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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206 views

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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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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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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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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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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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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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 ...
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Example for singular Lagrange dynamics giving rise to a primary constraint

There is vast literature about constraint dynamics, but the available material is quite abstract and it feels to read about things which are far from reality. Is there a simple example, where I can ...
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107 views

Issue of definition of potential energy in constraint force problem

I am having some confusion regarding Lagrange multipliers and constraint forces. Consider the simple pendulum with generalized coordinates $r$ and $\phi$. We have the constraint that $r=l$. Orienting ...
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Is the axiom of constraints always valid?

Theoretical Mechanics, S. Targ, p. 28, Axiom of Constraints: Any constrained body can be treated as a free body relieved from its constraints, provided the latter are represented by their ...
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Resources on constrained Hamiltonian field theory

I apologise in advance if the title is not clear, but I don't know the exact name of the subject I'm looking for. I started reading Dirac's "Lectures on Quantum Mechanics", and it sparked my interest ...
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Why are Hamiltonian Mechanics well-defined?

I have encountered a problem while re-reading the formalism of Hamiltonian mechanics, and it lies in a very simple remark. Indeed, if I am not mistaken, when we want to do mechanics using the ...
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Why is transverse delta distribution needed for Poisson brackets of classical EM field components?

During reading a German lecture notes (Quanten Optik by Dirk–Gunnar Welsch) about quantization of the EM field, I stumbled over a statement I cannot reproduce in detail: Generally, it is argued, that ...
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What are Lagrange Multipliers, regarding holonomic constraints in classical mechanics?

To make it plain and simple, if I have a holonomic constraint, that I want to treat using a lagrange multiplier, in any textbook I concern, they are just expressed as "$\lambda$" (omitting possible ...
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Deriving the Lagrangian in constraint motion- where do Multipliers come from

Say we have N bodies satisfying the following constraint : $h(\vec x_1 , ... ,\vec x_N)=0$.I'm aware of the fact that we then have to introduce the Lagrangian multipliers, but I want to see how we can ...
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Inconsistency in Lagrangian vs Hamiltonian formalism?

Can both Lagrangian and Hamiltonian formalisms lead to different solutions? I have a simple system described by the Lagrangian \begin{equation} L(\eta,\dot{\eta},\theta,\dot{\theta})=\eta\dot{\theta}+...
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Motion of a particle constrained on a rotating rod

A sphere of neglectable radius is placed on a very long and frictionless rod (which we can approximate to a straight line) on which it is able to move. The rod rotates around one of its end points ...
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Is there a modified Least Action Principle for nonholonomic systems?

We know that one can treat nonholonimic (but differential) constraints in the same manner as holonimic constraints. With a given Lagrange Function $L$, the equations of motion for a holonomic ...
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Quantizing one real fermion

It is well-known how to canonically quantize the Lagrangian $$L = i \bar{\psi} \dot{\psi} - \omega \bar\psi \psi$$ I now wonder how one quantizes the Lagrangian with one real fermion $$L = i \psi \...
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Deflating wheel rolling without slipping

For a wheel of radius $R$ on an horizontal plane, the "rolling without slipping condition" is given by $$\Delta s =\Delta l$$ with $\Delta l$ being the distance travelled on the plane. $\Delta l$, ...
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Canonical formalism in light-cone coordinates

Consider scalar field theory $$ \mathcal L = \frac{1}{2} (\partial \phi)^2 -V(\phi).$$ I want to understand Hamiltonian formalism in light-cone coordinates. I choose convention $$x^{\pm}=\frac{1}{\...
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bead on rotating wire, constraint force [closed]

I'm asked to find the force that a horizontal wire that rotates with constant velocity $\omega$ exerts on a bead on it, neglegting gravity. Attempt: Using two coordinates: $r$ as the distance from ...
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Lagrangian and KKT Constraint for Falling Ball with a Floor

I am trying to use KKT conditions and slack variables (pg. 118) to put a floor constraint into my Lagrangian for a falling ball with a floor. \begin{align} \text{extrem}\ \ &\int dt\ \frac{1}{2} ...
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Time-reparametization invariance in classical Hamiltonian mechanics

This post considers an aspect of time-reparametization invariance in classical Hamiltonian mechanics. Specifically, it concerns the use of Lagrange multipliers to rewrite the action for a classical ...
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A query on Hamiltonian formulation as explained in 3rd edition of Goldstein's “Classical Mechanics” book

In 3rd edition of Goldstein's "Classical Mechanics" book, page 335, section 8.1, it is mentioned that : In Hamiltonian formulation, there can be no constraint equations among the co-ordinates. ...
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String Constraint Relation Doubt

consider the diagram i have attached for reference guys. so in the diagram, the author has taken all the length of the strings from some single datum, the ceiling here. my question is, if i take the ...
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“Interlocking constraints” in Golwala's Classical Mechanics lecture notes

In order to obtain d'Alembert's principle, one must exclude situations in which constraint forces do virtual work. Actually, not individual constraint forces, but (according to the notes mentioned in ...
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What is wrong with my argument to derive the Hamiltonian in relativity?

