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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Which of the Physics textbooks would you recommend I read this quarter (Analytical Mechanics)? [duplicate]

My Analytical Mechanics class this quarter has one required textbook: "Classical Dynamics of Particles and Systems" by Thornton & Marion and three recommended readings: "Mechanics" by Landau ...
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29 views

Solving Lagrangian equations of motion for two point-bodies with gravitational interaction

I would like to solve the equations of motion with the Lagrangian function for two point-bodies that interact gravitationally via the potential $$V= {-Gm_1m_2 \over r_{12}} $$ where $$r_{12} = **r_1 ...
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1answer
40 views

Sign of matter Lagrangian term in curved space

In field theory the (matter) Lagrangian $\mathcal{L}_m$ is uncertain upto an overall constant multiplying factor (i.e. $\mathcal{L}_m$ and $a\mathcal{L}_m$ yield the same field equation(s) on ...
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3answers
57 views

Lagrangian/Hamiltonian mechanics at high school?

Has anyone developed an approach to teaching mechanics based on Lagrangian/Hamiltonian mechanics from the ground up. I mean from high school on up. This is akin to explicitly not talking about ...
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51 views

Schwarzschild metric circular orbits and kepler's 3rd law

I have been looking at the Schwarzschild metric presented to me as the following within lectures: ...
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32 views

Information contained in Lagrangians and actions [duplicate]

I've been looking into analytical mechanics with the intention of finding out more about Lagrangians and actions. As far as I currently understand it, the Lagrangian is formed with positions and ...
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40 views

Why is the strong CP term $ \theta \frac{g^2}{32 \pi^2} G_{\mu \nu}^a \tilde{G}^{a, \mu \nu}$ never considered for $SU(2)$ or $U(1)$ interactions?

The Lagrangian one would write down naivly for QCD is invariant under CP, which is in agreement with all experiments. Nevertheless, if we add the term \begin{equation} \theta \frac{g^2}{32 \pi^2} ...
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1answer
25 views

What is a “Reversed Effective Force”?

I have some confusion about the "Reversed effective force" as it appears in the derivation of D'Alembert's principle. First I have sources that seem to be contradictory. ...
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2answers
129 views

Non-relativistic QFT Lagrangian for fermions

Take the ordinary Hamiltonian from non-relativistic quantum mechanics expressed in terms of the fermi fields $\psi(\mathbf{x})$ and $\psi^\dagger(\mathbf{x})$ (as derived, for example, by A. L. Fetter ...
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0answers
46 views

Hamiltonian linearly proportional to momentum

In this question, it is discussed why, in Lagrangians we usually stick to first derivatives and quadratic terms we never see higher derivatives. The selected answer shows that, if a Lagrangian $L(q, ...
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2answers
79 views

Phase space Lagrangian?

Reading out of this lecture series we define a phase space Lagrangian $\mathcal L$ to be a function of $4n+1$ variables namely $q,\dot q,p,\dot p,t$. My question is, what space is this function ...
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1answer
77 views

Hamiltonian field equations constraints

Let's consider the Lagrangian $$\mathcal{L}~=~-\frac{1}{2}(\partial_\mu\phi^\nu)^2+\frac{1}{2}(\partial_\mu\phi^\mu)^2+\frac{1}{2}m^2\phi_\mu \phi^\mu,$$ with Minkowski metric $\eta_{\mu\nu}={\rm ...
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45 views

Why does the following contradiction arise in Lagrangian Formalism?

If we look at the Lagrange's equation $\frac{d}{dt}(\frac{\partial L}{\partial \dot{q_i}})- \frac{\partial L}{\partial q_i}=0$ It is clear that Lagrangian is invariant under a Transformation $L ...
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2answers
91 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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0answers
48 views

Lagrangian of constrained point mass [closed]

I'm trying to learn calculus of variations independently and I'm trying to solve this interesting looking problem 18.5.13 from Weber Essentials of Math Methods for Physicists (google preview here) but ...
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0answers
55 views

Noether's first and second theorems

My understanding of Noether's first theorem is as follows. Consider a set of infinitesimal transformations that leave the action invariant, that are indexed by $n$ linearly independent parameters, ...
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1answer
59 views

How to find equations of motion when potential is given by inverse-square? [closed]

When potential is $U=-\dfrac{a}{r^2}$ ($a>0$), how can I find $r=r(\phi)$? I'm trying to solve this problem during several hours. From $E=T+U$, and constant angular momentum $L$, I can get the ...
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0answers
70 views

When to use Hamiltonian vs Lagrangian?

