Abstract :
Linear transformations of special relativity considered in [Albert Einstein, Zur Elektrodynamik
der bewegte Körper. Annalen der Physik. 17 (1905) 891–921] are applied to Hamiltonian
systems and Dirac equations, as presented in [Paul A.M. Dirac, Lectures on Quantum
Mechanics, Belfer Graduate School of Science, New York, 1964]. To account for Weber’s
electro-dynamic law of particle attraction and for the relativistic increase of the mass in
particle accelerators, an extension of the second Newton’s law for motion subject to external
forces that may depend on accelerations and higher order derivatives of velocities is
considered and the Buquoy–Mestschersky generalization for motion of bodies with variable
masses is included. The causality of systems driven by such forces is assured by consideration
of left higher order derivatives in the right-hand sides of the equations of motion.
The consistency condition is presented, and existence of solutions for the equations of motion
driven by forces with left higher order derivatives is proved, leading to the generalized
Lagrange and Hamilton equations that incorporate those extensions. Then, non-holonomic
Hamiltonian systems with time-invariant constraints introduced by P.G. Bergmann and
P.A.M. Dirac are considered, as suggested by Dirac for atomic models. Since one and the
same process or particle may be observed as different images, the procedure is developed
for identification of processes in moving systems by inverse relativistic transformations
applied over small intervals of time to discrete experimental measurements obtained as
the images of those processes in the observed (relativistic) coordinates. It is demonstrated
that motions and processes evolving in still and/or moving systems can be described in the
proper (of a still system) and/or relativistic (of different moving systems) coordinates in
one common system of equations under the condition that all components of that system
are referred to one and the same time of a still or moving observer. The results open new
avenues for further research in relativistic systems theory and provide a basis for development
of computing software for process identification in the images transmitted from
distant or fast moving systems.
