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.
639
questions
3
votes
1
answer
60
views
Constraints Generating Gauge Transformations and BRST
Given a gauge-invariant point particle action with first class primary constraints $\phi_a$ of the form
$$S = \int d \tau[p_I \dot{q}^I - u^a \phi_a]$$
we know immediately, since first class 'primary' ...
- 3,967
2
votes
1
answer
87
views
Primary constraint of electrodynamics
I have some problems understanding the transition from the Lagrangian to Hamiltonian formalism of electrodynamics. I will use the metric $(-+++)$.
I want to start from the Lagrangian which is ...
5
votes
1
answer
132
views
+50
Understanding a supersymmetric quantum mechanical gauge theory model
I'm studying this paper on supersymmetric ground state wavefunctions. In section 5 "quantum mechanical gauge theories", it says:
"We begin with the ${\cal N} = 2$ gauge theory which ...
- 321
2
votes
0
answers
86
views
A Question of Pseudo-Classical Mechanics of Grassmann-Odd Variables [duplicate]
I would like to understand the following problem:
You have a classical fermion in one dimension. It has no mass, and no interactions. One can write its action as follows:
$$S=\int_{\mathbb{R}}dtL(\...
- 2,773
0
votes
1
answer
188
views
A Question of Pseudo-Classical Mechanics of Grassmann-Odd Variables
I would like to understand the following problem:
You have a classical fermion in one dimension. It has no mass, and no interactions. One can write its action as follows:
$$S=\int_{\mathbb{R}}dtL(\...
- 2,773
2
votes
1
answer
107
views
Similarity between holonomic/nonholonomic constraints and state/path variables in thermodynamics
I have recently learned about holonomic and nonholonomic constraints in analytical mechanics, and how they can be expressed as exact (Pfaffian form); for example:
$${\displaystyle df_{i}=\sum _{j}\ A_{...
- 104
0
votes
2
answers
154
views
Constraint equation for an elastic pendulum
I would like to know if you can help me determine the restraining force for an elastic pendulum. The problem is the following
A particle of mass $m$ is suspended by a massless spring of length $L$. ...
- 115
1
vote
1
answer
66
views
Hamiltonian analysis of relational $N$-Particle Dynamics
I am following "A Shape Dynamics Tutorial, Flavio Mercati" (https://arxiv.org/abs/1409.0105), and have problems understanding the hamiltonian formulation of $N$-particle dynamics as sketched ...
- 473
0
votes
0
answers
78
views
Secondary constraint imposed gauge fields
I am asking about aspects of quantizing electromagnetic field followed in Weinberg's Quantum theory of fields Volume I, in section 8.2 . I am able to understand the primary constraint that arises from ...
2
votes
2
answers
101
views
Help in understanding this derivation of Lagrange Equations in Non-Holonomic case
Whittaker, Analytical dynamics pg 215
I don't understand how we get the final equations relating $Q_r$ with $\lambda$ given the conditions above?
- 1,200
2
votes
1
answer
78
views
Independence of generalized coordinates in the derivation of Lagrange equations from d'Alembert's Principle
I am confused by this remark in the derivation of Lagrange equations from d'Alembert's principle in Goldstein:
I am not comfortable that I understand why, at this late stage of the derivation, they ...
- 31
0
votes
1
answer
55
views
Lagrange multipliers method for a bead on a parabola-shaped wire [closed]
I have this simulation problem that looks very easy at first look (and maybe it is, especially using Lagrange equations of the 2nd. kind). I am using the Lagrange multipliers and I am kinda not sure ...
- 69
0
votes
1
answer
58
views
Virtual work of constraints in Hamilton‘s principle
Goldstein 2ed pg 36
So in the case of holonomic constraints we can move back and forth between Hamilton's principle and Lagrange equations given as $$\frac{d}{d t}\left(\frac{\partial L}{\partial \...
- 1,200
1
vote
0
answers
24
views
Hamiltonian of the relativistic point particle [duplicate]
It is unclear to me why the Hamiltonian of a relativistic particle is zero. I know that, given a relativistic free particle, I can write the Lagrangian of it in this way: $$\mathcal{L} = mc^2\sqrt{1 - ...
- 11
1
vote
2
answers
108
views
How does Poisson bracket vanish in this equation?
I am studying the book "Lectures on Quantum Field Theory", by Ashok Das. I am stuck in the last step of equation (10.35) as I will explain below.
Under section 10.2 (Dirac method and Dirac ...
- 331
0
votes
0
answers
45
views
Constraint force using Lagrangian Multipliers
Consider the following setup
where the bead can glide along the rod without friction, and the rod rotates with a constant angular velocity $\omega$, and we want to find the constraint force using ...
- 183
1
vote
0
answers
51
views
Intuition behind energy not being conserved in Rheonomous mechanical system [closed]
firstly, this is what Rheonomous System means. So, in such a system, the kinetic energy is not exactly just a quadratic function of generalized velocities because one of the generalized coordinates ...
- 191
0
votes
1
answer
53
views
Why are constraint forces and gradient of constraint functions perpendicular?
