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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Write Equations of motion of a point particle that moves on the surface of a paraboloid [on hold]

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

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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Virtual Work and Arbitrary unit vectors

I was recently watching a lecture on MIT OCW where they were talking about Newton's Laws for a pulley system. The professor solved a problem pretty quickly using a method called virtual work. I tried ...
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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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147 views

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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35 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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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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Constraints in path integral and the Lagrange multiplier

I was reading some references on the slave-particle approach to the Kondo problem and Anderson model. It is known that the slave-particle is introduced in the large Hubbard $U$ limit of the system so ...
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What is Wedge Constraint? (Definition)

What is Wedge Constraint? How can we apply it in the following problem? The block of mass $m$ slides on a wedge of mass $m$ which is free to move on the horizontal ground. Find the accelerations ...
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In this pulley-block system, acceleration of blocks-A and B is same then how is their displacement different?

In the given arrangement, Block C begins to move down at a constant speed of $7\ \rm{cm/s} $ at time $t=0$. At the same instant, Block A is made to start moving down at constant acceleration. When it ...
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81 views

Constraint Forces - Motion of a bead on a wire

I am trying to understand the transition from Newtonian Mechanics to Lagrangian Mechanics. I have been looking at various examples of physical problems, starting from Newtonian Mechanics, and trying ...
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65 views

No explicit time-dependence in Lagrangian means constraints are explicitly time-independent?

Suppose a Lagrangian is not explicitly time-dependent. Does it mean that the constraint equations are also explicitly time-independent, and (as a result) the kinetic energy is necessarily a ...
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Why does the Hamiltonian of a Lagrangian that consists of only a coupled term become zero?

Let's say our Lagrangian looks something like this: $$L = \int dz\, Q\cdot \dot{A},\tag{1}$$ where $Q$ and $A$ are two generalized coordinates and $\dot{Q}$ and $\dot{A}$ would be the respective ...
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Radius of rain drop on ground

Just an observation I made this monsoon, When a raindrop falls on the ground its colour starts to fade, this may be due to the absorption of water by the ground. But gradually the radius of the water ...
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Constrained Motion [closed]

There are two bodies viz. A and B which are placed on a horizontal surface. The free end P is being pulled by a constant force of 1 N. Find the acceleration of free end P? My approach Sign ...
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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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1answer
58 views

Understanding the rolling constraint for one cylinder rolling inside another cylinder

This is the problem to find the equation of motion of 2 cylinders in which 1 cylinder is placed inside another cylinder with larger radius as shown in figure. The condition is that both are rolling. ...
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Constraining a Double Pendulum

My question is, how do I apply boundary constraints to a Lagrangian such as: $90 > \theta_1 > -45$ and $\theta_2 > 0$ I am trying to use a constrained double pendulum to simulate an ...
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Compute the Legendre transform for a singular Lagrangian

I'm given the lagrangian: $$ L(q,\dot{q}) = \frac{1}{2}(\dot{q_1}^2+\dot{q_2}^2+2\dot{q_1}\dot{q_2})-\frac{k}{2}(q_1^4+q_2^4). $$ I have to compute the Legendre transformation associated to it. The ...
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How to derive the Hamilton-Jacobi equation for the area of a minimal surface on a Riemannian manifold?

The action for a string in this background $$G_{IJ}\tag{1}$$ can be written as the Nambu-Goto action $$S_{NG}=\int d\sigma^1d\sigma^2\sqrt{g}\quad\quad\Rightarrow\quad\mathcal{L}=\sqrt{g}\tag{2}$$ ...
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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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Holonomic condition in book “The Variational Principles of Mechanics” by Lancroz

I have some difficulty in understanding the holonomic condition presented in Lancroz's book "The Variational Principles of Mechanics". The book has limited preview at google books which covers the ...
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1answer
63 views

Non-integrable differential equation and non-holonomic contraints

From the constraint $v=a\dot{\phi}$ of a rolling disk over a plane, where $a$ is the radius of the disk we can derive these two equations: we have two differential equations of constraint: $dx=asin\...
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Holonomic constraints and variables

Let's consider a holonomic constraint: $$f(q_{1},...,q_{n},t)=0$$ Must every term that compares in the equation be writable as a combination of the variables of a function? For example, in the ...
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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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1answer
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Trouble with grasping D'Alembert's Principle intuitively

