THE INVERSE PROBLEM FOR EULER’S EQUATION ON LIE GROUPS

1. THE INVERSE PROBLEM FOR EULER’S EQUATION ON LIE GROUPS Wayne Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 874-2749

2. RIGID BODIES Euler’s equation for their inertial motion angular velocity in the body inertia operator (from mass distribution) Theoria et ad motus corporum solidorum seu rigodorum ex primiis nostrae cognitionis principiis stbilita onmes motus qui inhuiusmodi corpora cadere possunt accommodata, Memoirs de l'Acad'emie des Sciences Berlin, 1765.

3. IDEAL FLUIDS Euler’s equation for their inertial motion velocity in space pressure outward normal of domain Commentationes mechanicae ad theoriam corporum fluidorum pertinentes, M'emoirs de l'Acad'emie des Sciences Berlin, 1765.

4. GEODESICS Moreau observed that these classical equations describe geodesics, on the Lie groups that parameterize their configurations, with respect to the left, right invariant Riemannian metric determined by the inertia operator (determined from kinetic energy) on the associated Lie algebra Une method de cinematique fonctionnelle en hydrodynamique, C. R. Acad. Sci. Paris 249(1959), 2156-2158

5. EULER’S EQUATION ON LIE GROUPS Arnold derived Euler’s equation that describe geodesics on Lie groups with respect to left, right invariant Riemannian metrics Mathematical Methods of Classical Mechanics, Springer, New York, 1978

6. LAGRANGIAN FORMULATION A trajectory is a geodesic for a left, right invariant Riemannian metric iff the associated angular velocity/momentum satisfies The momentum lies within a coadjoint orbit which has a sympletic structure and thus even dimension

7. GENERAL ASSUMPTIONS FOR THE INVERSE PROBLEM is a connected Lie group with Lie algebra is an inertia operator (self-adjoint and positive definite) satisfies Euler’s equation wrt Problem Compute ,up to multiplication by a constant, from the values of over an interval

8. A GENERAL SOLUTION Theorem If and are nondegnerate, then are invertible and A is determined, up to multiplication by a constant, from the following two equations

9. SOLUTION FOR RIGID BODIES Theorem (Lyle-Noakes, JMP, 2001) For G=SO(3), A can be determined iff is nondegenerate (not contained in a proper subspace) Proof If is degenerate then there exists such that then satisfies Euler’s equation for the inertia operator To complete the proof it suffices to show that if is degenerate then is degenerate. Consider

10. THREE DIMENSIONAL PROBLEM Define the scalar product and choose an orientation on a three dimensional Let denote the corresponding vector cross product Then Choose a basis so Construct a linear operator Let y, [L] denote wrt this basis Theorem Euler’s equation for y is

11. THREE DIMENSIONAL PROBLEM Assume that is also an inertia operator on Construct the operator Let denote the corresponding vector cross product Lemma and either is nonsingular Lemma Hom. Pol. Lemma on

12. THREE DIMENSIONAL PROBLEM Proof Clearly

14. UNIMODULAR GROUPS Theorem (Milnor) G is unimodular iff Then an orientation and basis can be chosen so that L = is diagonal and the signs determine G as below

15. NONUNIMODULAR GROUPS Theorem (Milnor) If G is unimodular for some basis