In General Relativity (and special too) the Lagrangian for a particle of mass $m$ in the absence of forces other than gravity is $$L=m\sqrt{g_{\mu\nu}U^\mu U^\nu}$$ where $U^\mu$ is the four-...
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Lagrange Multipliers for Simple Pendulum

Here we consider a simple pendulum that is being analyzed by Lagrange Multipliers. Shown in Fig. 1 is the pendulum of length $l$ and mass $m$. Let $U=0$ on the $x$-axis. Let the constraint equation be ...
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Four-divergence and Legendre transformation

As a study case, consider the following Lagrangian for a left-handed Weyl field $\chi \in \mathbb{C}^{2}$: $$\mathcal{L} = \chi^{\dagger} \mathrm{i} \overline{\sigma}^{\rho} \partial_{\rho} \chi$$ ...
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Is every constraint involving only two coordinates integrable?

There is a footnote on Goldstein's Classical Mechanics (3rd ed., page 15) which says the following: In principle, an integrating factor can always be found for a first-order differential equation ...
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Total vs extended action on constrained dynamics

Studying the electromagnetic hamiltonian dynamics, I used the extended formalism (after finding all constraints using the primary hamiltonian, also following the Dirac's recipe) to calculate the ...
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Holonomic constraints and degrees of freedom

Wikipedia and other sources define holonomic constraints as a function $$ f(\vec{r}_1, \ldots, \vec{r}_N, t) \equiv 0, $$ and says the number of degrees of freedom in a system is reduced by the ...
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Generalized Forces and Potential Energy

Consider a (conservative) system of $N$ particles with $\vec{r}_i$ being their positions. In this system, there are attractive and repulsive forces between these particles. $\vec{F}_i$ shall be the ...
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Using Work Energy theorem to find acceleration

Here's a sample problem: A block of mass $m$ is free to move vertically on a wedge of mass $M$ and angle of inclination $\theta$ that rests on flat ground. If all surfaces are frictionless, then find ...
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Can we solve for $x(t)$ and $y(t)$ in closed-form with respect to time?

I would like to ask a question I was discussing the other day with a friend of mine. Suppose you have a point mass m, sliding on the friction-free curve $y = e^{-x}$ starting from position $x(0) = 0$ ...
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Why is the Hamiltonian zero in relativity?

I'm trying to understand something with the lagrangian and the hamiltonian formalisms in relativity theory, and why the following result cannot be the same in classical (non-relativistic) mechanics. ...
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How generalized and curvilinear coordinates are different?

I have read about Cartesian, polar, spherical polar and cylindrical coordinates. All these are generalized coordinates. But many a times they are written as general curvilinear coordinates. And I have ...
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Hamiltonian for a massless particle - formal definition of energy

Given a Lagrangian, is possible to calculate momenta and from them the Hamiltonian, if the system is regular enough. Today, I have realized that the Lagrangian of a massless particle in a ...
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Question about constrained mechanical system

I am reading the book "Classical Mechanics" by Douglas Gregory, and the author writes that using Newtonian equations for constrained systems runs into two difficuties. (1). The equations of motion ...
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Does a good path integral exist in Loop Quantum Gravity?

The Hamiltonian operator of Loop quantum gravity is a totally constraint system $$H = \int_\Sigma d^3x\ (N\mathcal{H}+N^a V_a+G)$$ Here, $\Sigma$ is a 3-dimensional hypersurface; a slice of ...
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How to find Hamiltonian from this simple Lagrangian? (tricky)

$$L~=~ \frac{1}{2} \dot{q} \sin^2{q} $$ Is it zero or not defined?
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Generalized coordinates

I am trying to understand generalized coordinates. When is it smart to use generalized coordinates? And what are some analytical examples that are too messy to answer without generalized ...
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A partial differential equation for a vector function and resulting constraint between its components

If we have an algebraic equation connecting 3-variables $x,y,z$, such as $x^2+y^2+z^2=2$, we can immediately conclude that all the 3 variabes are not independent. Now, consider the following Maxwell's ...
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Derivation of Euler-Lagrange equations from Hamilton's and D'Alembert's principle

Goldstein's book of Classical Mechanics derive the Euler-Lagrange equations from two different principles: Hamilton's principle states that $$\delta S = \delta\int_{t_1}^{t_2}L(q^{i},\dot{q}^{i},t)...
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Lagrange's Demon de-Conserves Angular Momentum

Monsieur Lagrange pulls a string down through a hole in a horizontal table thereby effecting a rotating (point) mass. A daemon sits on his shoulder and takes careful note of the proceedings. There is ...
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Work done by constraints on rotating rigid bodies

I am trying to understand why constraint forces do no work on extended, rotating bodies. For instance, consider the problem of a rigid rod falling on a frictionless surface (K&K 7.17) There are ...
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Lagrangian Mechanics, When to Use Lagrange Multipliers?

I've seen a few other threads on here inquiring about what is the point of Lagrange Multipliers, or the like. My main question though is, how can I tell by looking at a system in a problem that ...
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When does “solve and plugin” fail?

In Lagrangian mechanics, sometimes some symmetry is implicit, and we perform variation on the Lagrangian and find the conserved quantity, e.g. conservation of angular momentum in a central field. ...
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Why is the d'Alembert's Principle formulated in terms of virtual displacements rather than real displacements in time? [duplicate]

Why is the d'Alembert's Principle formulated in terms of virtual displacements rather than real displacements in time? EDIT In other words, which step of the derivation of D'Alembert's principle (or ...