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.

Filter by
Sorted by
Tagged with
1 vote
2 answers
43 views

How do I find the constraint relation in this question?

Let $\alpha_1$ and $\alpha_2$ be the angular acceleration of top pulley and bottom pulley respectively while $a$ is the acceleration of centre of mass of bottom pulley. Then, $$\alpha_2r+a=\alpha_1r$$ ...
user avatar
  • 181
9 votes
2 answers
295 views

Energy-Momentum tensor in the non-relativistic limit of Klein-Gordon Field

Assume we have a real Klein Gordon field $\phi(x,y,z,t)$, and we do the non-relativistic expansion of it in terms of a complex field $\psi(x,y,z,t)$ $$\phi=\frac{1}{\sqrt{2m}}(\psi e^{-imt}+\psi^* e^{...
user avatar
0 votes
0 answers
32 views

Lagrangian of a bead free to move on the spoke of a bike wheel

I am trying out this system for a final project in my Physics class. So the wheel is vertical as in a bike wheel and it is being driven so that the wheel is not a part of the system or the Lagrangian....
user avatar
1 vote
0 answers
30 views

Why is the anticommutation relation for the Dirac field between fields? [duplicate]

The commutation relation for neutral Klein Gordan field is $$[\phi(x,t),\pi(x',t)]=i\delta^3(x-x')$$ with all other commutators zero; The commutation relation for charged Klein Gordan field is $$[\phi(...
user avatar
1 vote
0 answers
34 views

Inequality constraint in Lagrangian

An example for an equality constraint: \begin{equation} x \geq x_a \end{equation} which can be used in the lagrangian: \begin{equation} \mathcal{L} = E(x) + \lambda(x-x_a) \end{equation} but ...
user avatar
  • 77
1 vote
1 answer
31 views

Can a primary constraint contain spatial derivative of the field?

I am currently studying the Hamiltonian formulation of GR and I have problems understanding this definition of primary constraint. In the textbooks, primary constraint occurs when a momentum conjugate ...
user avatar
  • 37
1 vote
1 answer
40 views

The constraint commute with Hamiltonian in Gauge theory

When canonical quantizing gauge theory, we find that the canonical momentum corresponding to $A_0$ vanish since the Lagrangian contains no $\dot{A_0}$ . Thus we need to choose a gauge, for example, $...
user avatar
0 votes
1 answer
44 views

Degrees of freedom for Constrained Motion

I'm starting to learn about Degrees of freedom, and the idea of 'constrained motion' seems strange to me, surely any particle with a predefined path is 'constrained' in its motion, We also had ...
user avatar
0 votes
2 answers
106 views

Newton vs Lagrange's equations for a variable length pendulum

Consider a pendulum with a variable string length $l=f(\theta)$. The Lagrangian is: $L = \frac{m}{2}(\dot{l} ^ 2 + l^2 \dot{\theta} ^ 2) + mgl\cos\theta$. Using Lagrange multipliers for the holonomic ...
user avatar
  • 1
0 votes
0 answers
42 views

Generating functional for a chiral superfield

In this paper by Petrov, he writes the action of the Wess-Zumino model for a chiral superfield $$S_J[\Phi,\bar\Phi,J,\bar J]=\int d^6z \Bigg(\frac{1}{2}\Phi\left(-\frac{\bar D^2}{4}\right)\bar\Phi+\...
user avatar
  • 1,954
-1 votes
1 answer
43 views

Can we take any physical quantity as a generalized co-ordinate in Lagrangian function?

Lagrangian function is a function of generalized co-ordinates $q_1,q_2,....$ & possibly of time $t$. i.e. $L=L(q_1,q_2,....;t)$ Consider a simple pendulum. Can I take $q_1$ = kinetic energy of ...
user avatar
  • 1,038
2 votes
2 answers
106 views

If quantum gravity is a TQFT, why isn't the Wheeler-De Witt equation satisfied automatically?

