Questions tagged [constrained-dynamics]

A constraint is a condition on the variables of a dynamical problem that the variables (or the physical solution for them) must satisfy. Normally, it amounts to restrictions of such variables to a lower-dimensional hypersurface embedded in the higher-dimensional full space of (unconstrained) variables.

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Vertical Disk rolling (Goldstein page 15 ) [closed]

Consider a disk rolling on the horizontal xy plane constrained to move so that the plane of the disk is always vertical. The coordinates used to describe the motion might be x,y coordinates of the ...
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What are the quantum consequences of the usual constraint used in sigma models?

Consider the sigma model in 2 dimensions with $N$ sigma fields $$ \mathcal{L}=\frac{N}{2f}\partial_{\mu}\sigma^a\partial^{\mu}\sigma^a. $$ We want these fields to obey the constraints $$ \sigma^a\...
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Negative eigenfrequencies and conserved quantities

Suppose one has the following diagonal Hamiltonian $$ \hat{H}=\omega_1 \hat{n}_1 - \omega_2 \hat{n}_2 $$ where $\omega_1$ and $\omega_2$ are greater than zero. Number operators $\hat{n}_1$ and $\hat{...
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168 views

How are second-class constraints handled in the path integral formulation?

A first-class constraint is typically associated with a gauge redundancy. In order to account for this in the path integral, we simply integrate over only gauge-inequivalent configurations. This is ...
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1answer
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Identify a Hamiltonian system consistent or not?

I'm sorry if my question is too classic and basic. As Dirac-Bergmann algorithm for Hamiltonian formalism, I find out that a Hamiltonian system is inconsistent if Poisson bracket of primary constraints ...
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256 views

Proof of Thomson's theorem on electrostatics using variational calculus

I'm following a proof of Thomson's theorem but I'm a bit confused when they use a lagrange multiplier to include the charge conservation constraint, could someone explain to me how is this multiplier ...
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Confusing Question on “Degrees of Freedom”

I am studying “Rosenberg’s Analytical Dynamics of Discrete Systems”. In chapter 4, he discussed on holonomic and nonholonomic constraints. At the end of the chapter, he asked a question that confused ...
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Can we write a Lagrangian for the classical system with $H=kqp$?

Say we have the following Lagrangian: $$ H = kqp. $$ The equations of motion are easy to find: $$ \dot{q} = kq \\ \dot{p} = -kp, $$ and to solve: $$ q=q_0 e^{kt} \\ p=p_0 e^{-kt}. $$ I'm curious ...
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Question about the applications of Gauss's principle of least constraint

Recently i've learned the formulation of Gauss's principle of least constraints, which states that the motion of a system of material points is in maximal accordance with free motion, or under least ...
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When does 'naively' quantizing classical constraints work?

Consider the generally covariant formulation of the relativistic point particle, where the configuration is specified by $x^\mu(\tau)$, and $\tau$ is an arbitrary parameter. In the Hamiltonian picture,...
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Why is Lorenz gauge $\partial_\mu A^\mu = 0$ not attainable for 'permissible boundary conditions'?

I'm reading Paul Townsend's string theory lecture notes and I'm confused about a paragraph. Below, the $\varphi_i$ are first class constraints and the $\chi_i$ are gauge fixing conditions. ...
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Generalised Coordinates in 3D Rotation

If you have N particles on a surface of a rigid body and the rigid body is rotating about some axis, we say there are six generalised coordinates for the system (N particles on the surface) and set up ...
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Quantum mechanics with non-holonomic constraints?

Are there any established theories for non-holonomic quantum mechanics? I have googled for d'Alembert's principle, non-holonomic constraints and quantum mechanics but only found an abstract talking ...
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Finding generalized coordinates when the implicit function theorem fails

Given some coordinates $(x_1,\dots, x_N)$ and $h$ holonomic constraints, it should always be possible to reduce the coordinates to $n=N-h$ generalized coordinates $(q_1,\dots, q_n)$. This is ...
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1answer
358 views

How to find the equations of motion with a constraint?

I am trying to find the equation of motion for a particle constrained to move on the surface defined by $S:z=\cos x+\sin y$ under the influence of gravity. I am working in the Cartesian Coordinate ...
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How to get the fourth component of EOM in a relativistic formulation of a charged particle in an electromagnetic field?