I currently studying the Lagrangian and Hamiltonian formalisms in classical mechanics, but something I'm not seeing is how do I know which one to use in a given problem? After I find the Lagrangian, ...
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1answer
83 views

Lagrangian, Kinetic & Potential energy with two masses connected to three springs

Two masses $m_1$ and $m_2$ are on a frictionless surface. They are connected by three springs with constants $k_1,k_2,k_3$. $k_1$ and $k_3$ are attached to walls and $k_2$ is between the masses. $k_1$ ...
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1answer
40 views

Lagrangian for small oscillations

For a double pendulum we can consider 2 generalised coordinates $\theta_1$ (angle between first mass and vertical axis) and $\theta_2$ (angle between second mass and vertical axis). The Lagrangian to ...
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1answer
39 views

Lagrangian formalism (demonstration)

My question is about the multiplicity of the Lagrangian to a Physics system. I pretend to demonstrate the following proposition: For a system with $n$ degrees of freedom, written by the ...
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0answers
37 views

Is it possible to formulate a Hamiltonian for a damped system?

I recently found out that it is possible to formulate a Hamiltonian for a system with time-dependent coordinates such that the Hamiltonian is not the same as the energy When is the Hamiltonian of a ...
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1answer
45 views

Lagrangian formalism application on a particle falling system with air resistance

I have this problem, with a first-step resolution: $$...$$ So, I just don't know why they put the term $\frac{\partial F}{\partial \dot{z}}$ in Euler-Lagrange's equations. Why? I know that the ...
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2answers
55 views

Scalar and vector defined by transformation properties

In Classical Mechanics, we are defining scalars as objects that are invariant under any coordinate transformation. Vectors are defined as objects that can be transformed by some transformation matrix ...
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1answer
57 views

What is a point transformation?

This problem comes from Goldstein. What does $s=e^{\gamma t}q$ mean? Do I just put $q=e^{-\gamma t}s$ into the Lagrangian? But I don't know what that means. I think the point transformation may ...
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2answers
151 views

Adding a total time derivative term to the Lagrangian

This is proof that $L'$ represents same equation of motion with $L$ through Lagrange eq. I understand $L'$ satisfies Lagrange eq, but how does this proof mean $L'$ and $L$ describe same motion of ...
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1answer
39 views

Oscillation of a vertical rod supported by horizontal spring [closed]

The system seems to oscillate with $\omega = \sqrt{\frac{\frac{3}{2}mgl + 3k a^2}{ml^2}}$ for small angle $\theta$, and in particular for whatever stiffness $k$ chosen relative to gravitational ...
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1answer
56 views

What is the phase of a gauge coupling?

We typically take gauge couplings to be real and positive. Why do we impose these two conditions? I assume this is a requirement because gauge theories without positive couplings are unphysical or is ...
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0answers
36 views

Translation of the Mechanique analytique [closed]

Is there an English translation of the Mechanique analytique by Lagrange that is free? I have tried searching up online, however I only get French originals. The English translations seem all to be ...
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2answers
119 views

An inconsistency in Hamiltonian formulation for non-local Lagrangian: what am I doing wrong?

This question is based on a previous question I asked, Q. [1] In this question, I proposed an example of a non-local Lagrangian (functional), I'm revisiting it here: $$\mathbb{L}=\frac{1}{2}\int^t_0 ...
4
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1answer
126 views

Legendre transform for non-local Lagrangians, or Hamiltonian of non-local Lagrangian and their properties

This is sort of a multi-part question, mostly dealing with how to treat non-local Hamiltonians and how the corresponding properties of Hamiltonians work in a non-local framework. I proposed an example ...
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1answer
41 views

Metric in Lagrangian and the minimum total potential energy principle

I was wondering why physical systems "like" to go to the minimum of potential energy and I found this question, that tries to justify the minumum total potential energy principle. I was also reading ...
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1answer
49 views

Potential Energy in modified Atwood Machine

The initial length of the spring is $l_0$. I need help understanding how the potential energy of this system comes to be. I know the answer: $$ U = ...
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1answer
61 views

What is the function type of the generalized momentum?

