Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 013, 22 pages      arXiv:1808.01889      https://doi.org/10.3842/SIGMA.2019.013

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

Claudia Maria Chanu and Giovanni Rastelli
Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto 10, Italy

Received August 07, 2018, in final form February 14, 2019; Published online February 23, 2019

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-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 $\mathbb E^3$.

Key words: Stäckel systems; partial separation of variables; position-dependent time parametrisation.

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