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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Covariant form of non-relativistic free particle

I have two questions about the action of free particle. $$S=\int dt~\frac{m}{2}~(\frac{d \vec{x} }{dt})^2 \tag{1}$$ The Covariant form is: (assume: $m=1$) $$S=\int d\tau \frac{1}{v(\tau)}~\...
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1answer
47 views

Homework central force Lagrangian formulation

For central force $$T = \frac{m}{2}(\dot{r^2} + r^2\dot{\theta^2} + r^2\sin^2{\theta}\dot{\phi^2}$$ Now applying the Lagrangian equation of motion, we get $$\frac{\partial{L}}{\partial{\phi}}=0\...
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2answers
91 views

Action max, min, or saddle?

It is well known that $\delta S = 0$ lays the foundation for variational mechanics. But I am confused as to whether or not this S is a minimum, a maximum, or a saddle point. Some books address this ...
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1answer
117 views

Quantum field theory with constraint: energy-momentum conservation?

Suppose I have a 2-form field $B$ and a Lagrange multiplier field $\lambda$, then the Lagrangian $S = \int (B \wedge \delta B + \lambda \delta B \wedge \delta B)$ with a Lie derivative operator $\...
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2answers
95 views

How do you find potential by Lagrangian formalism?

Suppose a ball is falling towards earth and hence by Lagrange equation we can find $T$ and $V$ where $V$ is $mgh$. But we know $V$ only because we know $F = mg$. Now since Lagrange equation doesn't ...
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95 views

Is the phrase “coupling constant” interchangable with “ strength of interactions”?

Can I use the terms coupling constant and strength of interactions, interchangeably, or are there more subtleties to the term coupling constant that I am not aware of? Coupling Constants from ...
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32 views

Lagrangian for $\mathcal{N}=4$ SQED in 3D

What is the Lagrangian for $3D$ $\mathcal{N}=4$ supersymmetric QED, with $N_f$ hypermultiplets? In particular, which is the form of the Fayet-Iliopulos terms, and the real mass terms?
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1answer
98 views

Help on understanding a concept in Noether's first theorem

Given a Lie group $G$, whose most general transform depends on $\rho$ parameters, under the action of which an integral $I$ is invariant, there are $\rho$ linearly independent combinations of the ...
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98 views

Why do Lagrange multipliers work in mechanics?

I understand that it is not always simple to find generalized coordinates that satisfy the constraint equations, so we try to find an alternative (more mechanical) method that yields curves that ...
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1answer
104 views

Trouble understanding Landau & Lifshitz writing about Lagrangians and Galilean Relativity [duplicate]

We have two inertial coordinate systems, $K'$ and $K$. $K$ is moving with infinitesimal velocity ${\epsilon}$ relative to $K'$. Using Galilean relativity we can transform this into $v'=v+{\epsilon}$. ...
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2answers
116 views

Derivation of Euler-Lagrange equations in Landau's and Lifshitz's “Mechanics”

There's an integral ${\int\limits_{t_1}^{t_2}}(\frac{\partial{L}}{\partial{q}}{\delta}q+\frac{\partial{L}}{\partial{v}}{\delta}v)dt=0$. [1.] $ {\delta}v={\frac{d{\delta}q}{dt}}$ [2.] I should get $ [...
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2answers
678 views

Estimate for energy dissipated by a damper/dashpot

I have a system with a mass $m$ attached to the end of a cable. The cable mass is assumed negligible. The cable is attached to the ground at the one end while the other, with the attached mass $m$, ...
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0answers
26 views

Negative Kinetic energy of normal modes in an elastic solid?

So I'm pretty stupid, admittedly, but I'm also pretty confused. I also fully realize this question is probably completely idiotic. The background is normal modes in linear elastic solids. Let the ...
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0answers
72 views

Lagrangians not related via a total time derivative lead to same Noether symmetries?

