A nonlinear differential equation for the polar angle of a point of an ellipse is derived. The solution of this differential equation can be expressed in terms of the Jacobi elliptic function dn u,k . If the polar angle is extended to the complex plane, the Jacobi imaginary transformation properties and the dependence on the real and complex quarter periods can be described. From the differenti...

The standard text-book Jacobi equation (equation of geodesic deviation) arises by linearizing the geodesic equation around some chosen geodesic, where the linearization is done with respect to the coordinates and the velocities. The generalized Jacobi equation, introduced by Hodgkinson in 1972 and further developed by Mashhoon and others, arises if the linearization is done only with respect to...

in this paper, a new analytical method to find a near-optimal high gain controller for the non-minimum phase affine nonlinear systems is introduced. this controller is derived based on the closed form solution of the hamilton-jacobi-bellman (hjb) equation associated with the cheap control problem. this methodology employs an algebraic equation with parametric coefficients for the systems with s...

The Benjamin-Bona-Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg-de Vries equation, and is another uni-directional wave equation with cnoidal wave solutions. Cnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics. Using a computerized symbolic computation tech...

In the Riemannian geometry of quantum computation, the quantum evolution is described in terms of the special unitary group of n-qubit unitary operators with unit determinant. To elaborate on several aspects of the methodology, the Riemannian curvature, geodesic equation, Jacobi equation, and lifted Jacobi equation on the group manifold are explicitly derived. This is important for investigatio...

We discuss a class of time-dependent Hamilton-Jacobi equations, where an unknown function of time is intended to keep the maximum of the solution to the constant value 0. Our main result is that the full problem has a unique viscosity solution, which is in fact classical. The motivation is a selection-mutation model which, in the limit of small diffusion, exhibits concentration on the zero leve...