Generalized Hamiltonian theory: Central coordinates

The basis of a generalized formalism of Hamilton-Lagrange mechanics for elementary (i.e., individual) open systems was introduced in a previous work. In the present paper a covariance theory is constructed for the proposed generalization. The result is an extended transformation theory, or geometry, for phase space, in which the behavior of generalized Hamiltonian equations under time-dependent general coordinate transformations is determined. The proposed mechanics subsumes the ordinary mechanics in a natural way. New features include generalization of "pictures" (coordinate-system classes for fixed values of the Hamiltonian) to "viewpoints," which are characterized by values of a phase-space "vorticity" tensor, that always vanishes in ordinary mechanics: analogous to extremes of Heisenberg and Schrödinger pictures are those of canonical and Hamiltonian viewpoints. The Liouville theorem receives a novel generalization, and the scheme possesses a (viewpoint) covariant time derivative with analogies to the covariant derivative in a Riemannian space. Viewpoint covariance in the new formalism is the natural extension of canonical covariance in the old. The role of dynamical gauge invariance, investigated in a second previous paper, is elaborated, and the example of the undriven damped oscillator worked out to illustrate the operation of the formalism, including that of a generalized Hamilton-Jacobi theory. A line-element-space (Finslerian) interpretation is given for the new scheme, and a corresponding generalization is proposed for nonlinear self-interacting systems.