General Solutions to the Leah Hamiltonian and the Imani Periodic Functions

The Leah dynamic system corresponds to a one-dimensional classical oscillator for which the nonlinear force is proportional to the one-third power of the spring extension beyond its equilibrium position. Its energy function, the Leah Hamiltonian, therefore has a potential energy term proportional to the four-thirds power of the extension. The mathematical solutions to these two equations are called, respectively, the Leah and Imani functions. We demonstrate, by means of an explicit construction, that the Imani functions can be calculated. However, only general mathematical properties may be determined for the Leah functions. It should be noted that  both sets of functions have the same general features as the standard cosine and sine functions.