It is often said that QG is a topological QFT: given a bordism between $D$-manifolds $\Sigma_1$ and $\Sigma_2$, QG assigns a unitary between the Hilbert spaces associated with $\Sigma_1$ and $\Sigma_2$...
user avatar
1 vote
1 answer
40 views

What is the constraint acceleration equation of the system? [closed]

In the question there is a hint given, $y_A$ is not constant. From this hint I began to look at the pulley that holds $m_1$ and $m_2$, and can see that it has the same tension as $m_3$. From this ...
user avatar
1 vote
2 answers
49 views

Generalized forces of constraint

When using the method of Lagrange undetermined multipliers, it's assumed that the constraint generalized force, $Q_j$, is given by: $$Q_j=\lambda \cdot \frac{\partial f}{\partial q_{j}}$$ Where $f$ is ...
user avatar
0 votes
2 answers
49 views

Are these the two constraints in this system?

Consider a mass $m$ restricted to moving along a helix of fixed radius $b$ placed along the $z$ axis. Using cylindrical coordinates, also consider $z=a\varphi$. To me it seems pretty clear that the ...
user avatar
  • 185
0 votes
1 answer
56 views

Lagrange Multipliers

In this Lagrangian (from the paper: https://arxiv.org/abs/1302.0192 - page 4), $\eta, \mu, \nu, \& \lambda$ are lagrange multipliers. My question is: why do they include $\nu$ and $\lambda$ ...
user avatar
  • 77
2 votes
1 answer
77 views

Why is this hamiltonian not the energy? [duplicate]

Let a pendulum of length $\ell$ be connected to a rod that rotates with constant angular velocity $\omega$. $\theta$ is the angle of the pendulum wrt $z$ axis ($z$ axis is parallel to the rod). I ...
user avatar
  • 451
1 vote
1 answer
27 views

Constraint rate change problem [closed]

Two small rings O and O' are put on two vertical stationary rods AB and A'B' , respectively. One end of the inextensible threads tied at point A'. The thread passes through ring O' and it's other end ...
user avatar
0 votes
1 answer
39 views

Path of least action subject to an initial and a final conditions

I would like to find the path of least action subject to a initial and final condition. I don't know whether this is possible and meaningful at all, but here goes: Let us say we have a particle moving ...
user avatar
  • 1,226
3 votes
1 answer
78 views

What does Thornton and Marion mean by "validity of Lagrange's equations"?

I am a bit confused about the 2nd statement below from Thornton and Marion 7.4: It is important to realize that the validity of Lagrange's equations requires the following two conditions: The forces ...
user avatar
4 votes
1 answer
96 views

Why are $p$ and $q$ independent variables in Hamiltonian formalism?

Let's say we have $(q, \dot{q})$ as the generalised coordinate and generalised velocity. If we have a Lagrangian given by $$L=Aq\dot{q}+Bq$$ where $A$ and $B$ are constants that give the right units ...
user avatar
  • 3,803
2 votes
2 answers
64 views

Why doesnt the gravitional force exert a moment around the $z$-axis?

I just solved the problem below, but I have some questions regarding my solution. In my solution, I assumed that the total angular momentum around the $z$-axis was conserved. Which was shown to be ...
user avatar
  • 305
2 votes
2 answers
60 views

Time dependence of generalized coordinates and virtual displacement

The Cartesian coordinates of particles are related to the generalized coordinates via a transformation (for the $x$ component of the $j$-th particle) as: $$x_j = x_j(q_1, q_2, \ldots, q_N, t)$$ What I ...
user avatar
1 vote
3 answers
99 views

Why should degrees of freedom be independent?

