INVERSE PROBLEMS IN MECHANICS Wayne Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore Tel (65)

DIRECT and INVERSE PROBLEMS Direct: given the coefficient vector compute the rootsof the equation Inverse: given the roots of the equation, compute the coefficient vector. Answer: up to a scalar multiple

RIGID BODY MODEL In 1765 the mathematician Leonard Euler used Newton’s laws and properties of rotations to derive a system of three nonlinear first order differential equations that describe the motion of a rigid body. In matrix notation, these equations are is the 3 x 3 inertia operator is the angular velocity in the body

DIRECT PROBLEM Attracted the attention of leading mathematicians and physicists for two and a half centuries, including Adler, Appel, Arnold, Audin, Bobylev, Cartan, Chaplygin, Delone, Haine, Hilbert, Jacobi, Jordan, Kirillov, Klein, Kovalevskaya, Lagrange, Lax, Painleve, Picard, Poincare, Poinsot, Poisson, Suslov, Steklov, Weil, and Whittaker. Motivated much applied and theoretical mathematics including dynamics, elliptic functions, integrable systems, solitons, and string theories.

INVERSE PROBLEM Algebraic versus numerical computation Geometric intuition versus symbolic manipulation if can be computed from (up to a multiple) only ifis not degenerate Result 1 (lies in a two dimensional subspace), since

INVERSE PROBLEM Angular velocity in space momentum in space Energyand angular are constant

INVERSE PROBLEM is degenerate if and only ifResult 2 is nongegenerate thenIf is bounded below by a positive number, is periodic and

INVERSE PROBLEM Converse of result 1.Result 3 We first observe that

INVERSE PROBLEM where for

INVERSE PROBLEM Therefore, it suffices to show that If it lies in the line and hence is unbounded since can not lie in a two dimensional subspace is bounded below by a positive number. This contradicts the fact that is bounded above.

TWO MORE ALGORITHMS Algorithm 2. Compute a minor eigenvector of Algorithm 3. Exploit relationships between and the null vector of the matrix computed from the moments of

EXTENSIONS 2. Compute A from two measurements of 1. Prove results 1 and 3 for geodesic flows on general Lie groups with invariant metrics, e.g. Euler-Poincare equations for the (polymer) metric 3. Inverse problems for general classes of differential equations, e.g. integrable, stochastic

REFERENCES [2] Arnold, V. I., Mathematical Methods of Classical Mechanics, Springer, New York, [3] Auden, M., Spinning Tops, Cambridge University Press, [4] Chang, Y. T., Tabor, M., Weiss, J., Analytic structure of the Henon-Heiles hamiltonian and integrable and nonintegrable regimes”, Journal of Mathematical Physics, 23, p. 531, [1] Abraham, R. and Marsden, J. E., Foundations of Mechanics, Benjamin, Massachusetts, 1978.

REFERENCES [6] V. V. Golubev, Lectures on the Integration of the Equations of Motion of a Rigid Body About a Fixed Point, State Publishing House of Theoretical Technical Literature, Moscow, [7] Lawton, W. and Noakes, L., “Computing the inertia operator of a rigid body”, Journal of Mathematical Physics, April or May, [5] Euler, L., Theoria motus corporum solidorum seu rigodorum, 1765.

REFERENCES [9] Marsden, J. E., Ebin, D. G. and Fischer, A. E., “Diffeomorphism groups, hydrodynamics and relativity”, Proc. 13 th biennial seminar of the Canadian Mathematical Congress (J. R. Vanstone, ed.), Montreal (1972), [10] Weiss, J., Tabor, M. and Carnvale, G., “The Painleve’ property for partial differential equations”, Journal of Mathematical Physics, 24, p. 522, [8] Lawton, W. and Lenbury, Y., “Interpolatory solutions of linear ODE’s”, Submitted.