Relative equilibria of mechanical systems with rotational symmetry (Q6576896)

Consider a mechanical system \((Q,V)\) with configuration space \(Q\) upon which \(\mathbf{SO}(3)\) acts freely and isometrically and a potential function \(V\) which is invariant with respect to the \(\mathbf{SO}(3)\) action. The main result of this paper provides an additional geometry structure on shape space \({\mathcal B} = Q/ \mathbf{SO}(3)\), called a \(3\)-web, by which relative equilibria can be classified. A \(3\)-web on a manifold is a collection of \(k\) regular foliations \(\{ {\mathcal F}_1,{\mathcal F}_2,{\mathcal F}_3\}\) defined almost everywhere and given locally by the level sets of \(3\) functions \(\{\Lambda_1,\Lambda_2,\Lambda_3\}\). The eigenvalues \(\lambda_j\) of the locked inertial tensor \({\mathbb I}_q\) (\(q\in Q\)) descend to \({\mathcal B}\) and their level sets endow \({\mathcal B}\) with a \(3\)-web. A point \(x\in {\mathcal B}\) is a normal relative equilibria if and only if \(2\nabla V(x) = \kappa \nabla \lambda_j(x)\) for some multiplicity one eigenvalue \(\lambda_j\) and \(\kappa\geq 0\). Of necessity, a normal relative equilibrium \(x\in {\mathcal B}\) is critical point of \(V\) restricted to a leaf of constant \(\lambda_j\). The angular momentum of a normal relative equilibrium is an eigenvector \(L_j\) of \({\mathbb I}_q\) associated to a multiplicity one eigenvalue \(\lambda_j\) where the eigenvector has magnitude \(\vert L_j\vert^2 = \kappa \lambda_j^2\).\N\NThe main result follows from a more general result for a Lie group \(G\) acting on \(Q\), where it is assumed that orbit map \(\pi :Q\to {\mathcal B}\) is a smooth principal \(G\)-bundle onto \({\mathcal B} = Q/G\). Then, for \({\mathfrak g}\) being the Lie algebra of \(G\) and \({\mathfrak g}^*\) the dual of \(g\), there holds: for every coadjoint orbit \({\mathcal O}\) in \({\mathfrak g}^*\) (every adjoint orbit \({\mathcal O}\) in \({\mathfrak g}\)) there is defined an amended (augmented) \(k\)-web on \({\mathcal B}\) defined by \(k\) functions \(\{\Gamma_1,\dots,\Gamma_k\}\) (\(\{\Lambda_1,\dots,\Lambda_k\}\)) such that a point \(x\in {\mathcal B}\) is a normal relative equilibrium with angular momentum belonging to \({\mathcal O}\) if and only if \(\nabla V(x) = -\kappa \nabla \Gamma_j\) (\(\nabla V(x) = \kappa \nabla \Lambda_j(x)\)) for some \(j\) and \(\kappa\geq 0\). Furthermore, of necessity, a normal relative equilibrium is a critical point of the potential restricted to a leaf of constant \(\Gamma_j\) (\(\Lambda_j\)).\N\NExamples of \(k\)-webs are given for the following problems: a rubber ball, a triatomic molecule, a body in a central force field, and the spherical \(3\)-body problem. An analysis of abnormal relative equilibria is given in the equal mass spherical \(3\)-body problem.