Guided by Nambu ( n +1)-dimensional phase space formalism we build a new system of dynamic equations. These equations describe a dynamic state of the corporeal system composed of n subsystems. The dynamic equations are formulated in terms of dynamic variables of the subsystems as well as in terms of dynamic variables of the corporeal system. These two sets of variables are related respectively as roots and coefficients of the n -degree polynomial equation. In the special n =2 case, this formalism reproduces relativistic dynamics for the charged spinning particles.
Invariant theory as a study of properties of a polynomials under translational transformations is developed. Class of polynomials with congruent set of eigenvalues is introduced. Evolution equations for eigenvalues and coefficients remaining the polynomial within proper class of polynomials are formulated. The connection with equations for hyper-elliptic Weierstrass and hyper-elliptic Jacobian functions is found. Algorithm of calculation of eigenvalues of the polynomials based on these evolution equations is elaborated. Elements of the generalized dynamics with n-order characteristic polynomials is built.
A numerical ordinary differential equation method, in conjunction with a finite-difference discretization, is applied to solve a dynamic motion partial differential equation of mixed type. This dynamic motion equation arises in the application of a towed acoustic antenna in the ocean. The equation describes the transverse dynamic motion of a cylinder of zero bending rigidity (infinitely flexible) in a viscous fluid flow. The numerical treatment well-poses the problem, whose numerical solution is shown to be general purpose and convergent. An efficient implementation is applied to reduce the cost of computation. The formulation, convergence, and some computational aspects are fully discussed. In order to demonstrate the validity and capability of this treatment, an application is presented.