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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Problems based on constrained motion and energy conservation

Referring to qs no.47(Fig.20), we are asked to find the M/m ratio for the situation stated.Here's the solution to it- (This is actually a question from the renowned S.S.Krotov problems book!).The ...
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Lagrangian Mechanics - Bead sliding on a rotating rod

Say i have a bead of mass $m$ sliding on a friction-less rode (or wire) that is rotating with a permanent angular velocity $ω$. The whole system is under the influence of a uniform gravitational field ...
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Constructing Lagrangian from Hamiltonian for Majorana fermions

The text gives the Hamiltonian density as \begin{equation}{\cal H}=\frac{v}{2}\Big(\psi^\dagger\frac{\partial\psi^\dagger}{\partial x}-\psi\frac{\partial\psi}{\partial x}\Big)+\Delta\Psi^\dagger\Psi \...
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D'Alembers Principle - further explanation [duplicate]

In question : Why is the d'Alembert's Principle formulated in terms of virtual displacements rather than real displacements in time? there is a response : ...
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Can you help me understand how constraints are mathematically expressed? [duplicate]

Self studying Lagrangian mechanics using Goldstein. Holonomic constraints, an example being the distance between two particles of a rigid body, can be expressed as $(r_i - r_j)^2 - c_{ij}^2 =0$ and ...
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How can we know that changing variables to conjugate momenta is possible?

I am reviewing the derivation of Hamiltonian mechanics from Lagrangian mechanics, but I simply cannot understand how we can 'change variables' from $\dot q$ to $p$. Even on a very simple level, how ...
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Why is this a non-holonomic constraint?

Wikipedia states: holonomic constraints are relations between the position variables (and possibly time1) which can be expressed in the following form: $$f(q_{1},q_{2},q_{3},\ldots ,q_{n},t)=0$$ ...
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Deriving Euler-Lagrange equations for generalized coordinates without “virtual work”?

I have been reading "Classical mechanics" by Goldstein, Poole, and Safko. In particular, the section on "D'alembert's principle and lagrange's equations", in which the principle of virtual work is ...
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Hamiltonian of non-regular Lagrangian is well-defined on phase space

In section 1.1.3 of Quantization of Gauge Systems by Henneaux and Teitelboim, it is stated that the Hamiltonian $$H=\dot{q}^np_n-L,\tag{1.8}$$ although trivially a function of $q$ and $\dot{q}$, can ...
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Hamilton Constraint of the WdW equation

Can someone explain specifically what the surface term of the hamilton constraint in quantum cosmology actually describes and how it creates time even though we start with a timeless universe? And why ...
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Is Lagrange multiplier an arbitrary gauge?

A related post might be found here: Is the gauge transform field in electromagnetism a Lagrange multiplier? Consider the case of Lagrange multiplier $L=f(x)+\lambda g(x)$ where $g(x)=0$ was a ...
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Poisson bracket of momentum constraints in general relativity

I wish to compute the Poisson bracket of the momentum constraints in general relativity. Unfortunately, I am not able to do it correctly and the answer I am getting is not a linear combination of the ...
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Understanding the equations of motion for the Polyakov action in string theory

I want to make a small numerical simulation of how strings in theory move under their equations of motion but I'm getting stuck at implementing the constraints. The Polyakov action reads $$S=-\frac 1{...
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Example of a single constraint force doing virtual work despite the sum of work done by constraints being zero

When deriving d'Alembert's Principle it must be assumed, that the total virtual work done by constraint forces vanishes. $$\sum_{j=1}^N\mathbf{C}_j\cdot\delta \mathbf{r}_j=0.$$ In the books I've read, ...
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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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How to find an underlying holonomic constraints from a differential constraint?

