Hamilton's Canonical Equations and Einstein's E=mc2 Relation

The history of Einstein's landmark introduction of the energy-mass equivalence relation is reviewed. Emphasis is placed on the role that Hamilton's Canonical Equations play in arriving at the famous E=mc2 formula. Although both Einstein and Planck derived the relativistic energy-momentum relationships by considering the effects of electromagnetic interactions, it is shown that a comprehensive theory can be formulated which is based exclusively on Hamilton's dE=vdp relation when used in conjunction with the assumption of light-speed constancy in free space. To this end, it is helpful to return to Voigt's derivation in 1887 of a relativistic space-time transformation that was the precursor of the Lorentz transformation. A parallel derivation of Voigt's space-time transformation which takes account of Hamilton's Canonical Equations is shown to lead in a straightforward manner to both the relativistic energy-momentum transformation and Planck’s definition of relativistic momentum as p=γμv, as well as the E=mc2 relation itself. The associated E=γE0 equation is found to be completely analogous to the relation between atomic clock rates in different inertial systems that is used in the “pre-correction” procedure of the Global Positioning System to insure that atomic clocks on satellites run at the 
same rate as their counterparts on the earth’s surface. Finally, Einstein’s faulty claim of “longitudinal mass” that he used in his original E=mc2 derivation is traced back to his improper application of the Relativity Principle to a non-inertial system.


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