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

TL;DR

This paper introduces a new form of partial separation of variables for natural Hamiltonians called block-separation, which generalizes classical St"ackel separation by using twisted products and block structures.

Contribution

It characterizes block-separation invariantly, relates it to St"ackel matrices, and classifies block-separable coordinates in three-dimensional Euclidean space.

Findings

Dynamics reduces to separate blocks with scalar potentials.

Block-separation generalizes classical St"ackel separation.

Classification of block-separable coordinates in 3.

Abstract

We study twisted products $H=\alpha^rH_r$ 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-St\"ackel 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 St\"ackel separation of variables. We classify the block-separable coordinates of $\mathbb E^3$.