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Nonholonomic double-bracket equations and the Gauss thermostat
Alberto G Rojo1, Anthony M Bloch
1Department of Physics, Oakland University, Rochester, Michigan 48309, USA. rojo@oakland.edu
Summary
This study introduces a new class of nonlinear equations derived from a constant kinetic-energy constraint on oscillators. These equations generalize double-bracket equations and offer insights into nonequilibrium molecular dynamics.
Area of Science:
- Nonlinear dynamics
- Statistical mechanics
- Mathematical physics
Background:
- Nonlinear nonholonomic constraints are crucial in modeling complex systems.
- Nonequilibrium molecular dynamics requires advanced theoretical frameworks.
- Double-bracket equations are significant in integrable systems and gradient flows.
Purpose of the Study:
- To investigate equations arising from a constant kinetic-energy constraint on 1D oscillators.
- To explore the connection between these equations and generalized double-bracket equations.
- To analyze the dynamics and solutions under specific potential conditions.
Main Methods:
- Imposing a nonlinear nonholonomic constraint (constant kinetic energy).
- Applying Gauss's law of least constraint for dynamics.
- Analyzing the resulting equations for specific external potentials (e.g., harmonic).
Main Results:
- The derived equations generalize the double-bracket equations.
- Specific external potentials lead to a symmetric bracket description.
- Periodic solutions were obtained for harmonic potentials.
Conclusions:
- The constant kinetic-energy constraint provides a novel route to generalized integrable systems.
- The findings are relevant for nonequilibrium molecular dynamics simulations.
- Periodic solutions highlight potential for stable dynamics in constrained oscillator systems.
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