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      Hamilton-Jacobi Theory and Superintegrable Systems

      Armstrong, Craig Keith
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      Armstrong, C. K. (2007). Hamilton-Jacobi Theory and Superintegrable Systems (Thesis, Master of Science (MSc)). The University of Waikato, Hamilton, New Zealand. Retrieved from https://hdl.handle.net/10289/2340
      Permanent Research Commons link: https://hdl.handle.net/10289/2340
      Abstract
      Hamilton-Jacobi theory provides a powerful method for extracting the equations of motion out of some given systems in classical mechanics. On occasion it allows some systems to be solved by the method of separation of variables. If a system with n degrees of freedom has 2n - 1 constants of the motion that are polynomial in the momenta, then that system is called superintegrable. Such a system can usually be solved in multiple coordinate systems if the constants of the motion are quadratic in the momenta. All superintegrable two dimensional Hamiltonians of the form H = (p_x)sup2 + (p_y)sup2 + V(x,y), with constants that are quadratic in the momenta were classified by Kalnins et al [5], and the coordinate systems in which they separate were found.

      We discuss Hamilton-Jacobi theory and its development from a classical viewpoint, as well as superintegrability. We then proceed to use the theory to find equations of motion for some of the superintegrable Hamiltonians from Kalnins et al [5]. We also discuss some of the properties of the Poisson algebra of those systems, and examine the orbits.
      Date
      2007
      Type
      Thesis
      Degree Name
      Master of Science (MSc)
      Publisher
      The University of Waikato
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