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Quantization of a nonholonomic system.

Anthony M Bloch1, Alberto G Rojo

  • 1Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA. abloch@umich.edu

Physical Review Letters
|September 4, 2008
PubMed
Summary
This summary is machine-generated.

Quantizing nonholonomic systems is challenging due to their non-Hamiltonian nature. This study introduces a method using a coupled field to quantize the Chaplygin sleigh, addressing internal dissipative dynamics.

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Area of Science:

  • Theoretical Physics
  • Quantum Mechanics
  • Classical Mechanics

Background:

  • Nonholonomic systems, characterized by non-integrable constraints, pose significant challenges for quantization.
  • Traditional Hamiltonian or variational approaches are not directly applicable to these systems.
  • The Chaplygin sleigh serves as a representative model for studying nonholonomic dynamics.

Purpose of the Study:

  • To develop a viable method for quantizing nonholonomic systems.
  • To investigate the quantization of a specific nonholonomic system, the Chaplygin sleigh.
  • To address the complexities introduced by internal dissipative dynamics in nonholonomic systems.

Main Methods:

  • Coupling the nonholonomic system to an auxiliary field that enforces constraints in a limiting case.
  • Formulating the full system as a Hamiltonian system amenable to quantization.
  • Analyzing the quantized system, including its dissipative properties.

Main Results:

  • A novel approach to quantize nonholonomic systems by introducing a constraint-enforcing field.
  • Successful quantization of the Chaplygin sleigh, demonstrating the applicability of the method.
  • Identification and analysis of internal dissipative dynamics within the quantized nonholonomic system.

Conclusions:

  • The proposed field-coupling method provides a pathway for quantizing nonholonomic systems.
  • The quantization of the Chaplygin sleigh highlights the importance of addressing dissipative effects.
  • This work offers a foundation for further research into the quantum behavior of constrained systems.