To define the position of a system of $N$ particles in space, it is necessary to specify $N$ radius vectors, i.e. $3N$ co-ordinates. The number of independent quantities which must be specified in ...
user avatar
0 votes
0 answers
55 views

Deriving the equation of motion of a compound pendulum using d'Alembert's principle

After having derived the equation of motion of a simple pendulum using d'Alembert's principle, I'm trying to do the same with a compound pendulum, however I have stumbled upon a problem. Consider the ...
user avatar
  • 13
1 vote
1 answer
66 views

Troubles with signs using d'Alembert's principle

I am trying to find the equation of motion for the system below using d'Alembert's and virtual work principles, however I must be making a mistake since my equation doesn't match the solution that I ...
user avatar
  • 13
0 votes
1 answer
44 views

Why do we maximize according to the values of Lagrange multipliers?

In some Lagrangian problems, when we use the lagrange multipliers to minimize a function $f(x)$ they write: \begin{equation} \max_{\lambda,\mu} \min_{x}\mathcal{L} = \max_{\lambda,\mu}\min_{x} \Big( ...
user avatar
  • 45
3 votes
2 answers
345 views

What is the difference between constraint and boundary condition?

Sometimes, in Lagrangian mechanics, we need to identify the constraints of the problems and using the Lagrange multiplier technique, we got to the equilibrium equations of the problem under study and ...
user avatar
  • 45
0 votes
0 answers
33 views

Lagrangian: difference between 2 different Definitions for first kind

Let me introduce an exemplary system first for which the Lagrangian and equation of motions are already a given, a particle moving freely inside the surface of a cone pointing downwards and ...
user avatar
  • 267
9 votes
5 answers
1k views

Are constraint forces infinite?

A lot of authors claim that mechanical constraints are idealizations obtained by allowing enforcing forces to be infinite. But I either disagree or don't know what they mean. The only case where I ...
user avatar
  • 91
2 votes
0 answers
60 views

Why is the Poisson bracket of the YM Hamiltonian with the secondary constraint zero?

Suppose we have the following Hamiltonian density $$\mathcal{H} = \frac{1}{2}\pi_i^a \pi_i^a + \frac{1}{4}F_{ij}^a F_{ij}^a $$ where $$F_{ij}^a = \partial_i A_j^a - \partial_j A_i^a + gf^{abc}A_i^b ...
user avatar
2 votes
1 answer
68 views

Lagrange equations and moving constraints

I'm learning about Lagrange equations and was wondering if the lagrange equations would hold with moving constraints. I've searched a lot on the internet, but never came across my exact question. I ...
user avatar
2 votes
1 answer
68 views

Euler's Equations with auxiliary conditions - "why is $\frac{\delta y}{\delta \alpha}$ and $\frac{\delta z}{\delta \alpha}$ no longer independent?"

let $J(\alpha)$ be a functional of the parameter $\alpha $ such that: \begin{equation}J(\alpha) = \int_{x_1}^{x_2}f\{y,y',z,z';x\}dx \end{equation} and let \begin{equation}f = f\{y,y',z,z';x\} \end{...
user avatar
  • 227
1 vote
0 answers
123 views

Proving that the relative angular velocity of any particle with respect to any other particle is the same in a rigid body

Claim: The angular velocity of any point mass of a rigid body relative to any other point mass is the same, i.e., $\vec{\omega_{i,j}} = \vec{\omega}\;\,\forall{i}\,\forall{j}$, where $\vec{\omega}$ is ...
user avatar
2 votes
1 answer
68 views

What is a flow generated by a function in phase space?

I am reading the book "Quantum Field Theory" by Jean-Bernard Zuber and Claude Itzykson. I encounter great difficulties from page 457 section 9-3-1, which introduces Dirac's constrained ...
user avatar
0 votes
1 answer
56 views

Is there something like acceleration constraint in a rigid body? [closed]

Suppose we have two points on a body which having acceleration a1 and a2 direction we know and magnitude as well , is there some sort of relation between them we can establish , like for velocities ...
user avatar
  • 485
0 votes
0 answers
37 views

I don't understand this text about relativistic Liouville equation here

This slideshow here https://itp.uni-frankfurt.de/~hees/transport-meeting/ws12/marty.pdf I don't understand, because there are introduced time constraints and I don't know why these are there and what ...
user avatar
  • 3,203
0 votes
0 answers
27 views

Seeking examples of non-ideal constraints in analytical mechanics

What would be some examples of non-ideal constraints in classical mechanics? Here ideal constraints are by definition those for which the work done by the constraint forces under a virtual ...
user avatar
  • 101
1 vote
1 answer
36 views

Can Lagrange's equation be used if the virtual work done by constraint forces is not zero?

