Constant locked inertia tensor trajectories for simple mechanical systems with symmetry

Relative equilibria of Lagrangian systems with symmetry

“…We discuss the relation between this result and a different criterion for relative equilibria given by Lewis [9]. We also consider in some detail two special cases, namely the case where the configuration space is a Lie group [5,7,16] and the case of a simple mechanical system [11], and in the context of the latter we make some remarks about the so-called Saari conjecture [8]. One of our purposes in this paper, indeed, is to provide a single framework for a variety of results about the conditions for the existence of relative equilibria both in general and in particular circumstances.…”

“…The matrix-valued function (g ab ) is called the locked inertia tensor. It has been conjectured (see [5,8]), on the basis of certain results for the N -body problem, that 'a Lagrangian simple mechanical system with symmetry is at a point of relative equilibrium if and only if the locked inertia tensor is constant along the integral curve that passes through that point'. The original version of this conjecture, in the context of the N -body problem, was formulated by Saari; the version above is called the naive generalization of Saari's conjecture.…”

“…So it seems unlikely that Saari's conjecture holds in all generality; and indeed it is known to be false. In a refined version of Saari's conjecture formulated in [5,8] it is required only that g ab ξ b is constant along the integral curve. It is clear that when ξ does define a relative equilibrium point, g ab ξ b is constant along the integral curve, because g ab ξ b = µ a is the value of the momentum at the relative equilibrium point and the integral curve through the point lies in the same level set.…”

“…A relative equilibrium of a Lagrangian system which is invariant under a Lie group G is a motion of the system which coincides with an orbit of a 1-parameter subgroup of G. Relative equilibria are of interest in particle dynamics [8,9,14] of course, and also in Riemannian and Finsler geometry [5,7,15], though there they are often studied under different names.…”

Saari's conjecture is true for generic vector fields

“…The general case remains open. Several researchers have worked to generalise Saari's conjecture appropriately, and to find counter-examples to these generalisations, including G. Roberts [Rob06]; M. Santoprete [San04]; A. Hernández-Garduño, J. K. Lawson, and J. E. Marsden [HLM05]; and J. K. Lawson and C. Stoica [LSt07].…”