An Inverse Problem of Hamiltonian Dynamics

We study the question of whether for a natural Hamiltonian system on a two-dimensional compact configuration manifold, a single trajectory of sufficiently high energy is almost surely enough to reconstruct a real analytic potential. Consider a compact configuration manifold M equipped with a finite Borel measure (essentially we deal with the dimension n = 2) and a natural Hamiltonian system thereon, with the Hamiltonian H(p, q) = 〈p, p〉q + U(q), (p, q) ∈ T ∗Mn. Above, 〈·, ·〉q is a Riemannian metric on M and U a potential. The direct problem of dynamics on M is finding the trajectory q(t) ⊂ M, with initial conditions q(0) = q0 and q̇(0) = v0, moving in the known force field f(q) = −∇qU(q) on M, where the gradient ∇q has been associated with the metric 〈·, ·〉q. Let us call the inverse problem of dynamics the problem of reconstruction of the potential by observing the system’s trajectories q(t). The first problem of this type was explored in Newton’s Principia, in a quest for a physical law determining the planetary motion compatible with observational data. In the general case, knowledge of infinitely many trajectories is required to completely solve the problem. In this note we show that in the special case when M is two dimensional, compact and topologically non-trivial, a single trajectory with sufficiently large energy would almost surely suffice to reconstruct the potential. In the sequel, we assume that M as well as all the quantities involved are realanalytic. Also suppose, there is an a-priori estimate |U(q)| < C0, ∀q ∈ M, and we consider only the trajectories q(t) with total energy E ≥ C0. Theorem. Let n = 2 and suppose M is not diffeomorphic to S or RP . Almost every trajectory q(t), t ≥ 0, with energy E ≥ C0, suffices to reconstruct the potential U as a real-analytic function on M. Let us recall the definition of a key set, or set of uniqueness; see e.g. [3]. Definition. LetD be a domain in R and C(D) the class of real-analytic functions inD. A setK ⊂ D is a key set if any f ∈ C(D) vanishing identically onK vanishes identically on D. Received by the editors February 14, 2005 and, in revised form, May 26, 2005. 2000 Mathematics Subject Classification. Primary 37J05, 70H12. 1Note that the term “inverse problem of mechanics” has also been used to address the problem of deciding whether a given system of second order ODEs on Mn has a Lagrangian; see e.g. [5]. c ©2006 American Mathematical Society Reverts to public domain 28 years from publication