Having answered my initial two questions (v1), I now consider a third possibility. Consider two Lagrangians that both lead to equivalent equations of motion. Suppose that they are not related via a ...
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1answer
165 views

Derivatives with upper and lower indices

I'm studying classical and quantum field theory, but evaluating derivatives of fields (scalar and/or vector) described with upper and lower indices is somewhat new to me. I'm trying to evaluate $$\...
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3answers
474 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 be ...
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2answers
98 views

Is there any loss or gain of information if a physics law is changed from one form to another? [closed]

Is there any loss or gain of information if a physics law is changed from one form to another such that the parameter appearing in them is changed from vector to a scalar? For example, consider the ...
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2answers
348 views

Goldstein's derivation of the 'principle of least action'

I want make an punctual question ands it's about The derivation of the expression $$ \Delta\int_{t_1}^{t_2} Ldt=L(t_2)\Delta t_2-L(t_1)\Delta t_1 + \int_{t_1}^{t_2} \delta L dt. \tag{8.74}$$ You can ...
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1answer
66 views

Is it strictly necessary to require gauge invariance of the action and equations of motion?

When writing down an action for a gauge theory, we require that the action be gauge invariant. This is typically taken to mean that the action must be written explicitly in terms of gauge invariant ...
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2answers
79 views

From Newtonian systems to Lagrange mechanics using Euler - Lagrange equations [closed]

I am reading some notions about Lagrangian Mechanics from Holm's book.(page 12). There is: Every Newtonian system, $$m_i \ddot{q_i}=\frac{\partial V}{\partial q_i}, \mbox{ } i=\overline{1,N},...
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1answer
110 views

Finding the Hamiltonian for sound vibrations in a gas in the momentum representation

I am working on a problem where I have been given the following Lagrangian density for describing sound vibrations in a gas: $\mathcal{L}=\frac{1}{2}[\rho_0\dot{\eta}^2+2P_0\nabla\cdot\eta-\gamma P_0(...
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36 views

Strain energy density (potential energy of elastic continua)

This question has to do with writing down the potential energy of an elastic body, which obeys a generalized Hooke's law [; \sigma_{ik} = \sum_{klm} \lambda_{iklm} u_{lm} ;] Where $\sigma$ is the ...
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1answer
77 views

Potential Energy of Lagrangian system

I want to clear my basic understanding of Lagrangian system. I am confused about the potential energy for the Lagrangian system. According to this picture, m is suspended from a rigid massless ...
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32 views

The Hamilton-Suslov principle

Building on the introduction given in the following paper, how does the Hamilton-Suslov principle generalise Hamilton's principle to consider contained mechanics? \begin{equation} \int ^{t_1}_{t_0}\...
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2answers
113 views

qualitative explanation of Principle of Least Action (vertical movement)

Consider the following situation I want to understand what the PLA means here from an intuitive and qualitative point of view. I understand the mathematical approach. Combining $L(y,\dot{y})=\...
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1answer
82 views

Independence of generalised coordinates and momenta in Hamiltonian mechanics [duplicate]

I am told that in Hamiltonian mechanics, we put the generalised coordinates $q_i$ and generalised momenta $p_i$ on equal footing, and treat them as being independent from one another. But I'm ...
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1answer
207 views

Why are the Euler-Lagrange equations invariant if we add a surface term to the action?

In the lecture on Noether's theorem and the Lagrange formulation of classical field theories, my professor wrote A symmetry is a field variation that maps solutions to solutions, which is true if ...
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1answer
127 views

Why does a system have to be holonomic?

So I'm doing some work from Taylor's mechanics book. He says for the problems in the book, we require the system to be holonomic - that is the number of generalized coordinates = number of Deg. of ...
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1answer
64 views

Deriving velocity after elastic and inelastic collision via the Principle of Least Action

I am reading on the Principle of Least Action from a historical perspective. I am also trying to make sense of it from a contemporary point of view -- though my training in contemporary physics is ...
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1answer
152 views

Locality defined in terms of the Lagrangian density

I've been reading through Matthew Schwartz's book "Quantum Field Theory and the Standard Model" and in chapter 24 there is a section on locality (section 24.4). In it he defines locality in terms of ...
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1answer
91 views

Allowed Virtual Displacements

I am having trouble understanding the kinds of virtual displacements which are permitted for a given constrained system. I have a specific example in mind: A block of wood resting on a table parallel ...
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25 views

Ostrogradsky instability [duplicate]

I am interested in Lagrangian Mechanics and I want to know why higher derivative theories are inconsistent with the nature. I heard that there is something called Ostrogradsky instability for higher ...
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1answer
132 views

Mathematics of the Virtual Displacement

So I'm pretty certain this question has been asked to death here, but I still can't find a good explanation of a very particular aspect of the virtual displacements in physics. Background For ...
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5answers
736 views

When/why does the principle of least action plus boundary conditions not uniquely specify a path?

