Master of Science (MS)
Hamilton-Jacobi (HJ)-theory is an extension of Lagrangian mechanics and concerns itself with a directed search for a coordinate transformation in which the equations of motion can be easily integrated. The equations of motion of a given mechanical system can often be simplified considerably by a suitable transformation of variables such that all the new position and momemtum coordinates are constants. A particular type of transformation is chosen in such a way that the new equations of motion retain the same form as in the former coordinates; such a transformation is called canonical or contact and can greatly simplify the solution to the equations of motion. Hamilton (1838) has developed the method for obtaining the desired transformation equations using what is today known as Hamilton's principle. It turns out that the required transformation can be obtained by finding a smooth function S called a generating function or Hamilton's principal function, which satisfies a certain nonlinear first-order partial-differential equation (PDE) also known as the Hamilton-Jacobi equation (HJE). Unfortunately, the HJE being nonlinear, is very difficult to solve; and thus, it might appear that little practical advantage has been gained in the application of the HJ-theory. Nonetheless, under certain conditions, and when the Hamiltonian is independent of time, it is possible to separate the variables in the HJE, and the solution can then always be reduced to quadratures. Thus, the HJE becomes a useful computational tool only when such a separation of variables can be achieved. However, in this thesis we develop another approach for solving the HJE for a large class of Hamiltonian systems in which the variables may not be separable and/or the Hamiltonian is not time-independent. We apply the approach to a class of integrable Hamiltonian sytems known as the Toda lattice. Computational results are presented to show the usefulness of the method.
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Aliyu, Mohammad Dikko, "A parametrization approach for solving the Hamilton-Jacobi-Equation and application to the A2 Toda lattice" (2003). LSU Master's Theses. 2539.