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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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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30 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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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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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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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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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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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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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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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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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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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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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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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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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://...
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Confusion about valid set of virtual displacement

The answer of the problem below makes me confused about my thinking about virtual displacement... Consider the system in the figure below, similar to mechanisms used for aerial filming of sporting ...
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How many degrees of freedom in a massless $2$-form field?

Consider the Kalb-Ramond field $B_{\mu\nu}$ which is basically a massless $2$-form field with the Lagrangian $$ \mathcal L = \frac{1}{2}P_{\alpha\mu\nu}P^{\alpha\mu\nu}\,, $$ where $P_{\alpha\mu\nu} \...
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Meaning and Origin of an Expression which Involves Virtual Displacement

As an additional point of confusion related to the answer given here: Confusion with Virtual Displacement I have encountered the following expression in my study of virtual displacements. $$\delta{...
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Confusion with Virtual Displacement

I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs ...
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How to deal with no-slip non-holonomic constraints in Lagrangian?

I'm solving a dynamical system of a ball of mass $m$ and radius $R$ rolling on a rotating platform ("turntable") for which I found the Lagrangian to be: $$L=\frac{1}{2} m (\dot{x} - \Omega y)^2 + \...
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D'Alembert's principle and equation of motion

Is obtaining proper equation of motion from D'Alembert's principle a mere coincidence or there is some logic behind this? This is asked because while we are finding the equations in a regime where ...
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Block sliding down hemisphere problem (harder) (need pleb Newtonian explanation)

So this is from a pretty recent (freshman-level) physics university exam. Part (a) is relatively simple and standard for a freshman physics course. What I am finding much more difficult is part (b). ...
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Constrained Hamiltonian systems: spin 1/2 particle

I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action \begin{equation} S=-m\int ...
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Geodesic equations from action with auxiliary field

A textbook says that the geodesic equations (for both massive and massless) can be derived from the following action: $$ S = -\frac{1}{2} \int d\tau \:\eta \: (\eta^{-2} \dot{x}^\mu \dot{x}^\nu g_{\...
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Hamiltonian for relativistic free particle is zero

One possible Lagrangian for a point particle moving in (possibly curved) spacetime is $$L = -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu},$$ where a dot is a derivative with respect to a parameter $\...
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Does Noether's theorem apply to constrained system?

The Lagrangian of a constrained system will be $$L-\lambda_1f_1-\lambda_2f_2-...\lambda_kf_k.$$ If a transformation will not affect the constrained Lagrangian, the there is some corresponding ...
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If two surfaces slip on each other then their velocities along the common normal are equal?

If two surfaces slip on each other then their velocities along the common normal are equal?
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Rigorous definition of generalized coordinates

In Goldstein's classical mechanics and in many other books I haven't seen a rigorous definition of generalized coordinates. In a system of $N$ particles described by $\textbf{r}_1, \dots, \textbf{r}...
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111 views

Dirac Brackets in General Relativity

I want to calculate Dirac brackets of different phase space variables in gravity. In case of electrodynamics, one does the same using the following steps: Looking at the momenta to find that $\Pi^...
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1answer
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Constraints in general relativity

In this review on inflation, on Pg. 135, Baumann talks about the energy and the momentum constraints for gravity. Are these equations the $G_{00} = T_{00}$ and $G_{0i} = T_{0i}$ components of the ...
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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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When the constraints are not holonomic, why is it not possible to find such $q_i$s that $\delta q_i$s are independent of each other?

In the book of Classical Mechanics by Goldstein, at page 20, it is given that However, I cannoot understand from what has been presented so far that when the constraints are not holonomics, why is it ...