A few months ago I was telling high school students about Fermat's principle. You can use it to show that light reflects off a surface at equal angles. To set it up, you put in boundary conditions, ...
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0answers
67 views

Why should physical theories always have a Lagrangian formalism? [duplicate]

I've often heard that every physical theory has some kind of Lagrangian formalism, or a formalism in terms of a principle of stationary action. The Standard Model has one, General Relativity has one, ...
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2answers
132 views

Equivalence of functional and partial derivatives

I am trying to derive Newton's second law from the principle of least action, that is, setting the functional derivative $\frac{\delta S}{\delta x(t)}$ equal to 0. $$S = \int dt' \left[ \frac{m}{2} ...
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2answers
234 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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2answers
86 views

Independent Variables of a Lagrangian

Let us consider a particle in one spatial dimension $x$ and one temporal dimension $t$. Its Lagrangian $L$ is given by \begin{eqnarray*} L &=& T- V \\ &=& \frac{1}{2} m\dot{x}^2 - ...
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115 views

Two Particles in a Harmonic Oscillator with repulsive short-range potential

Do bear with me, I am attempting to learn to write some simulations on the computer and learn some simple MD, so I defined sort of a toy problem. I have two particles confined in a Harmonic Potential ...
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1answer
118 views

$U(1)$ local gauge invariance in QED [duplicate]

While constructing Lagrangian of QED, we don't add the mass term for photon $\dfrac{1}{2} m^{2}A_{\mu}A^{\mu}$ because gauge invariance does not allow. I want to ask, whether "$\bf{Theoretically}$", ...
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1answer
679 views

Obtain the Lagrangian from the system of coupled equation [closed]

In this particular paper, "Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation" by C.K.Law, PhysRevA.51.2537 \begin{equation} \ddot{Q}_{k}=-\omega^{2}_{k}Q_{k}+2\...
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0answers
65 views

Is there any reason for principle of least action to be true? [duplicate]

My question is not rigidly related to physics. The principle of least actions says that for any dynamical system there exists a function parameterized by $q$'s and $\dot{q}$'s such that the line ...
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2answers
107 views

Why Kink can not tunnel to vacuum, and is topologically stable?

Why the kink $$\phi(x)=v\tanh(\frac{x}{\xi}) ,$$ can not tunnel into vacuum $+v$ or $-v$ (Spontaneous symmetry breaking vacuum). From the boundary condition ($x\rightarrow \pm\infty, \phi(x)\...
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3answers
132 views

Lagrangian from Path Integral

Suppose I somehow know propagator for a given quantum mechanical system but I don't happen to know either the Lagrangian or Hamiltonian. (For simplicity, assume that this is non-relativistic.) Is ...
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2answers
230 views

Appearance of the Jerk Term in Dynamics of Mass-Spring-Damper System

I am coming from the computer science territory and have not a long trace in mechanics. My background in derivation of the system dynamics could be summarized with utilization of the ...
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62 views

Noether's 2nd Theorem and Local Gauge Identities

I am trying to derive the so called Gauge Identities: \begin{equation} D_\nu\frac{\delta S}{\delta\phi} = 0 \end{equation} Where $D_\nu$ is an operator involving derivatives and $\frac{\delta S}{\...
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1answer
115 views

Do time-invariant Hamiltonians define closed systems?

In classical mechanics, every time-invariant Hamiltonian represents a closed dynamical system? Can every closed dynamical system be represented as a time-invariant Hamiltonian? Or are there closed ...
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71 views

Finding the configuration space and degrees of freedom of spherical pendulum

Suppose we have a spherical pendulum tethered to the origin in $\mathbb{R}^3$ where the length of the rod is a time varying function $l(t)$. What is the configuration space of this system, and how ...
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1answer
72 views

Is local chiral symmetry qualitatively the same as gauge symmetries?

I am confused by the role that local chiral symmetry plays in chiral perturbation theory. For the case of chiral QCD with three quark flavors, the Lagrangian is invariant under global $SU(3)_L\times{}...
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1answer
125 views

Hermiticity of Dirac operator in curved spacetime

The Dirac Lagrangian in curved spacetime is usually given by \begin{equation} \mathcal{L} = i\bar{\Psi}\gamma^a e^{\mu}_a(\partial_\mu + \frac{1}{4}\omega_{\mu bc}\gamma^b\gamma^c)\Psi \end{equation} ...