Exact solvability of potentials with spatially dependent effective masses

Biochemical systems theory (BST) is the foundation for a set of analytical andmodeling tools that facilitate the analysis of dynamic biological systems. This paper depicts major developments in BST up to the current state of the art in 2012. It discusses its rationale, describes the typical strategies and methods of designing, diagnosing, analyzing, and utilizing BST models, and reviews areas of application. The paper is intended as a guide for investigators entering the fascinating field of biological systems analysis and as a resource for practitioners and experts.

https://downloads.hindawi.com/archive/2013/897658.pdf

Biochemical Systems Theory: A Review

Invariance principles are used, in physics, in two distinct manners. First, they are used as superlaws of nature in that, once their validity has been suggested by their consistency with the known laws of nature, they serve as guides in our search for as yet unknown laws of nature. Second, they can serve as tools for obtaining properties of the solutions of the equations provided by the laws of nature. It is desirable for the first use to give a formulation of invariances directly in terms of the primitive concepts of physical theory, i.e., in terms of observations, or measurements, and their results. Invariances which can be so formulated are called geometric invariances. The present paper contains an attempt at such a formulation of geometric invariances. This formulation is then applied, in detail, to the classical mechanics of point particles, to a relativistic mechanics of interacting point particles, and to quantum theory. With the exception of the relativistic mechanics of point particles, these applications form a review, from a single point of view, of earlier work on this subject. The Last part of the paper contains a review of the second use of invariances.

The conceptual basis and use of the geometric invariance principles

Radiation Reaction and Angular Momentum Loss in Small Angle Gravitational Scattering

This paper presents a structure-exploiting nonlinear model reduction method for systems with general nonlinearities. First, the nonlinear model is lifted to a model with more structure via variable...

Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition

It is shown that very general nonlinear ordinary differential systems (embracing all that arise in practice) may, first, be brought down to polynomial systems (where the nonlinearities occur only as polynomials in the dependent variables) by introducing suitable new variables into the original system; second, that polynomial systems are reducible to ’’Riccati systems,’’ where the nonlinearities are quadratic at most; third, that Riccati systems may be brought to elemental universal formats containing purely quadratic terms with simple arrays of coefficients that are all zero or unity. The elemental systems have representations as novel types of matrix Riccati equations. Different starting systems and their associated Riccati systems differ from one another, at the final elemental level, in order and in initial data, but not in format.

Universal formats for nonlinear ordinary differential systems

The time‐symmetrical interaction of charges (Wheeler‐Feynman electrodynamics) is shown in principle to yield second‐order Newtonian‐type equations of motion under the restriction that the motions be analytic extensions of free‐particle motions. The means for explicitly generating the electrodynamic equations of motion describing invariant world‐lines are given. These physically relevant equations do not fit into Dirac's (1949) formulation of Hamiltonian relativistic particle dynamics, where either world‐line invariance is given up, or only trivially straight world lines can be described (``zero‐interaction theorem'' of Currie, Jordan, Sudarshan, 1963). The misfit is due to the requirement in Dirac's scheme that position x be canonical. Under the Lie‐Konigs theorem, however, Hamiltonian statements of dynamics with invariant world lines remain possible when suitable Q(x, ẋ) are introduced instead of x as canonical position variables. A group of necessary conditions on the structure of any dynamics that permits x to be canonical are worked out to indicate how stringent is this permission in general.

Can the Position Variable be a Canonical Coordinate in a Relativistic Many‐Particle Theory?

The Wheeler‐Feynman scheme of classical electrodynamics, using half‐retarded+half‐advanced fields between pairs of interacting charges, is written in a differential form which expresses the motion of the charges at any inertial observer's single present time through infinite‐order differential equations of motion. The conversion of field variables to mechanical ones in this way does not imply an infinite number of degrees of freedom; rather it is shown how to construct a Hamiltonian, depending solely on ordinary particle‐position coordinates and certain canonically conjugate momenta, which is both the energy and the generator of the ``correct'' motions. The latter are taken to be those continuously developable from free‐particle motions as the strength of interaction increases from the value zero. It is just this criterion that establishes particle‐position+conjugate momentum to be sufficient for the description of motion. The highness of the order of the equations of motion apparently gets to be expresse...

Hamiltonian Formulation of Action‐at‐a‐Distance in Electrodynamics

The old problem of how to represent uniquely a prescribed classical Hamiltonian H as a well‐defined quantal operator Ĥ is shown to have a clear answer within Feynman's path‐integral scheme (as expanded by Garrod) for quantum mechanics. The computation of Ĥ involves the momentum Fourier transform of a coordinate average of H. A differential equation for a reduced form of the Feynman propagator giving Ĥ from H is found; and the example of polynomial H worked out to give the Born‐Jordan ordering rule for Ĥ in this case.

Unique Hamiltonian Operators via Feynman Path Integrals

Plank’s recent discussion in this Journal of Hamiltonian structure of Lotka–Volterra dynamics is shown to have roots going back many years. This is briefly sketched together with an infrastructure of the Lie–Koenigs theorem and Gibbs ensemble theory.

Edward H. Kerner

Papers

Universal formats for nonlinear ordinary differential systems

Can the Position Variable be a Canonical Coordinate in a Relativistic Many‐Particle Theory?

Hamiltonian Formulation of Action‐at‐a‐Distance in Electrodynamics

Unique Hamiltonian Operators via Feynman Path Integrals

Comment on Hamiltonian structures for the n-dimensional Lotka–Volterra equations