THE INVERSE PROBLEM FOR EULER’S EQUATION ON LIE GROUPS Wayne Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore Tel (65)

RIGID BODIES Euler’s equation inertia operator (from mass distribution) angular velocity in the body for their inertial motion 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.

IDEAL FLUIDS Euler’s equation pressurevelocity in space for their inertial motion Commentationes mechanicae ad theoriam corporum fluidorum pertinentes, M'emoirs de l'Acad'emie des Sciences Berlin, outward normalof domain

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),

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

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

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 ofover an interval

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

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

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

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

THREE DIMENSIONAL PROBLEM Proof Clearly

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

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