Let $$L:{\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}\to {\mathbb R}$$ denote the Lagrangian (it should be differentiable) of a classical system with $n$ spatial coordinates. In the action ...
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1answer
30 views

A question on the functional dependence of the Lagrangian density

I understand that in classical mechanics the state of a particle at a given instant in time is given by its position $q$ and its velocity at that point $\dot{q}$, and given that, for any given point ...
2
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1answer
62 views

In general, can a Lagrangian density depend on space-time explicitly?

In an exercise on classical field theories, I'm trying to derive the general formula of the Energy-momentum tensor. According to the formula in the lecture notes, this tensor includes a term of minus ...
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1answer
42 views

How to calculate the Hamiltonian from the Lagrangian for a non-relativistic charged point particle in an EM field?

I was given the equation of the Lagrangian: \begin{equation} L~=~\frac{1}{2}m \dot{x}^2+\frac{e}{c}\vec{\dot{x}}\cdot \vec{A}(\vec{x},t)-e\phi (\vec{x},t). \end{equation} I proceeded to use the ...
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1answer
47 views

Lagrange equation and a force derivable from a generalized potential

I was reading the solution of this exercise and I have a doubt: A point particle moves in space under the influence of a force derivable from a generalized potential of the form $$U(r,v) = ...
2
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1answer
40 views

Full time derivative of the Frank-Oseen energy. Mathematical problem

I am studying liquid crystal theory with the book Kleman, Lavrentovich, Soft Matter Physics. In the Ericksen-Leslie theory, Frank-Oseen energy density is: $$ f=0.5*(K_1*div^2 (n)+K_2 ...
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1answer
119 views

Calculus of variations and string theory

In Polchinski's String theory book, Vol 1., in chapter 1, p. 18, he is deriving the Lagrangian in the light cone gauge (that's not necessary to know in order to answer this question), and he gets ...
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1answer
30 views

Under what cases is the Batalin-Vilkovisky (BV) operator nilpotent?

It is understood that when we deal with gauge algebras which close on-shell only after using equations of motion or where the space-time is curved, we can no longer just do away with BRST ...
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2answers
151 views

Kepler's Laws to Determine Radius of Circular Orbit

"In nonrelativistic limit of general relativity there is a correction to the Newtonian gravitational potential energy $−h/r^3$ with $h = αL^2/(mc)^2$, where c is the speed of light, α = GMm and L is ...
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1answer
49 views

Show $\frac{\partial T}{\partial \dot q_j} = m_i \dot r_i^T\frac{\dot r_i }{\partial \dot q_j} $ [closed]

This is a basic result in lagrangian mecanics. Let $T$ be the kinetic energy, $r_i$ be the position of the $i^{th}$ particle in the system I need to show $$\frac{\partial T}{\partial \dot q_j} = ...
3
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1answer
80 views

Lagrangian vector field expression

The Lagrangian vector field $X_L$ on the tangent bundle is given in page 4 of Marco Mazzucchelli's "critical Point Theory for Lagrangian systems" to be; \begin{equation} ...
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1answer
129 views

Solve hanging rope question by using Lagrangian density [closed]

for this one, I can only do the first(a) $$V_{total}=\frac{mg}{l} \int_{x_0}^{d-x_0}{y(x)\sqrt{1+(\frac{dy(x)}{dx})^2}}dx$$ and have no idea about others.
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0answers
55 views

The principle of least action [duplicate]

I have read about the principle of least action. This principle suggests that nature would allow a particle to travel in a path along which the integral of the difference between kinetic energy and ...
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1answer
50 views

Showing time-invariance of Lagrangian with time-displacement operator

I am trying to show that the time-invariance of the Lagrangian of a simple one-particle system implies energy conservation for that system. The first step is, well, to show that the Lagrangian is ...
2
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2answers
106 views

Hamiltonian from a Lagrangian with constraints?

Let's say I have the Lagrangian: $$L=T-V.$$ Along with the constraint that $$f\equiv f(\vec q,t)=0.$$ We can then write: $$L'=T-V+\lambda f. $$ What is my Hamiltonian now? Is it $$H'=\dot q_i p_i ...
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1answer
51 views

Dissipated Energy from Falling Object using Lagrangian [closed]

A plate of mass $M$ moves horizontally with initial speed $v$ on a frictionless table. An object of mass $m$ is dropped vertically onto it from the height $h$ and smashes. How much energy is ...
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1answer
48 views

Euclidean classical action

This is the Euclidean classical action $S_{cl}[\phi]=\int d^{4}x\ (\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi))$. It would be nice if somebody could explain the structure of the potential. I don't ...