I have been reading through "The Variational Principles of Mechanics" by Lanczos (if anyone is familiar with this text), and I am currently reading through the section discussing holonomic constraints....
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Find the time period for pulley spring system [closed]

What I tried to do was first consider this system at equilibrium, assuming the pulley to be a disc of radius $'r'$ and I calculated the extensions of the 2 springs as $e(=mg/k)$ for lower spring and $...
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Conversion of non-holonomic constraints to holonomic

In the case of a disc rolling without slipping, we have a constraint $\dot{x}=a\dot{\theta}$ where $a$ is the radius of the disc. Note that I have considered $x$ and $\theta$ as the generalized ...
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Constrained Hamiltonian dynamics [closed]

Suppose that I have a state $q$ that is constrained to satisfy $\mu(q)=0$. Assuming that $\nabla_q \mu(q)$ is a matrix of full-rank, I can use the Jacobian of the constraint function to define tangent ...
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Equations of motion for a certain constrained system

As an exercise in Lagrangian and Hamiltonian mechanics, I am looking at a system with the following Lagrangian: $$L=\dot R \cdot\dot R-\theta\dot R\cdot (SR)+\lambda(R\cdot R-1) $$ $R$ is a vector ...
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Performing Legendre transformation when conjugate momentum is independent of time derivative of generalised co-ordinates [duplicate]

Suppose there is a Lagrangean $L = \frac{1}{2} m c \dot x - \frac{1}{2}kx^2$ where $c$ is a constant to keep the dimensions right. The conjugate momentum is then $ p_x = \frac{\partial L} {\...
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Difference between finite and infinitesimal motion

I am studying Arnold Sommerfeld mechanics. Here they talk about finite and infinitesimal motion. Quoted from the text: The simplest example of a non-holonomic condition is furnished by a sharp ...
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Can constraint forces be parallel to virtual displacements?

It is often stated that virtual displacements must always be orthogonal to constraint forces. However, an example where this seems to fall apart is that of the Atwood machine, where the virtual ...
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D'Alembert's principle and the work done by constraint forces in Atwood's machine

From what I understand, constraint forces do no work because they are perpendicular to the allowed virtual displacements of the system. However, if you consider an unbalanced Atwood machine, in which ...
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Why is it important that there is no variation of time $\delta t=0$ in the definition of virtual displacement?

In Goldstein's Classical mechanics I found a proposition that I don't understand: Similarly, the arbitrary virtual displacement $\delta \mathbf{r}_i$ can be connected with the virtual displacement $...
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Determine force of particle along trajectory [closed]

The problem: I have a trajectory in cartesian plane defined by points. A particle with mass $m$ runs through this trajectory, with a defined initial velocity. At every moment, there is a force $F$ ...
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During BRST quantization of non Abelian gauge fields, is it necessary for the quantized particles to be transverse? [duplicate]

While covariantly quantizing non-Abelian gauge theories, we first impose the condition that the action of the BRST charge on physical states must yield zero. Then we further demand that such states ...
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Is a brachistochrone a straight line in curved space?

Please bear with me, and don't get upset if i have lack in knowledge about spacetime. Brachistochrone: Given two points A and B in a vertical plane, what is the curve traced out by a point acted ...
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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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Why the total virtual work done by forces from constraints vanishes? (Perpendicularity of two or more particles)

My mechanics book claims that the total force on the $i$-th particle is $$ F_i=K_i+Z_i \tag{2.5} $$where $Z_i$ is the force due to constraints and $K_i$ the real, dynamic force. Then, the book states ...
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Procedure for reducing the degrees of freedom of an arbitrary system

I'm trying to understand how it's possible to always reduce the degrees of freedom of a system represented by $n$ coordinates $q_{i}$ for which holds $f(q_{1},...,q_{n})=0.$ For example, let's ...
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Reducing degrees of freedom of a system described by an equation [duplicate]

Let's consider a system described by this simple equation: $x^{2}+y^{2}=5$ this represents a holonomic constraint. Suppose I would like to reduce the degrees of freedom of this system, I should be ...
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Covariant mechanics problem in Rovelli's 2.86

I'm trying to solve and understand a provided solution to a problem in Rovelli - Quantum Loop Gravity on page 56, eq. number (2.86). In the problem among other things I'm asked to compute the ...
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Misunderstanding normal force?

