Poisson Algebras and 3D Superintegrable Hamiltonian Systems
Poisson Algebras and 3D Superintegrable Hamiltonian Systems
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DOI：
10.3842/sigma.2018.022
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发表时间：
2017-08
影响因子：
0.9
通讯作者：
A. Fordy;Qing Huang
中科院分区：
文献类型：
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作者：
A. Fordy;Qing Huang
使用李代数sl(2)的三维泊松括号表示，我们首先使用最高权重表示将其嵌入到更大的李代数中。然后将其解释为与二次卡西米尔函数相关的“动能”的对称和共形对称代数。然后，我们考虑可以添加的势，同时保持可积，导致可分离系统的族，依赖于单个变量的任意函数。在超可积的情况下，添加进一步的积分，将这些函数限制为特定的形式，依赖于有限数量的任意参数。研究了这些超可积系统的泊松代数。将动能对称代数的自同构推广到全泊松代数，使我们能够建立完整的泊松关系集。
Using a Poisson bracket representation, in 3D, of the Lie algebra sl (2), we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras of the “kinetic energy”, related to the quadratic Casimir function. We then consider the potentials which can be added, whilst remaining integrable, leading to families of separable systems, depending upon arbitrary functions of a single variable. Adding further integrals, in the superintegrable case, restricts these functions to specific forms, depending upon a finite number of arbitrary parameters. The Poisson algebras of these superintegrable systems are studied. The automorphisms of the symmetry algebra of the kinetic energy are extended to the full Poisson algebra, enabling us to build the full set of Poisson relations.