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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“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 ...
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If I evaluate degree of freedom and got some number $n$, then how can I know what are those $n$ independent coordinates?

Using $3N-f=d$ we can evaluate the degree of freedom or independent coordinates of a system. But how can we know which coordinates are actually independent? (Here $n$ = number of particles, $f$ = ...
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Is this action a Hamilton's principal function?

In Ref. 1, in section 3, they wrote: \begin{equation} L'(q,\dot{q},\ddot{q},t)~=~L(q,\dot{q},\ddot{q},t)-\frac{dS(q,\dot{q},t)}{dt}.\tag{14} \end{equation} Then the Hamilton-Jacobi equation is \begin{...
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Are the degrees of freedom of a system decreased when the system is subjected to a non-holonomic constraint?

Are the degrees of freedom of a system decreased when the system is subjected to a non-holonomic constraint? I know when a system is subjected to a holonomic constraint then its degrees of freedom ...
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Can we think of holonomic constraints as constraints which are state functions?

While nonholonomic constraints are path functions. Because as holonomic constraints are those constraints which are the same for any system with all equivalent position coordinates, while nonholonomic ...
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Equations of motion for a free particle on a sphere

I derived the equations of motion for a particle constrained on the surface of a sphere Parametrizing the trajectory as a function of time through the usual $\theta$ and $\phi$ angles, these equations ...
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What does d'Alembert's principle actually tell us?

I have read in my book that d'Alembert's principle states: The reversed effective forces and the impressed forces at any point of a system are always in equilibrium. Now when I opt to study ...
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In 2D machines, why does higher pair joints deduct 1 D.O.F.?

I have been taught that higher pair joints (e.g. Gears, cams, rollers) deduct 1 d.o.f. Due to the fact that they still allow 2 motions translation along the tangent surface rotation around the ...
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Why does “Riemannian Geometry cease to play the role it played before” while working with rheonomic constraints?

I am reading Scleronomic and rheonomic systems: The law of conservation of energy in The Variational Principles of Mechanics by Cornelius Lanczos. This is the concerned excerpt: Rheonomic systems ...
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Is it always possible to define/find conjugate variables? And if yes how one can find it?

My question is in the context of both classical and quantum mechancis and field theory. Generally, how can one define/find the (canonically) conjugate of some variable/operator/field? Examples ...
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Why is the equation of constraint scleronomous when the wire moves “as a reaction to the bead's motion”?

So, I was reading Classical Mechanics by Goldstein where he just introduced scleronomous and rheonomous constraints. He writes: Constraints are further classified according to whether the ...
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Free body diagram of block on accelerating wedge

Consider the following system: I am thoroughly confused about certain aspects of the situation described in this diagram in which a block is placed on a wedge inclined at an angle θ. (Assume no ...
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Virtual displacement

Currently I am reading Classical Dynamics written by Donald Greenwood. I have a doubt in the section discussing about virtual displacement. As far as I understood, Virtual displacement (virtual or ...
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Lagrange multipliers - isothermal-isobaric ensemble

I know that the entropy of isothermal-isobaric ensemble is given by: $$S = -k \sum_{i=1}^M p_i \ln p_i \quad \textrm{where $p_i$ must be normalized} \quad \sum_{i=1}^M p_i = 1 \, .$$ The average ...
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Forces (Using Lagrange multipliers) of a fixed bicycle wheel

I'm having doubts about my Lagrangian when I release my constraints: I'm using Euler angles and using a system which is referenced to the wheel. It's quite straightforward to get the Lagrange ...
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Constraints in classical mechanics

I am self-studying classical mechanics and I have a couple of questions about constraints. Goldstein in his book Classical Mechanics writes in the beginning of Section 1.3 that: It is an overly ...
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Lagrangian exercise from Goldstein

Question 14 from the 1st chapter of H.Goldstein's book "Classical Mechanics": Q: Two points of mass $m$ are joined by a rigid weightless rod of length $l$, the center of which is constrained to ...
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Doubt on the consistency condition of the secondary constraint of electrodynamics

On the Dirac's method for the electromagnetism, demanding consistency on the secondary constraint $X$ (which should be identically achieved since there are no further constraints), $$ X_{\mathbf{x}}[\...
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Is this constraint holonomic or non-holonomic?

$$f(q,q^\prime, t) = 0, ~\mathrm df = \frac{\partial f}{\partial q}~\mathrm dq + \frac{\partial f}{\partial q^\prime}~\mathrm dq^\prime+ \frac{\partial f}{\partial t}~\mathrm dt = 0$$ I really want ...
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How to find a constraint equation to relate the two tension forces in this arrangement?

Consider this arrangement of pulley: The problem is to find the acceleration of $m_1$. This is my solution: The problem explicitly wants $\ddot y_1$. So starting by applying Newton's second law we ...
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Hamilton's principle and virtual work by constraint forces

I have a question about the following pages(pg 47 and 48) from Goldstein's "Classical Mechanics" I do not understand how (2.34) shows that the virtual work done by forces of constraint is zero. How ...
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Confusion about virtual displacements

I am self-studying Goldstein's book "Classical Mechanics", and I need some help understanding the part where Goldstein discusses using Hamilton's principle to solve systems with holonomic constraints (...
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What is difference between variations of the work and virtual work?

I really want to know whether or not both equations are the same mathematically. I think that they are the same, I just want to be sure. (Reference: this website.)
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Work done by internal forces of a rigid body

I am reading Goldstein's Classical Mechanics book, and I came across that: In a rigid body the internal forces do no work Is this statement based on the assumption that the internal forces are ...
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Question on the constraint force of a string

I would like to know if the following statement is true or false even if I expect that it is true. Notation: I will consider a string that has no mass and that can not be extended. Saying that a ...
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Virtual Work- Is the presentation in Cornelius Lanczos wrong?

Book: The Variational Principles of Mechanics by Cornelius Lanczos Edition: 4th Chapter: 3, The Principle of Virtual Work I am on the second page of the 3rd chapter (pg 75; it has the Eqn. 31.1). ...
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Why does the 'Jacobian of at least one combination of $n$ functions shall be different from zero'?

I've started reading The Variational Principles of Mechanics by Cornelius Lanczos; here is the concerned excerpt from p. 11: The generalized coordinates $q_1,q_2,\ldots, q_n$ may or may not have a ...

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