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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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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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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The consistency conditions of constrained Hamiltonian systems

I am studying the Hamiltonian description of a constrained system. There are some questions puzzled me for days, which I have been stuck on it. From the lagrangian, we can obtain the primary ...
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235 views

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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Intuition behind the principle of virtual work

To derive Lagrange's Equations we need the principle of virtual work first. This principle states that whenever a system of $K$ particles is constrained to a submanifold $\mathcal{M}\subset \mathbb{R}^...
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Has the conjecture of Guillemin-Sternberg been proven for relevant physics cases?

From a working physicist's perspective, the conjecture of Guillemin-Sternberg (and its generalisations) seems to state in a highly technical manner that quantization commutes with gauge-fixing. In ...
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A question about the constraints in BRST-Fock theories

In BRST Symmetry in the Classical and Quantum Theories of Gauge Systems, Henneaux says the Fock representation is not applicable to an odd number of constraints. Then he goes on to say that the Kugo-...
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242 views

Lagrangian with vanishing conjugate momentum, independent variables

Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes: $$\pi_n=\frac{\partial\...
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Constraint equation for Einstein-Hilbert action in Light-cone gauge

In Light cone coordinate system $(+,-,i)$, where $i=1,2$, the light cone coordinates are defined as $x^{\pm}=\frac{x^0 \pm x^3}{\sqrt{2}}$, if we consider the $+$ coordinate to be our "timelike" ...
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418 views

How is the method of Lagrange multipliers used for multiple constraints of multiple variables?

Let's say for example that I have two constraints $f(x,\dot{x},y,\dot{y})$ and $g(x,\dot{x},y,\dot{y})$ and a Lagrangian $L(x,\dot{x},y,\dot{y})$. What are the Euler-Lagrange equations of the first ...
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Inconsistency? Lagrangian with its Euler–Lagrange equation as condition

Consider the action $$A_{1} = \int{L(q, \dot{q})}{dt}\tag{1}$$ and the corresponding Euler–Lagrange equation $$\frac{\partial{L}}{\partial{q}} - \frac{d}{dt}\left(\frac{\partial{L}}{\partial{\dot{q}...
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Lapse and shift inside or outside the Poisson bracket?

For general relativity in the 3+1 ADM formulation, one has $H=\int dx [N{\cal H}+N^a{\cal H}_a]$ with $N$ and $N^a$ the lapse and shift which are undetermined Lagrange multipliers. The dynamical ...
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Constraint forces do no virtual work: does this always apply?

The thread title is my main question, but to give some context, I'll include a particular example that made me ask the question in the first place. In Hand and Finch, a small block is on a ...
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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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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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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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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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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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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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47 views

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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331 views

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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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 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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285 views

Calculation of the Poisson bracket of a (Classical) Yang-Mills generator

This question might be too technical or minute, but I believe someone can give me the right advise. What I want to calculate is a Poisson bracket algebra of classical YM gauge generators, \begin{...
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1answer
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Why friction force is force of constraint?

My understanding about constraint force is that it is a force which limits the geometry of particle's motion. For example, situations such as the particle trapped in a track or limited in domain can ...
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How to solve this one-sided constraint problem?

As shown in the figure below, ball 1 and ball 2 are connected by a rigid rod without mass. The wall and floor are absolutely smooth, and the mass of ball 1 and ball 2 are all $m$. The other geometric ...
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Automatically embedded in Lagrangian formulation

Since bond forces are automatically embedded in Lagrangian formulation, while in Newton they are not, and you have to operate directly with them [you have freedom from generalized coordinates]. That's ...
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Is there a “quick” way to visualize constrainsts on elementary classical Lagrangian Mechanics?

The very basic ideias of Lagrangian Mechanics with some introduction to Variational Calculus isn't a great big deal to understand, I mean, aren't so difficult to understand the point of view of ...
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1answer
75 views

Non-holonomic constraints, degree of freedom and generalized coordinates

If a system has $N$ coordinates and $M$ number of holonomic constraints then number of degree of freedom $=N-M$ and generalized coordinates $=N-M$ too. But if there are $k$ non-holonomic constraints ...
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1answer
48 views

On the use of Lagrange multipliers in deriving the Lagrange eqn. in classical mechanics

Can one derive the Lagrange eqn based on the methods of Lagrange multipliers? That is, we need to minimize the action with respect to the trajectory keeping the net energy of the body in motion ...
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1answer
65 views

Lack of Constraint equations

I was trying to find how a uniform string of length $L$ fixed at a point (I assumed $(0,0)$) bends under gravity. I tried to minimise the potential energy within the constraint of the length of the ...
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Insufficiency of Newton's third law to solve constrained motion problems

In The Variational Principles of Mechanics Lanczos describes what he calls 'vectorial mechanics': the process of solving mechanical problems by recourse to the immediate consequences of Newton's laws, ...
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55 views

Field degrees of freedom from equations of motion and higher spin

It is my understanding that we compute the number of degrees of freedom of a quantum field as the number of its components minus the number of non trivial equations we get by taking the divergence of ...
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1answer
195 views

Physical Constraints

In physics, what does one mathematically mean by constraint in classical mechanics? What are the the different types/cases and how do people deal with them?
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174 views

Ambiguity in d'Alembert's principle

It seems to me that many different momenta $\dot{\bf p}_j $ can satisfy d'Alembert's principle: $$\tag{1} \sum_{j=1}^N ( {\bf F}_j^{(a)} - \dot{\bf p}_j ) \cdot \delta {\bf r}_j~=~0 $$ in a ...
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How to derive the Hamiltonian of general relativity (ADM formalism without surface terms)?

Given that $$ds^{2} =−N^{2}dt^{2} +h_{ij}(dx^{i} +N^{i}dt)(dx^{j}+N^{j}dt)$$ $$S=\int dt d^{3} x\sqrt{h} N(^{3}R+K_{ij}+K^{ij}-K^{2})$$ where $^{3}R$ is the Ricci scalar of $hij$, $h$ the ...
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Variation with respect to a traceless symmetric tensor

Suppose we have an action variation like $$\delta S[G]=\int \mathfrak{H}^{\mu\nu}\delta G_{\mu\nu} \,\, d^Nx,$$ where $\mathfrak{H}^{\mu\nu}$ is a tensor density. If the variation with respect to $...
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338 views

Proof of holonomic constraints for a wheel on a track

I'm faceing a problem of a thin wheel of radius R rolling without slipping on a track (y = f(x); on xy-plan). The wheel plane stays vertical and tangent to the track at the contact point P. $\alpha$ ...
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108 views

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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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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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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“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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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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2answers
700 views

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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2answers
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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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1answer
280 views

Membrane Theory

I'm not a physicist or educated mathematician, so please excuse me if my question is scientifically rudimentary. It concerns Membrane Theory. If all open strings are attached to the surface of the D ...
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Non-null hessian condition for regular dynamical systems

I'm "researching" on unquantised Yang-Mills theory. For that I'm studying the Dirac's method for singular constrained systems and having problems to follow the first considerations on that matter. I ...
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562 views

Finding the constraint equation

I am trying to solve a problem on Constraint equations for a triple pendulum model, but was not able to derive a constraint equation for the last mass. I solved constraint equations for Masses 1 and ...
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When the equations of motion are not unique (eg. when they are given by eigenvectors), which will the free particle adhere to?

For this question I think it will be easier to express the usual equation describing the motion of a "free particle,"--viz. $g_{ij}\dot{x}^i\dot{x}^j$--in matrix form as follows: $$g_{ij}\dot{x}^i\...