Toda链的通用化Eisenhart升降机
Generalised Eisenhart lift of the Toda chain
摘要：
直线上最近邻相互作用粒子的户田链既可以用带有一个额外维度的流形上的测地运动来描述，也可以用带有几个额外维度的对称空间中的测地运动来描述。我们研究这两个实现之间的关系，并发现对称空间是一个广义的，多粒子的原始问题，减少到标准的Eisenhart电梯。这种广义的Escherart升降机作为一个逆卡鲁扎-克莱因减少，促进耦合常数的动量在更高的维度。特别是，等距的广义升力度量对应于能量保持变换，混合坐标和耦合常数。分析的一个副产品是户田拉克斯对的提升可以用来为标准和广义提升度量构造更高秩的Killing张量。
英文摘要：
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.
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