In Newtonian mechanics we all learned that a mass accelerates when it is under the influence of a nonzero net force. However, I learned of D'Alembert's Principle today and it seems to oppose what I ...
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Lagrange Multiplier with inequality conditions

How can Lagrange multiplier method be used when inequality conditions are given instead of equality conditions?
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Constrained Curve in 3 Dimensions [closed]

I have a particle in a 3D space that moves on a curve of the function $$r(x)=\begin{bmatrix}x \\ x\sin(x) \\ \exp(x^2)\end{bmatrix}$$ I know that there must be 1 degree of freedom left thus $S = 3N-P$...
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Sign of Lagrange multiplier

Hello I have a short question. Say I would consider a pendulum and define the Lagrangian as usual being \begin{align} L = \frac{1}{2} m(\dot{x}^2 + \dot{y}^2) - mgy \color{red}{-} \lambda (x^2 + y^2 ...
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Lagrange Undetermined Multipler [closed]

Q1) Write down the Lagrangian of the system in terms of y(t) Q2) Obtain the Eqn of motion Q3)Using Lagrange Multiplier method find the forces of constraints 1) We have a constraint such that $$f=y-r\...
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32 views

Galilei group and Constrained QM

Let's assume spin-0 for simplicity. So far as I understand the issue, the Galilei simmetries constraints the possible hamiltonians of a quantum systems so that the only possible interactions of a ...
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1answer
63 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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What is the algorithm for finding the constraint force using the method of Lagrange multipliers? [duplicate]

Is there a general procedure one can follow to find the force of constraint for a classical holonomic system with the nonconstraint forces derivable from a potential energy $U\left(\mathbf{r}_1, \dots ...
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1answer
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Stress-Energy Tensor and Conformal Invariance in String Theory

Since the Euler-Lagrange Equations corresponding to the Polyakov Action implies no dependance on the auxillary metric we arrive at the constraint $T_{ab}=0$. We then change to lightcone coordinates $++...
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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
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Equation of constraint - Falling disc unrolling from an attached string

Where does the equation of constraint below come from? I've tried to rationalize it, but the angle will be 0 more than one time as the string unrolls, even though y will keep going down (right?), not ...
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What objective function is Lagrange's equation of the first kind based on?

In Lagrangian mechanics, Lagrange's equation of the first kind states that $$ \frac{\partial L}{\partial r_k} - \frac{d}{dt}\frac{\partial L}{\partial \dot{r_k}} + \sum_{i=1}^C \lambda_i \frac{\...
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1answer
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Ambiguous Constraint equation

This is from the solution manual of some problem of Kleppner's book. I didn't understand how the constraint equation came about to be. First of all, I don't see how that equation is equal to the ...
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4answers
156 views

How to resolve velocity components?

In the arrangement shown in the figure, the ends P and Q of an inextensible string move downwards with uniform speed $u$, pulleys A and B are fixed. With what speed does the mass M move upwards? My ...
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157 views

Non-relativistic limit of Hamiltonian for a free particle in general relativity

The Hamiltonian for a particle moving in a gravitational field can be taken as $$\mathcal{H} = \frac12 \sum_{\mu,\nu=0}^3g^{\mu\nu}(x)p_\mu p_\nu\tag{1}$$ as long as the parametrization is affine. ...
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Confusion about virtual displacement

From Goldstein: A virtual (infinitesimal) 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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How to determine whether a set of coordinates are independent and sufficient to determine the system completely?

In Analytical mechanics, when we formulate our principles, in general, it is assumed that we start with a cartesian coordinate system, and then find some generalised coordinates $q_j$s they are all ...
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Question about holonomic constraints

Goldstein says that when a system of $N$ particles is subject to $k$ holonomic constraints, the positions $\mathbf{r}_1, \dots, \mathbf{r}_N$ can be parameterized by $3N - k$ independent coordinates $...
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1answer
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Constraints and time derivative

Consider a system of $N$ particles. There are $C$ holonomic time independent constraints, $$ \begin{aligned} f_1(\mathbf{r}_1,\dots,\mathbf{r}_N) & =0 \\ f_2(\mathbf{r}_1,\dots,\mathbf{r}_N) & ...
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2answers
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Concerning Constraint Equations for Lagrangian Formalism

I was working on a problem studying for a classical mechanics class and came across an idea I'm not sure about concerning the formalism of Lagrangian mechanics concerning constraint problems. https://...