Block-Separation of Variables: a Form of Partial Separation for Natural Hamiltonians

Abstract

We study twisted products $H={\alpha }^r{H}_r$ of natural autonomous Hamiltonians $H_r$, each one depending on a separate set, called here separate $r$-block, of variables. We show that, when the twist functions ${\alpha }^r$ are a row of the inverse of a block-Stäckel matrix, the dynamics of $H$ reduces to the dynamics of the $H_r$, modified by a scalar potential depending only on variables of the corresponding $r$-block. It is a kind of partial separation of variables. We characterize this block-separation in an invariant way by writing in block-form classical results of Stäckel separation of variables. We classify the block-separable coordinates of ${\mathbb{E}}^3$.

References in corpus (3)

• Complex variables for separation of Hamilton-Jacobi equation on real pseudo-Riemannian manifolds
• Eigenvalues of Killing Tensors and Separable Webs on Riemannian and Pseudo-Riemannian Manifolds
• Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables

Cited by in corpus (3)

• Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations
• Hamiltonian integrable systems in a magnetic field and Symplectic-Haantjes geometry
• Higher Haantjes Brackets and Integrability