This thesis is devoted to creating a systematic way of determining all inequivalent
orthogonal coordinate systems which separate the Hamilton-Jacobi equation for a
given natural Hamiltonian defined on three-dimensional spaces of constant, non-zero
curvature. To achieve this, we represent the problem with Killing tensors and employ
the recently developed invariant theory of Killing tensors.
Killing tensors on the model spaces of spherical and hyperbolic space enjoy a
remarkably simple form; even more striking is the fact that their parameter tensors
admit the same symmetries as the Riemann curvature tensor, and thus can be
considered algebraic curvature tensors. Using this property to obtain invariants and
covariants of Killing tensors, together with the web symmetries of the associated orthogonal
coordinate webs, we establish an equivalence criterion for each space. In
the case of three-dimensional spherical space, we demonstrate the surprising result
that these webs can be distinguished purely by the symmetries of the web. In the
case of three-dimensional hyperbolic space, we use a combination of web symmetries,
invariants and covariants to achieve an equivalence criterion. To completely solve the
equivalence problem in each case, we develop a method for determining the moving
frame map for an arbitrary Killing tensor of the space. This is achieved by defining
an algebraic Ricci tensor.
Solutions to equivalence problems of Killing tensors are particularly useful in the
areas of multiseparability and superintegrability. This is evidenced by our analysis
of symmetric potentials defined on three-dimensional spherical and hyperbolic space.
Using the most general Killing tensor of a symmetry subspace, we derive the most
general potential “compatible” with this Killing tensor. As a further example, we
introduce the notion of a joint invariant in the vector space of Killing tensors and use
them to characterize a well-known superintegrable potential in the plane.
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| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:NSHD.ca#10222/14191 |
| Date | 09 June 2011 |
| Creators | Cochran, Caroline |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
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