We consider in Lorentz spacetime, $(x^0,x^1,x^2,x^3)=(t,x,y,z)$, choose the unit of time such that $c=1$. Given a four vector $A_\mu$, and let the Lagrangian $$L(x^i,\dot x^i,t)=-m\sqrt{1-\dot x_i\...
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(Anti)commutation of ghosts and fermions

I would like to ask whether fermionic Grassmann fields in a gauge theory path integral (say in QCD) should be chosen to commute or anticommute with ghost and anti-ghost fields. The way most textbooks ...
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Adding gauge fixing directly by hand is different from by Lagrange multiplier?

Why is adding gauge fixing directly different from doing so by Lagrange multiplier? For simplicity, we don't use field model. Direct method Consider a system $$L(x,\dot x,y,\dot y)=\frac{\dot x^2}{...
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Showing that Gubta-Bleuler doesn't work in non-Abelian case

In my class the teacher tried to show that Gupta-Bleuler doesn't work in non abelian gauge theories. But I didn't really understood what he did. Here is what we did: We have: $$ \mathcal{L}=\...
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Lagrangian mechanics - constraint forces & virtual work

The constraint forces have the dot product 0 ($\textbf C \cdot \textbf x=0$, where $\textbf C$ and $\textbf x$ are vectors, x being the virtual displacement. But the dot product is 0 if $\textbf C$ ...
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What is the major difference between Dirac and BRST quantization of point particle?

I have derived the action for the bosonic point particle and now I want to quantize it but there are two formalism: one is Dirac and the other one is BRST. I want to know what is the major difference ...
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Exact differential equations and holonomic constraints

I understand that if a constraint equation given on a differential form is exact, that means it is also holonomic since I can find a solution. But there are other types of differential equations, like ...
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How to impose this constraint on two particles on General Relativity?

Let $(M,g)$ be spacetime and consider a system composed of two particles of masses $m_1$ and $m_2$ connected by a massless rod whose lenght varies in time. This constraint is quite easy to implement ...
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Frictionless motion along a surface defined by a function

I have recently been learning Lagrangian Mechanics and wanted to apply it to a case I came up with in my head. Say you have a function f(x) that is always > 0. If a box or some solid particle was ...
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Why are constraint forces assumed to be perpendicular?

In Taylor, it is assumed that the constraint forces are perpendicular to the constraint surface. I am kind of wondering what the justification for this is. Sure, if you mention typical constraint ...
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Lagrangian of the Schrödinger equation: Ambiguity in legendre transformation? [duplicate]

The Lagrangian-density (as a function of the "independent fields" $\Psi$ and $\Psi^*$ which leads to the Schrödinger-equation is: $$ \frac{i \hbar}{2}(\Psi^* \dot{\Psi} - \Psi \dot{\Psi^*}) - \frac{\...
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1answer
309 views

Condition that the Lagrangian energy function $h\equiv\sum_i\frac{\partial L}{\partial\dot q_i}\dot q_i-L$ would be same as the mechanical energy $E$

I'm studying Classical Mechanics by Goldstein. I solved a problem but I have a question. Pro 2.18 A point mass is constrained to move on a massless hoop of radius a fixed in a vertical plane ...
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Why use constraints at start in the Hamiltonian expression?

For example, consider the following situation: I have a simple plane pendulum consisting of a mass $m$ attached to a string of length $\ell$. After the pendulum is set into motion, the length of the ...
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Supersymmetry of action with constraints in terms of unconstrained fields - Witten's topological sigma model

The following is a continuation of this question. The action of Witten's topological sigma model (defined on a worldsheet, $\Sigma$, with target space an almost complex manifold denoted $X$) takes the ...
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Statistical Treatment of Constrained Systems in the Microcanonical Ensemble

Consider a constrained classical Lagrangian $ \mathcal{L}' = \mathcal{L}(q, \dot{q}) + \lambda f(q) $ where $ \lambda $ is the lagrange multiplier for the constraint. We can get a Hamiltonian for this ...
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Equation of motion of a suspended weight along an elliptic arc [closed]

I have a weight moving between two poles on a cable which is always under tension. The fact that the cable has constant length means that it moves along an elliptic arc from periapsis to apoapsis. The ...
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Can Lagrange multipliers be tensorial quantities?

Let's say I have a many-rigid-bodies-system subject to the constraint that some of the distances must be constant over time. The usual Lagrangian for the system is $$\mathcal L=T(\{\dot q_n\})-V(\{q_n\...
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Virtual displacements

From Goldstein's Classical Mechanics, A virtual displacement of a system refers to a change in the configuration of the system as the result of any arbitrary infinitesimal change of the coordinates ...
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1answer
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What are the properties of constraint forces? [closed]

I've just started studying mechanics. I need to find the properties of the constraint forces. I've gone through many books and also searched on internet but do not find any thing useful.
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Atwood Machine - with pulley attached to one mass. What's the constraint?