My question is about the general relationship between the constraint functions and the constraint forces, but I found it easier to explain my problem over the example of a double pendulum:
Consider a ...
- 183
3
votes
4
answers
175
views
Rigid bodies: proof of existence of internal forces that preserve the distances [duplicate]
I am new to Physics and I have a pure Math background. I am currently studying mechanics and I have the following question regarding rigid bodies. I am posting here the 2D version of the question. If ...
- 147
1
vote
1
answer
52
views
Requirement of Holonomic Constraints for Deriving Lagrange Equations
While deriving the Lagrange equations from d'Alembert's principle, we get from $$\displaystyle\sum_i(m\ddot x_i-F_i)\delta x_i=0\tag{1}$$ to $$\displaystyle\sum_k (\frac {\partial\mathcal L}{\partial\ ...
- 183
5
votes
2
answers
223
views
Why can't we "simply" quantize Maxwell's equations without a Lagrangian to create a quantum theory of electrodynamics?
Useful quantum field theories like quantum electrodynamics (QED) suffer from a litany of problems related to the fact that, at least in their usual Lagrangian formulation, interactions between the ...
- 19.6k
5
votes
2
answers
258
views
Dirac procedure for Wheeler De Witt equation
After computing the Hamiltonian constraint and the momentum constraint in general relativity the Hamiltonian constraint is turned into an operator equation and solved in a manner similar to a ...
- 898
0
votes
1
answer
66
views
Classical Mechanics proof Lagrangian constraint forces
I've got a simple mathematical question. I was studying the Lagrangian approach of classical mechanics and in this part I had the intention of proving that the differential of the Lagrangian is equal ...
2
votes
1
answer
62
views
Derivation of the Equation for Constraint Forces
How do we derive the relationship $$F_\text{constraint}=\displaystyle\sum_i \lambda_i\nabla g_i$$ where $g$ is the constraint function, from the following relationship? $$\frac d {d\epsilon}S|_{\...
- 183
3
votes
3
answers
134
views
Is there a proof that a constraint force between two particles must be parallel to the vector joining them?
I'm in the process of self-studying (classical) analytical mechanics, and in various textbooks I have come across the assumption that if two particles are constrained to remain at a fixed distance ...
0
votes
0
answers
20
views
How to calculate angular velocity from forces on spherical pendulum head?
I have a bunch of forces that add up the force vector $(a_x, a_y, a_z)$ which is applied to the head of a spherical pendulum with given angles and angular velocities $ (\phi, \theta, \phi', \theta') $ ...
- 235
0
votes
1
answer
51
views
How to universally validate the method of virtual work for all pulley systems?
How can it be proven that the method of virtual work for pulley constraints is true for all pulley systems, including complex pulley systems (such as block and tackle)?
Method of virtual work : The ...
- 53
0
votes
0
answers
65
views
I’m not sure if this mathematical formulation for normal force is correct
Consider an arbitrarily defined surface
$$f(x,y)\in\mathbb{R}^3$$
The normal vector to any given point on the graph of f is given by the gradient of f. Since the normal force at a point on the surface ...
- 101
3
votes
1
answer
71
views
Derivation of constrained Euler-Lagrange equation
I've seen the formula for a constrained Lagrangian several times and while its somewhat intuitive, coming up with a rigorous proof has stumped me. I think I might misunderstand part of it and hope to ...
- 99
3
votes
2
answers
140
views
Is the magnetic Lorentz force $\vec{F} = q(\vec{v}\times\vec{B})$ a force of constraint?
I am currently studying the Lagrangian mechanics, and as far as I've understood, forces of constraint are the forces that are perpendicular to the surface of the movement of the object, thus do not ...
- 183
1
vote
1
answer
142
views
Forces of constraint and Lagrangian in a half Atwood Machine with a real pulley
I was thinking about this problem and had some trouble about the constraint equation.It's just a pulley with mass and moment of inercia $I$ that is atached to two blocks, just like in the picture. And ...
- 95
6
votes
1
answer
250
views
Geometric interpretation of constrained dynamical systems
Below are two pictures from Bojowald's book Canonical Gravity. The author tries to present a geometrical picture of a constrained system, however, the description regarding this seems quite scant to ...
- 616
1
vote
1
answer
56
views
An explicit form for the co-BRST operator?
Take a theory with 1st class constraints $M_{\alpha}$. We gave ghosts $c^\alpha$ and their conjugates $b_\alpha$ for every constraint. The BRST operator $\Omega$ has ghost number $+1$ and has an ...
- 732
2
votes
2
answers
73
views
Does the following limit exist in the BRST formalism?
Consider the BRST operator $\Omega$ (which has ghost number $+1$) and the gauge fermion operator $\rho$ which has ghost number $-1$. Given an exact state $|\Phi\rangle$ (i.e. $|\Phi\rangle=\Omega|\Psi\...
