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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41 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 ...
1
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1answer
46 views

Derivation of Maxwell stress tensor from EM Lagrangian

From Noether's theorem applied to fields we can get the general expression for the stress-energy-momentum tensor for some fields: $$T^{\mu}_{\;\nu} = \sum_{i} (\frac{\partial \mathcal{L}}{\partial ...
3
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0answers
42 views

Classical toy models of particles with intrinsic spin

Related to my question here (spacetime torsion, the spin tensor, and intrinsic spin in einstein cartan theory), I'd like to be able to put test particles on a manifold with non-zero torsion and see ...
1
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1answer
65 views

Variational principle for a point particle (massive or massless) in curved space

We know that for a point particle, the action is $$ S[x,e] ~=~ \frac{1}{2}\int_{\lambda_A}^{\lambda_B} d\lambda\left[e^{-1}(\lambda)~g_{\mu\nu}(x(\lambda))~\dot{x}^\mu(\lambda)~\dot{x}^\nu(\lambda) ...
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1answer
134 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. ...
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0answers
60 views

Lagrangian mechanics is different form Newtonian? [duplicate]

I am a post graduate student and had completed my classical Mechanics with Newtonian mechanics to Hamiltonian mechanics. I have better understanding of Newtonian, Lagrangian and Hamiltonian. But I ...
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1answer
68 views

What are great circles of 2-sphere?

What exactly are great circles, and how does one derive them? Given that the Lagrangian is: $$ L =\frac {1}{2}(\dot\theta^2 + \sin^2\theta\dot\phi^2)$$ it was written that the great circles were ...
2
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1answer
56 views

Connection between conserved charge and the generator of a symmetry

I'm trying to understand the connection between Noether charges and symmetry generators a little better. In Schwartz QFT book, chapter 28.2, he states that the Noether charge $Q$ generates the ...
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2answers
58 views

Ball Bearing Inside a Hollow, Spinning Rod: where is the logical flaw?

As described in the title, suppose we have a frictionless, hollow rod that is rotating in the $xy$-plane with some fixed angular velocity $\omega$. The rod is pivoting around its midpoint. Suppose we ...
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0answers
53 views

Geodesic equation from Euler - Lagrange

There are several ways to derive the geodesic equation. One of which is the variational method which I seemed to understand it because it was written in great details. Then it was mentioned that the ...
3
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0answers
118 views

How is the method of Lagrange multipliers used for multiple constraints of multiple variables? [closed]

Let's say for example that I have two constraints $f(x,\dot{x},y,\dot{y})$ and $g(x,\dot{x},y,\dot{y})$ and a Lagrangian $L(x,\dot{x},y,\dot{y})$. What are the Euler-Lagrange equations of the first ...
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1answer
66 views

Why is the Hamiltonian the Legendre transform of the Lagrangian?

So, as the title says, why is the Hamiltonian the Legendre transform of the Lagrangian? I know that from quantum mechanics, one can start with the Hamiltonian defined as the generator of time ...
3
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1answer
58 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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2answers
107 views

Why are these two definitions for symmetries in the Lagrangian equivalent?

I have heard the following two definitions for a symmetry of the Lagrangian: If under a coordinate transformation the form of the Lagrangian remains unchanged then there is a symmetry. If $\delta ...
4
votes
1answer
221 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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1answer
127 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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2answers
78 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 ...
2
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2answers
64 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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3answers
521 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 ...
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2answers
477 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 ...
13
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2answers
289 views

Determine if Theory is Unitary from Lagrangian

Question: Given a quantum theory specified with a Lagrangian and the degrees of freedom to be varied, what is the procedure to determine if the theory is unitary or not? Concrete example to aid ...
2
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4answers
162 views

Poincare invariant Lagrangians

The Lagrangian density of a Poincare invariant theory should not depend explicitly on the space-time coordinates. Does this mean $$ \partial_\mu \mathcal{L}=0~? $$ If this is the case doesn't the ...
2
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1answer
84 views