2 problems here: What is the force pulling $m_2$ when $m_1$ falls? I can only think of tension. When $m_1$ does fall (maybe a height $H$), does $m_2$ move $H/2$ to the right? Since the other end of ...
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1answer
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What is the precise relationship between a non-invertible Hessian matrix for the Lagrangian and the presence of a gauge symmetry?

Consider a system described by $q^i(t)$ and its derivatives, by means of a Lagrangian $L=L(q,\dot q)$ and possibly $t$. We say the system is degenerate if $$ \det\left(\frac{\partial L}{\partial \dot ...
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Derivation of Lagrange Equations from Newton's Second Law for a Non-holonomic System of Particles

I am interested to write down a derivation of Lagrange equations from Newton's second law for a non-holonomic system of particles. Here, I mention my derivation where I am stuck right at the last step....
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2answers
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How to decide between vectors and it's components?

Suppose we have following situation assume that both are kept over each other. As $m$ experiences $N$ normal due to $M$ and $mg$ due to gravity. There no motion hence we can write: $mg-N\cos A=0\...
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2answers
258 views

Obtaining Euler-Lagrange equation from action with constraint - Witten's topological sigma model

The action of Witten's topological sigma model (defined on a worldsheet, $\Sigma$, with target space an almost complex manifold denoted $X$) takes the form $$ S=\int d^2\sigma\big(-\frac{1}{4}H^{\...
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2answers
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Why can we not set each applied force equal to zero?

With reference to page 17 of "Classical Mechanics" by Goldstein, Safko and Poole, the small paragraph after eq. 1.43, $$\sum_i \mathbf{F}^{(a)}_i \cdot \delta \mathbf{r}_i ~=~ 0.\tag{1.43}$$ I do not ...
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Are generalised coordinates truly independent?

Say we have a system with two generalised coordinates $x$ and $y$. When we solve the equations of motion we find $x=x(t)$ and $y=y(t)$. I can invert one of these solutions to find $t=t(y)$ and ...
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Confused about the definition of holonomic constraints [duplicate]

I'm reading Goldstein's Classical Mechanics and he defines a constraint on particles having radii $\mathbf{r}_i$ to be holonomic if it can be written as $f(\mathbf{r}_1, \mathbf{r}_2, \dots , t) = 0$. ...
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366 views

Does d'Alembert principle hold for non-conservative forces?

I know that D'Alembert's principle doesn't hold for sliding friction. But does it hold for any non-conservative force (other than sliding friction) or not? Could you give some examples of non-...
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1answer
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What is “irreversible displacement”?

In this Wiki page on D'Alembert Principle they say that "The principle does not apply for irreversible displacements, such as sliding friction, and more general specification of the irreversibility is ...
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1answer
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How to determine the equations of motion of a rigid body's center of mass from a constraint?

Picture a rigid square with one of its vertices attached to the end of a massless rigid rod whose other end is attached to a point fixed in space. The motion is restricted to the plane containing the ...
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1answer
532 views

Hamilton's principle with semiholonomic constraints in Goldstein

I am studying from Goldstein's Classical Mechanics, 3rd edition. In section 2.4, he discussed Hamiltion's principle with semiholonomic constraints. The constraints can be written in the form $f_\alpha(...
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1answer
42 views

2 Pulleys moving towards each other

2 stands B, with pulleys mounted on them move towards each other with velocity V A rope is passing over the pulleys and a block A is attached to the rope Find the velocity of A when rope is at angle ...
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1answer
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Deriving the commutation relations in Maxwell-Chern-Simons theory

I am learning quantization of MCS theory. $$L_{MCS}=-\frac{1}{4}F^{\mu \nu}F_{\mu\nu}+\frac{g}{2} \epsilon^{\mu \nu \rho}A_\mu\partial_\nu A_\rho$$ I have reached the commutation relation $$[A_i(\...
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1answer
147 views

Problem with constraint (Lagragian mechanics) [closed]

I'm working on this problem where there is a small cylinder with radius $r$ rolling inside a bigger cylinder with radius $R$. I'm asked to solve the lagrange equation. It looks like this: I looked at ...
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1answer
57 views

Deriving a Formula related to magnetic braking

So recently I came about this report on magnetic braking which I mostly got. However, there was a proof that they skipped over and went straight to the solution that I would like to understand. Would ...