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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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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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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
29 views

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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1answer
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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
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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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1answer
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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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1answer
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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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4answers
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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://...
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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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1answer
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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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2answers
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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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1answer
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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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1answer
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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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2answers
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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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5answers
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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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3answers
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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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1answer
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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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1answer
106 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 ...
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1answer
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D'Alembert's principle when the mass of the particles are changing

In the book of Classical Mechanics by Goldstein, at page 19, while deriving D'Alembert's principle, the author assumes that $$\dot p = m \ddot r.$$ However, when the mass of the bodies also changes, ...
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2answers
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Finding the value of the holonomic constraint forces

So let's say I have a Lagrangian augmented with some holonomic constraints. $$L' = L + \sum_i \lambda_i(t) f_i(q,t).\tag{i}$$ The solutions is the system of differential equations: $$\frac{\partial ...
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2answers
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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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0answers
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Going downhill - what's the constraint force? [closed]

We are going downhill on a path $$y=a(1-x^2),$$ and I need to calculate the constraint force-position function. What I've done is this: The lagrangian of the system is $$L=\frac{1}{2}(\dot{x}^2+\dot{...
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0answers
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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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1answer
54 views

Restriction Forces in Lagrangian Mechanics

I was recently preparing for a test on Classical Mechanics and a friend of mine started wondering if there was any method through which we could obtain the restriction forces acting on a certain ...
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0answers
62 views

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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1answer
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Deriving the geodesic equation using a Lagrange multiplier to fix affine parametrisation

The geodesic equation can be derived using the action $$S_0 ~=~ \int d\tau \sqrt{-\dot{x}_\mu\cdot \dot{x}^\mu}\tag{1}$$ (I am using the (-+++) convention and $\dot{x} = \frac{dx}{d\tau}$). To ...
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1answer
79 views

Polyakov Lagrangian and Lagrange multipliers

I'm reading Polchinski's Introduction to String Theory (volume I) and something got me quite puzzled in the beginning (At the top of page 19 to be precise). This part is about the open string and the ...
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1answer
171 views

Lagrangian of point mass on rod inside ring with holonomic constraint

I am given a massless ring of radius $R$ that is rolling along a flat plane without slipping. There is friction. A massless rod of length $\frac {R} {2} $ is attached to the inside edge of the ring ...
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1answer
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Hamiltonian Structure of Chern Simons Electrodynamics

I am reading the review paper "Aspects of Chern-Simons Theory" by Gerald Dunne https://arxiv.org/abs/hep-th/9902115 Starting from p. 17, Dunne works on the Hamiltonian structure of the CS ...
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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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1answer
114 views

Intuition for free-particle action principle?

Consider a particle constrained to a manifold $\mathbb Q$ embedded within standard Euclidean 3-dimensional space that experiences no forces other than constraint forces keeping it within $\mathbb Q$. ...