Constant locked inertia tensor trajectories for simple mechanical systems with symmetry

Saari’s Conjecture, generalized from its usual context of the NN-body problem to a simple mechanical system with symmetry, says roughly that a condition of constant locked inertia tensor (interpreted appropriately) along a solution curve should guarantee that the curve is a relative equilibrium. Using a local Lagrangian slice parametrization about a non-symmetric point in phase space, we offer the motion in the form of a reduced Euler–Poincaré–Lagrange system together with the reconstruction equation. We state necessary and sufficient conditions for the existence of relative equilibria in this parametrization. These conditions allow us to relate curves with constant locked inertia tensors to relative equilibria. We find a class of simple mechanical systems with symmetry for which Saari’s Conjecture is true. We also show that if a simple mechanical system with nn degrees of freedom is symmetric under the free linear action of a kk-dimensional Lie group where k(k+1)/2≥(n−k)k(k+1)/2≥(n−k), then a version of Saari’s Conjecture holds except at specific isolated points. We apply our results to the three-dimensional three-body and four-body problems and to the nn-dimensional general relative two-body problem.