Quasi-Holonomic Dynamical Systems

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Abstract
When a dynamical system contains both mechanical and electromagnetic energies, the dynamical equations assume special forms. It is shown that when the dynamical system is nonholonomic, or when it is holonomic but is referred to nonholonomic reference axes, the equations still have the same form as those of a holonomic dynamical system and can be solved as such without any auxiliary equations. Following a suggestion of Lorentz, such dynamical systems are called here ``quasi‐holonomic'' dynamical systems. Because of the special form of the equations, their development in tensor notation allows the introduction of concepts from the absolute calculus that apparently have not been hitherto applied in classical dynamics. It is shown here that the equation of motion of quasi‐holonomic systems in tensor symbolism may be interpreted as specifying the trajectory of a point moving in an (n‐dimensional) non‐Riemannian space, referred to holonomic or nonholonomic reference axes. When the field equations of Maxwell in tensor form are formulated for the electromagnetic part of the quasi‐holonomic system, it is found that the electromagnetic field must be expressed in terms of its components along an orthogonal ennuple, the field lying in a local Euclidean space, tangent to the underlying non‐Riemannian space. By the use of such a nonholonomic reference frame the dynamical equations of Lagrange and the field equations of Maxwell are correlated. The equations show formal analogies with those of the five‐dimensional unified field theory of relativity dynamics, where combined electromagnetic and gravitational systems are analyzed. Practical examples of quasi‐holonomic systems are the numerous types of rotating electrical machinery used in industry.