In class, we recently learned about banking angles and centripetal force. The teacher said that $N\cos\theta = mg$ after finding the vertical component of normal force and setting it equal to the ...
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Physical significance of virtual displacement

I have found many posts on this topic but I am not really able to understand what virtual displacement means in classical mechanics. Is there any simple way of explanation without going into rigorous ...
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How many degrees of freedom does a system have if we only eliminate some of the constraints?

At the very bottom of page 11 in his The Variational Principles of Mechanics, Lanczos says the following: We sometimes prefer to eliminate only some of the kinematical conditions, and to leave the ...
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What is mean by the statement that constraint forces do no work?

Consider a system consisting of two particles of a given mass, floating in space, aligned horizontally, constrained to remain at a fixed distance from each other by a massless thin rod (or string). If ...
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Can the Lagrange multiplier method be used with non-holonomic constraints?

The confusion for me comes from page 46 of Goldstein, where he says "However, it has been proven that no such varied path can be constructed unless [the differential equations of constraint] are ...
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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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Principled approach to arrive at geodesic Hamiltonian $H = g^{\mu \nu} p_\mu p_\nu$?

The background: If we have a spacetime path $x^\mu(t)$ parameterized by arbitrary parameter $t$, the proper time along the path between $t_1$ and $t_2$ is $$ \int_{t_1}^{t_2} (g_{\mu \nu} \dot x^\mu \...
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Write Equations of motion of a point particle that moves on the surface of a paraboloid [closed]

I'm not a physics major student and I call myself a total noob in physics, however, I have this issue to solve. The problem is that I don't even know where to start from. One little thing to add: this ...
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Infinity problem in basic classical mechanics problem [closed]

My question arises from a classical mechanics problem from a Hong Kong physics training programme: This is not a homework question as I am not asking about how to solve the problems in the image. As ...
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Conservation of total energy for a system with holonomic constraints

Consider a system with generalized coordinates $u_1, u_2$ and $u_3$ such that $u_1$ and $u_2$ are dependent through the following holonomic constraint \begin{equation} G(u_1, u_2)=0. \end{equation} It ...
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Particle sliding on a sphere with friction

This is a generalization of the question Particle sliding on a sphere when we also have friction given by $F_f = \mu N$. See the following figure: Before doing anything, we can imagine what friction ...
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Arnold's holonomic constraints being limits of potential energy

The following quote comes from Arnold's "Mathematical methods in mechanics" book: "We consider potential energy $U_N = Nq_2^2 + U_0(q_1, q_2) $, depending on parameter $N$ (which we will tend to ...
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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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Forces in a barbell bench press and other similar movements - what's really happening?

This should probably be solvable using Lagrangian mechanics, but I haven't learned that yet and so I would appreciate an explanation of what happens without referring to it, if possible, and ideally ...
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Some clarifications about ADM Hamiltonian constraints

I have some trouble with refreshing ADM split and Hamilton formalism of GR in context of introducing Wheeler-de-Witt equation. Having Lagrangian in form: $$\mathcal{L}_{ADM}=\sqrt{h}N(G^{abcd}K_{ab}...
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Does rigid body rotation always add a new independent variable?

I want to talk about the constrain added by introducing rotation of a rigid body to a simple case: An homogeneous ring at rest is dropped from height $H$ of an declined surface without any kind of ...
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What is the reasoning that leads one to postulate this second form for the relativistic particle action?

The action for the free relativistic particle with worldline $\gamma : I\subset \mathbb{R}\to M$ is $$S[\gamma]=-m\int d\lambda\sqrt{-\dot{\gamma}^a(\lambda)\dot{\gamma}_a(\lambda)}\tag{1} $$ Now, ...

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