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Dynamics on Differential One-Forms by Troy Story

By Troy Story

Dynamics on Differential One-Forms proposes a unifying precept for mathematical versions of dynamic structures. In "Thermodynamics on One-Forms (chapter I)", the long-standing challenge of deriving irreversibility in thermodynamics from reversibility in Hamiltonian mechanics, is solved. Differential geometric research exhibits thermodynamics and Hamiltonian mechanics are either irreversible on consultant prolonged section areas. "Dynamics on Differential One-Forms (II)" generalizes (I) to Hamiltonian mechanics, geometric optics, thermodynamics, black holes, electromagnetic fields and string fields. Mathematical versions for those structures are published as representations of a unifying precept; specifically, description of a dynamic method with a attribute differential one-form on an odd-dimensional differentiable manifold leads, via research with external calculus, to a collection of differential equations and a tangent vector defining procedure changes. Relationships among versions utilizing external calculus and traditional calculus indicate a technical definition of dynamic equilibrium. "Global research of Composite debris (III)" makes use of differential topology to boost the idea of huge vibration-rotation interactions for composite debris. an international classical Hamiltonian and corresponding quantum Hamiltonian operator are derived, then utilized to the molecular vibration-rotation problem."Characteristic Electromagnetic and Yang-Mills Gauge (IV)" makes use of differential geometry to take away many of the arbitrariness within the gauge, and exhibits how gauge capabilities for electromagnetic and Yang-Mills fields stick with an analogous differential equation.

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Phys. 23, 388 (1982). [7] V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978), pp. 233-237 . Tr o y L . S t o r y [8] T. L. Story, Int. J. Quantum Chem. 17, 915 (1980). 25 Chapter II ———————▼——————— Dynamics on Differential One-Forms Mathematical models of dynamics employing exterior calculus are shown to be mathematical representations of the same unifying principle; namely, the description of a dynamic system with a characteristic differential one-form on an odd-dimensional differentiable manifold leads, by analysis with exterior calculus, to a set of characteristic differential equations and a characteristic tangent vector which define transformations of the system.

Note that differential equations obtained in this way are mathematically equivalent to the Maxwell equations in the following sense: If the inverse of the contraction F( X ) is performed , F and, through duality, *F are obtained again; then, the exterior derivatives dF and d*F can be contracted on a triple of tangent vectors and set equal to the unique scalars zero and 4π*J(X ,Y, Z ), respectively, thereby generating Maxwell’s equations. In the present analysis characteristic differential one-forms are used as a starting point, thereby leading to vortex vectors, differential equations equivalent to Maxwell’s equations and the use of the contractions dSF (R ) and dS*F (R ) to obtain expressions for the Lagrangian.

Phase changes and chemical reactions In this section another form of Eqs. (17) is given by utilizing familiar definitions of characteristic function S . Then, as a prelude to directly applicable forms of these latter equations, a discussion of the difference in the manner in which physical processes are perceived theoretically through differential geometry and the manner in which they are perceived 18 Dynamics on Differential One-Forms empirically through measurements, is presented. Finally, a discussion is given of the use of these equations.

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