplicate]

I've been looking for a textbook in classical mechanics that's readily available (like can be found in the library of James Cook University of Townsville, Australia) and full of fully-answered ...
5
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1answer
262 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 ...
2
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1answer
496 views

Hookes law and objective stress rates

Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ...
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0answers
67 views

How can we describe the trajectory of an object projected into a viscous fluid?

Can we quantitatively describe the motion of a particle inside a fluid? Let us consider the case when an object (say a sphere, so that stoke's law holds true) is projected into a fluid of viscosity ...
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0answers
196 views

Naive questions on the classical equations of motion from the Chern-Simons Lagrangian

Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial ...
5
votes
2answers
611 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) ...
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0answers
77 views

Another Power Counting/ mass dimension question

Are the mass dimension of the Dirac field different from those of the Klein-Gordon field, or is this just another issue of "cannonical normalization?" For instance if $\mathcal{L}_{KG}=\int ...
3
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0answers
295 views

Question about an integration by parts in Feynman's Quantum Mechanics [closed]

I have begun reading Feynman & Hibbs Quantum Mechanics and Path Integrals. Knowing little about variational calculus or Lagrangians I found the following integration by parts opaque. I think if I ...