[Paper Review] Dynamical systems with internal degrees of freedom in non-Euclidean spaces

Jan J. Sławianowski, Vasyl Kovalchuk+3 more|arXiv (Cornell University)|Feb 21, 2008

Elasticity and Wave Propagation16 references3 citations

TL;DR

This paper formulates the kinematics and dynamics of material points with internal degrees of freedom—specifically infinitesimal gyroscopes and affinely-rigid bodies—moving in Riemannian manifolds. It derives equations of motion showing how internal angular momenta couple to spatial curvature and torsion, demonstrating that the system has five conserved quantities and one active rotational degree of freedom in the angular momentum space, with the total motion dependent on the inertial moment I.

ABSTRACT

Presented is description of kinematics and dynamics of material points with internal degrees of freedom moving in a Riemannian manifold. The models of internal degrees of freedom we concentrate on are based on the orthogonal and affine groups. Roughly speaking, we consider infinitesimal gyroscopes and homogeneously deformable gyroscopes (affienly-rigid bodies) in curved manifolds. We follow our earlier models of extended rigid and affinely-rigid bodies moving in a flat space. It is well known that in curved spaces in general there is no well-defined concept of extended rigid or affinely-rigid body. Our infinitesimal models are mathematically well defined and physically they may be interpreted as an approximate description of "small" rigid and affinely-rigid bodies. We derive equations of motion and show how internal degrees of freedom interact with spatial geometry, first of all with the curvature but also with the torsion. Integrability and degeneracy problems are discussed.

Motivation & Objective

• To develop a mathematically consistent framework for extended rigid and affinely-rigid bodies in non-Euclidean spaces where standard definitions fail.
• To model internal degrees of freedom using orthogonal and affine groups as symmetries, treating them as infinitesimal gyroscopes and homogeneously deformable bodies.
• To derive equations of motion that describe the interaction between internal dynamics and the geometry of curved manifolds, particularly curvature and torsion.
• To analyze integrability and degeneracy in such systems, identifying conserved quantities and reduced dynamical behavior.

Proposed method

• Uses a fibre bundle framework with base manifold M (physical space) and total configuration space Q = ⋃ₓ∈M Qₓ^int, generalizing the Cartesian product Q = M × Q^int.
• Applies Lagrangian and Hamiltonian formalisms on the total configuration space, with kinetic energy expressed in terms of canonical momenta S(R) and S_rl.
• Derives equations of motion via Poisson brackets, showing that only the interference term in the Hamiltonian contributes to time evolution.
• Introduces canonical angular momenta S(R) and S_rl, with explicit inverse relations involving mass m and moment of inertia I.
• Uses 3D vector notation in ℝ³ to express dynamical equations as coupled vector differential equations involving cross products.
• Performs Legendre transformation to obtain the Hamiltonian, showing explicit dependence on I and all kinetic energy terms.

Experimental results

Research questions

• RQ1 How can extended rigid and affinely-rigid bodies be consistently modeled in curved Riemannian manifolds where standard definitions break down?
• RQ2 What is the nature of the coupling between internal degrees of freedom and the curvature and torsion of the underlying space?
• RQ3 How many conserved quantities exist in the system, and what is the structure of the remaining dynamical degrees of freedom?
• RQ4 To what extent does the inertial moment I influence the time evolution of configuration variables despite not appearing in the Poisson bracket equations?
• RQ5 Can the dynamics of internal angular momenta be reduced to a single active rotational degree of freedom?

Key findings

• The system exhibits five independent constants of motion: |S(R)|², |S_rl|², and the scalar product δ^{AB}S(R)_A S_rl_B, which are functionally dependent on the first two.
• The vector (R/2)S(R)^A + S_rl^A is a conserved quantity, implying that the angle between the two angular momentum vectors is constant.
• Only the interference term in the kinetic Hamiltonian contributes to the time evolution of the canonical momenta, as the first and third terms are Casimir invariants.
• The dynamical system reduces to a single active degree of freedom: the polar angle of the plane spanned by S(R) and S_rl, rotating about the conserved vector (168).
• The time evolution of configuration variables (r̄, κ̄) explicitly depends on the inertial moment I, despite I not appearing in the Poisson bracket equations.
• The Legendre transformation and its inverse explicitly involve I, confirming its role in mapping between velocity and momentum variables.