Abstract
In the covariant Hamiltonian mechanics with action-at-a-distance, we compare the proper time and dynamical time representations of the coordinate space world line using the differential geometry of nongeodesic curves in 3+1 Minkowski spacetime. The covariant generalization of the Serret-Frenet equations for the point particle with interaction are derived using the arc length representation. A set of invariant point particle kinematical properties are derived which are equivalent to the solutions of the equations of motion in coordinate space and which are functions of either the proper time or the dynamical time. Expressions for the quantities are given for the example of the covariant harmonic oscillator and comments are offered regarding the measurability of the dynamical time.
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Trump, M.A., Schieve, W.C. Off-mass-shell dynamics in flat spacetime. Found Phys 27, 389–414 (1997). https://doi.org/10.1007/BF02550163
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DOI: https://doi.org/10.1007/BF02550163