I'm learning analytical mechanics and was just introduced to d’Alembert’s principle, which I know is only valid when constraint forces' virtual work is zero. My question is, does this restriction also ...
user avatar
1 vote
1 answer
59 views

How to derive infinitesimal gauge transformations from constraints?

I am reading some papers about quantizing the gravitational fields, for example, here, here, and here. Since the classical actions for gravitational fields are singular, they contain some constraints. ...
user avatar
4 votes
1 answer
129 views

Can I write the Hamiltonian $H$ in the standard way $p\dot{q}-L$ for a general QFT?

I have read some questions (and the Wikipedia article) about the hamiltonian formulation of a QFT, but the only example that seems to be brought up is the scalar case, saying that $$\mathcal{H}_S=\Pi\...
user avatar
0 votes
1 answer
49 views

Confusion in constraint equation [closed]

I have to find the relation between $a$ and $b$ using constraint equation of the string. (Note: a and b are accelerations.) (Note: Friction is to be ignored, string and pulley is ideal and all ...
user avatar
  • 33
1 vote
1 answer
119 views

Triangulation of the Hamiltonian constraint in Loop quantum gravity

Im trying to obtain regularized (and triangulated) version of Hamiltonian constraint in the LQG. However, one step remains unclear to me. I am starting with the Euclidean Hamiltonian: $H_E=\frac{2}{\...
user avatar
1 vote
1 answer
36 views

Consistency of substitution of a canonical variable from EoM back into (momentum-less) action

I was reading this answer, where the issue of substituting equations of motion (eoms) into the action is addressed. I am fine with the basic idea that the action principle is destroyed when the eoms ...
user avatar
  • 53
1 vote
0 answers
37 views

Representation of Holonomic Constraints by independent generalized coordinates

Say we have a system with N particles described by N position vectors: $\{\vec{r_{i}}\};$ $i=1,...N$ Say we have a holonomic constraint: $$f(\{\vec{r_{i}}\},t)=0 \tag{1}$$ Since we have one holonomic ...
user avatar
0 votes
0 answers
23 views

When is Hamilton's principle valid? [duplicate]

When is Hamilton's principle$$\delta \int L d t=0$$ valid? Is it only valid for monogenic and holonomic systems? What about monogenic and non-holonomic systems? (I'm asking this because I got confused ...
user avatar
  • 688
1 vote
2 answers
183 views

Lagrange equations for non-holonomic monogenic system

For monogenic and a special case of non-holonomic constraints where we have$$ \sum_{k} a_{l k} d q_{k}+a_{t t} d t=0 \tag{2-20} $$ we use lagrange multipliers and hamiltons principle to reach the ...
user avatar
  • 688
4 votes
1 answer
72 views

Pulley system with constraints

Now I need to find out the acceleration of pulley A. I know that (acceleration of a) = 2(acceleration of c) and the accelerations of A and C are in opposite directions using constraints. The solution ...
user avatar
  • 111
0 votes
0 answers
33 views

Virtual displacement consistent with constraints or not

Are virtual displacement consistent with the constraints? It is claimed that (i) a virtual displacement δr is consistent with the forces and constraints imposed on the system at a given instant --1; ...
user avatar
  • 688
1 vote
0 answers
25 views

How to pick a constraint function for Lagrangian mechanics? [duplicate]

Motivating Example Consider a system which consists of two masses $m_1$ and $m_2$ at positions $x_1$ and $x_2$ respectively joined together by a rigid rod of negligible mass and length $l$ . We have ...
user avatar

1
2 3 4 5
11