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.
736 questions
1
vote
1
answer
113
views
Constraints in Dirac's Lectures
Currently, I am reading section 3 of a 1950 lecture/paper (PDF) by Dirac, about general hamiltonians and dynamics in the formalism. He defines
$$H= \mathfrak{H(q,p)},\tag{7}$$
weakly (as in only holds ...
- 125
2
votes
1
answer
202
views
Equation of motion of a point sliding down a parabola [closed]
I have a frictionless parabola $ (t,t^2) $ on the $x,y$ plane. I was having difficulties deriving the equations of motion for a point P placed at a height h on the parabola and let go of without any ...
- 21
6
votes
0
answers
199
views
How to deal with Equation of Motion with singularities?
I was trying to describe the movement of a ball rolling on bowl.
The degrees of freedom of the system are the following:
The Position of the Center Of Mass (where $r$ is the distance from the origin ...
- 2,069
0
votes
2
answers
129
views
Why the election of the reference frame make constraints appear or not?
Maybe this is a dumb question, but imagine we have the following system:
If we work on the red reference frame (the inclined plane frame, denoted by $I$), this system is easily solvable:
$$
\mathbf ...
- 2,069
1
vote
0
answers
121
views
Finite Symmetry Transformations in Classical Mechanics
This is not a homework exercise. I graduated from univerisity more than 10 years ago. I ask questions from my self-study.
There're two types of symmetry transformations in classical mechanics. One is ...
- 3,227
1
vote
0
answers
111
views
Variations with Non-Holonomic Non-Integrable Constraints
I am reading this paper by Flannery. He considers a non-holonomic non-integrable set of constraints
$$g_k=g_k(q_i,\dot q_i,t)=0, \quad k=1,\ldots,c,\tag{3.1}$$
where the $i$ index runs to $n$. He ...
- 345
5
votes
1
answer
181
views
The Grassmann-Odd Part of the Lagrangian of a Spinning Particle
I have the following series of questions from the lecture notes "Constrained Hamiltonian Systems and Relativistic Particles" by Fiorenzo Bastianelli. On page 15, section 2.2 the Lagrangian ...
- 3,227
6
votes
1
answer
275
views
Prove Noether Charge of Gauge Transformations is Zero on Constraint Surface
Noether's theorem states the following fact:
Consider the action
$$S[p,q]=\int dt\left\{p_{i}\dot{q}^{i}-H(p,q)\right\}.$$
Suppose under a global variation, the action changes by a total derivative
$$\...
- 3,227
0
votes
1
answer
96
views
What is the rigorous justification for using a single scalar acceleration in multi-axis Atwood systems?
In solving Atwood-type systems—especially those involving inclines or pulleys—we’re usually taught the following approach:
Choose a direction of motion (e.g., “block A moves down the incline”).
...
- 313
5
votes
1
answer
194
views
Commutation Relations in BRST Quantization
In Chapter 15 of Weinberg's The Quantum Theory of Fields, Weinberg states that the commutation relations for the creation and annihilation operators associated with the $A^\mu$ gauge fields of pure ...
- 345
1
vote
0
answers
93
views
Possible Typo in Poisson Brackets in Weinberg QFT Vol. 1, Section 8.3
In Chapter 8.3 of Weinberg's The Quantum Theory of Fields-Volume I, he gives the following Poisson bracket relations:
$$\begin{aligned}
\left[A^i(\mathbf{x}), \chi_{1\,\mathbf{y}}\right]_P &= -\...
- 345
2
votes
0
answers
176
views
How to write the Lagrangian Mechanics constraint for a circle that can't enter a rectangle?
I'm making a 2d game with a ball on a plank (rotated by the user pressing arrow keys) and trying to work out the physics of the ball's motion.
I started approaching it with Newton's laws, using ...
- 131
14
votes
1
answer
663
views
Physical Description of a Coin. Equations and constraints
I've been trying to describe the behaviour of a coin that can roll, spin and fall with Lagrangian Mechanics. The coin can roll without slipping with it's only "knowledge" of the floor being ...
- 2,069
2
votes
2
answers
196
views
Understanding Hamilton's principle with constraints in section 2.4 of Goldstein, 3rd edition
I'm working through Goldstein's Classical Mechanics, 3rd edition. In section 2.4, we are extending Hamilton's Principle to a system with constraints. In the beginning of the section he makes a couple ...
- 119
3
votes
2
answers
425
views
What are the constraints of rolling without slipping on a rotating disk?
Given the following system: a disk rotating with constant angular velocity and a ball rolling without slipping on the disk.
Imagine three diferent reference frames, $S, S', S''$. The $S$ frame is ...
- 2,069
3
votes
0
answers
79
views
Faddeev-Jackiw canonical quantization
In the context of quantization singular systems, the Faddeev-Jackiw symplectic formalism transforms a pre-symplectic space into a regular symplectic space (phase space) by resolving constraints ...
0
votes
1
answer
162
views
About the generalized coordinates of a pure rolling disc on a 2D plane being holonomic vs. semi-holonomic
This particular question is from eq. (1.39) in Goldstein "Classical mechanics".
I've seen 2 kinds of solutions for a pure rolling disc on a 2D plane (i) using "differential 1-form"...
2
votes
1
answer
190
views
Why Virtual Work for a System at Equilibrium is Zero when given a infinitesimal Virtual Displacement?
Please make it sensible for me that in Goldstein's Classical Mechanics book in the section 1.4 of d'Alembert's Principle and Lagrange's Equations, it is stated that if we give a infinitesimal virtual ...
- 47
-5
votes
1
answer
152
views
Predicting the position of a particle in spherical motion given two prior positions [closed]
I'm working on a problem involving a particle moving in 3D space under the following constraints:
a. The particle maintains a constant distance R from the origin (moves on a sphere)
b. There is no ...
- 91
4
votes
2
answers
263
views
Mathematical questions about equivalence of actions (1d Liouville and Schwarzian)
https://arxiv.org/abs/1705.08408 says the following action
\begin{align}
L = \pi_\phi \dot{\phi} + \pi_f \dot{f} - (\pi_\phi^2 + \pi_f e^\phi)\tag{2.1}
\end{align}
reduces to a Schwarzian action $L = \...
- 131
1
vote
1
answer
128
views
Goldstein’s section 2.4 Extending Hamilton’s Principle to systems with constraints
Goldstein in section 2.4 starts with an equation that I can’t wrap my head around for extending Hamilton’s Principle to systems with constraints. The integrand in the action is the Lagrangian plus the ...
- 11
1
vote
1
answer
132
views
Non-holonomic generalised forces of constraint
I am currently studying Analytical Mechanics having among one of the many references the book “Classical Mecanics” by Goldstein. And it is with respect to this that I have a doubt. My goal at the ...
- 41
3
votes
2
answers
311
views
Does Poincaré's lemma hold for on-shell equalities?
Given a manifold $M$, Poincaré's lemma states that every closed differential form is locally exact. That is, if $\omega$ is a $p$-form such that $d\omega=0$, then there exists (locally) a $(p-1)$-...
- 393
1
vote
0
answers
132
views
Lagrangian for a free particle with parametrized time
There are two seemingly contradictory Lagrangians for a free particle with a parametrized time. The first is the one given by Dirac in his lectures on quantum mechanics. Take the action of a free ...
- 17k
2
votes
1
answer
162
views
Catenary and non-holonomic constraint
I'm looking at Problem 1.6 in Mathematics for physics: A guided tour for graduate students by Stone & Goldbart, which rederives the equation for the catenary by parametrizing the arc by $x(s)$ ...
- 69