Use of the term first order dependency

In a question I am doing it says: Show explicitly that the function $$y(t)=\frac{-gt^2}{2}+\epsilon t(t-1)$$ yields an action that has no first order dependency on $\epsilon$. Also my textbook ...
2
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1answer
79 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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1answer
33 views

General potential of rotating system

I'm new here at the physics site, and not really that deep into the area of which i'm going to ask a question about now. Therefore please feel free to ask clarifying questions. I'm trying to deal ...
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0answers
44 views

Boundary term in Einstein-Hilbert action

Why is the boundary term in the Einstein-Hilbert action, the Gibbons-Hawking-York term, generally "missing" in General Relativity courses, IMPORTANT from the variational viewpoint, geometrical setting ...
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0answers
43 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 ...
4
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2answers
117 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} ...
3
votes
3answers
252 views

Does the variation of the Lagrangian satisfy the product rule and chain rule of the derivative?

I have seen wikipedia use the product rule and maybe the chain rule for the variation of the Langragin as follows: \begin{align} \dfrac{\delta [f(g(x,\dot{x}))h(x,\dot{x})] } {\delta x} = \left( ...
2
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2answers
104 views

The variation of the Lagrangian density under an infinitesimal Lorentz transformation

I'm trying to introduce myself to QFT following these lectures by David Tong. I've started with lecture 1 (Classical Field Theory) and I'm trying to prove that under an infinitesimal Lorentz ...
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1answer
59 views

The principle of stationary action?

In proving that the action $$S\equiv \int^{t_2}_{t_1}L(x, x',t)dt$$ has a has a stationary point $x_0$ that satisfies the following: $$\frac{d}{dt}(\frac{\partial L}{\partial x'_0})=\frac{\partial ...
2
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3answers
106 views

Virtual Work: How is the applied force related to the coordinates chosen?

I have a question after reading a section from Goldstein's Classical Mechanics. The question deals with equation 1.43 in the text (given below): $$ \tag{1.43} \sum\limits_{i} {\bf F}_i^{(a)}\cdot ...
2
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1answer
87 views

Example of Hamilton's Principle to Systems with Constraints (Goldstein)

I'm currently studying Goldstein's Classical Mechanics book and I can't get my head around his reasoning in section 2.4. (Extending Hamilton's principle to systems with constraints). I'd like to ...
2
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0answers
95 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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0answers
55 views

What's the value of the coupling constant in interacting field theories?

Consider this Lagrangian : $L = \frac{1}{2}(\partial_\mu \Phi)^2 - \frac{M^2}{2}\Phi^2 +\frac{1}{2}(\partial_\mu \phi)^2 -\frac{m^2}{2} \phi^2 -\mu\Phi\phi^2$ Its interaction term is given by : ...
2
votes
2answers
120 views

Momentum vector transformation

I am confused about the way momentum vector transforms in the following case: $$q_k \to q_k'= q_k + \epsilon f_k(q)$$ The Jacobian is thus $\Lambda_{ij} = \frac{\partial q'_i}{\partial q_j} \approx ...
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1answer
78 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 ...
3
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0answers
61 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 ...
6
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1answer
119 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 ...
4
votes
2answers
264 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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1answer
59 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
67 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 ...
5
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0answers
57 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 - ...
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0answers
33 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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2answers
113 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 ...
4
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1answer
98 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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0answers
40 views

Treating $\psi$ and $\psi^{*}$ as independent variables when varying the action [duplicate]

Consider the following Lagrangian (deinsity): $${\mathcal{L}} = \partial_{\mu}\psi^{*} \partial^{\mu}\psi - V(|\psi|^2) $$ In my notes it says that " it's easier (and equivalent) to treat $\psi$ and ...
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0answers
30 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
81 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 ...
5
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4answers
2k views

What is canonical momentum?

What does the canonical momentum $\textbf{p}=m\textbf{v}+e\textbf{A}$ mean? Is it just momentum accounting for electromagnetic effects?