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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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 ...
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On gauge theories and redundant degrees of freedom
Given an action or Lagrangian with the additional information that it is a gauge system, how do we know this field has how many physical or redundant degrees of freedom? Is there any systematic method ...
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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 ...
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A question related to torque at the molecular level
My argument is: When you nudge the molecule at the end of a rigid body, that molecule must move in a straight line initially in order to transfer force to the pivot. However, due to Pythagoras’ ...
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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 ...
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Is the d'Alembert's principle arbitrary and redundant? [duplicate]
D'Alembert principle states that: $$ \sum_i \left (\mathbf F_i^{(a)}- \mathbf{\dot p}_i \right) \cdot \delta \mathbf r_i = 0 $$ but I'm not quite getting it. The derivation seems trivial for me, ...
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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 ...
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Commmuting variations and derivatives in the context of field theory
For point particles, a holonomic constraint is of the form, $$C(q,t)=0. \tag{1}$$ Following [1], I can show the following for holonomic constraints on point particles: $$\delta(\dot{q})=\frac{d}{dt}(\...
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About the generalized coordinates of a pure rolling disc on a 2D plane being holonomic vs. semi-holonomic
This particular question is from eq. (1.39) in Goldstein "Classical mechanics". I've seen 2 kinds of solutions for a pure rolling disc on a 2D plane (i) using "differential 1-form"...
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Why can we choose the Lagrange multiplier in Electrodynamics? [duplicate]
Consider a classical theory described by a Lagrangian $\mathscr{L}$ under the constraint $C=0$. We may make use of the Lagrange multipliers method and write the following, $$\mathscr{L}\mapsto\mathscr{...
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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 ...
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What if we leash the Moon? [closed]
It is understood that the Moon's kinetic energy is being chipped away at by the friction of the tides it generates with the Earth below, and that this causes its velocity to slow down and consequently ...
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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 ...
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1answer
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Semi-Holonomic Constraint Forces Derivation Using d'Alembert's Principle
The other day I was in a lecture of Analytical Mechanics about d'Alembert's Principle, and specifically about semi-holonomic constraint forces. At the lecture, my professor stated that the constraint ...
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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 = \...
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