The Inverse Problem for Euler’s Equation on Two and Three Dimensional Lie Groups

ABSTRACT Euler’s equation, that describes geodesics for a left-invariant Riemannian metric on a Lie group associated with an inertia operator on its Lie algebra, plays an important role in mechanics. We examine the inverse problem of computing the inertia operator, up to multiplication by a constant, from a single solution of Euler’s equation. We prove that, with exactly two exceptions, every two and three dimensional Lie group has the property that this inverse problem has a solution if and only if the angular velocity in the body does not lie in a proper subspace of the Lie algebra. The two exceptions are the group of Euclidean transformations of the plane and the product of the two-dimensional affine group with the group of real numbers.