2013/12/06 by Marco Cariglia, Gary Gibbons · 1 citation
Physics and Astronomy · Mathematics · #math-ph #gr-qc #hep-th #math.MP
paper · pdf · doi:10.1063/1.4866318
published as J. Math. Phys. 55, 022701 (2014) · 8 pages, no figures
arxiv created 2013/12/06 · arxiv updated 2014/03/26
The Toda chain of nearest neighbour interacting particles on a line can be described both in terms of geodesic motion on a manifold with one extra dimension, the Eisenhart lift, or in terms of geodesic motion in a symmetric space with several extra dimensions. We examine the relationship between these two realisations and discover that the symmetric space is a generalised, multi-particle Eisenhart lift of the original problem, that reduces to the standard Eisenhart lift. Such generalised Eisenhart lift acts as an inverse Kaluza-Klein reduction, promoting coupling constants to momenta in higher dimension. In particular, isometries of the generalised lift metric correspond to energy preserving transformations that mix coordinates and coupling constants. A by-product of the analysis is that the lift of the Toda Lax pair can be used to construct higher rank Killing tensors for both the standard and generalised lift metrics.