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

Abstract

We study twisted products H=αʳHᵣ of natural autonomous Hamiltonians Hᵣ, each one depending on a separate set, called here a separate r-block, of variables. We show that, when the twist functions αʳ are a row of the inverse of a block-Stäckel matrix, the dynamics of H reduce to the dynamics of the Hᵣ, modified by a scalar potential depending only on the 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 E³.