- 732
2
votes
2
answers
138
views
Square of BRST operator
The BRST operator $\Omega$ can be expanded in powers of the ghost fields $c^{\alpha}$ and their conjugates $b_{\alpha}$ (which satisfy $\{c^\alpha,b_\beta\}=\delta^{\alpha}_{\beta}$):
$$
\Omega=c^{\...
- 732
3
votes
1
answer
107
views
Massive vs. massless relativistic point particle in einbein form: Difference in the gauge structure?
The action for the relativistic point particle with mass $m \geq 0$ in a curved background is given by:
\begin{equation}
S[X] = \int_{\tau_0}^{\tau_1} d\tau \left[ e(\tau)^{-1} g_{\mu \nu}(X(\tau)) \...
- 374
1
vote
0
answers
113
views
Why introduce Lagrange multipliers? [duplicate]
For a non-relativistic particle of mass $m$ with a conservative force with potential $U$ acting on the particle and a holonomic constraint given by $f(\mathbf{r},t)=0$, the system can be incorporated ...
- 1,017
0
votes
0
answers
24
views
Simple Lagrangian with free bound and constraint
Let $\alpha,\beta$ non-zero real numbers, $f$ a function of time. I define $L_1=\alpha f + \beta$ and $L_2=p(t) L_1$.
I want to minimize $\int_0^T L_2$ under the constraint $\int_0^T L_1=v$, with $T$ ...
2
votes
1
answer
111
views
Why is it ok to set the dilaton to its classical value in Jackiw-Teitelboim (JT) gravity?
I am reading arXiv:1606.01857 (Maldacena-Stanford-Yang), one of the main papers on Jackiw-Teitelboim (JT) gravity. To derive the Schwarzian action, they use the classical solution for the dilaton, eq....
- 48
1
vote
0
answers
47
views
Conceptual problem with incorporating constraints to a particular variational principle problem
Consider the following problem:
A vector field $\boldsymbol{F}(x)$ is defined over a finite region $V$. A functional of the form
\begin{equation}
U = \int_V u(\boldsymbol{F})\ d^3x
\end{equation}
is ...
- 85
3
votes
1
answer
105
views
Barut-Zanghi (BZ) Lagrangian derivation
I'm trying to derive the BZ Lagrangian (density) from the Dirac Hamiltonian density and some questions popped up.
BZ Lagrangian is
$$\mathcal{L} = \frac{i}{2}(\dot{\bar{\psi}}\psi - \bar{\psi}\dot{\...
- 133
2
votes
2
answers
93
views
Constraint forces in Lagrangian mechanics
Some Background
I was reading up on some elementary Lagrangian mechanics from David Morin's Classical Mechanics while also going through Goldstein's Classical Mechanics for further clarification. ...
- 63
1
vote
2
answers
85
views
Constraint forces and the Lagrangian [closed]
Consider a scenario where a particle is placed on the surface of a dome-shaped object (a hemisphere) and moves on the surface of the dome due to gravity. My question was, how do I find the equation of ...
- 63
1
vote
1
answer
40
views
Lagrangian energy equation with a nonholonomic constraint?
Problem 6.8 on p. 39 in David Morin's The Lagrangian Method gives a stick pivoted at the origin and rotating around the pivot with constant angular velocity $\dot{\alpha}$ (which is given as $\omega$ ...
- 51
0
votes
0
answers
71
views
Variation vs. derivative wrt a symmetric and traceless tensor
Consider a Lagrangian, $L$, which is a function of, as well as other fields $\psi_i$, a traceless and symmetric tensor denoted by $f^{uv}$, so that $L=L(f^{uv})$, the associated action is $\int L(f^{...
- 11
3
votes
1
answer
71
views
Confused about the relation between BRST invariant states and 'group averaging'
https://arxiv.org/abs/hep-th/0111270 by O.Y. Shvedov tries to relate the 'group averaging procedure' (see https://ncatlab.org/nlab/show/group+averaging and references therein) to the BRST formalism so ...
- 732
0
votes
1
answer
164
views
What diffeomorphism does the Hamiltonian constraint generate?
Consider the Hamiltonian constraint $\mathcal H(x)$ in the ADM formalism. What diffeomorphism does this generate?
- 732
2
votes
4
answers
255
views
Why do we put factors of zero in a Lagrangian that is to be extremized?
According to the Wikipedia page on Lagrange multipliers under the section - Example 3: Entropy, it is written that:
$$f(p_1,p_2,\ldots,p_n) = -\sum_{j=1}^n p_j\log_2 p_j$$
For this to be a ...
- 620
0
votes
0
answers
48
views
Why does thermal equilibrium between two systems $A$ and $C$ imply the constraint $f_{AC}(A1, A2, · · · ; C1, C2, · · ·) = 0$?
I am trying to understand the mathematical formulation of the zeroth law of thermodynamics.
I get that if a system A is in equilibrium with a system C and a system B is in equilibrium with the same ...
- 65
0
votes
3
answers
102
views
Been confused about string constrained motion for a while
Let's have a look at the following set up:
Suppose Block C is moving downwards with a velocity "u" and we are required to find the velocity of bead "v" . So what we were taught in ...
