The successful formulation of quantum energy was preceded by a formalism using canonical conjugation. This is accomplished by generalized coordinates of position and velocity differentiable over a Lagrangian function and by generalized position and momentum differentiable over a Hamiltonian function. Before the use of operators, complex numbers, and complex matrices (spinors), these two formulations describe the natural interaction of real potential energy and real kinetic energy. The Lagrangian describes the difference; while the Hamiltonian, the sum. For a 1-particle system, the Hamiltonian is given by H1 = (1/m1)(1/2)(p1) + V1, while the Lagrangian is given by L2 = (m2)(1/2)(v2) - V2. In a 2-particle system, and if V1 = V2 then H1 + L2 = (1/m1)(1/2)(p1) + (m2)(1/2)(v2). Furthermore, if m = m1 = m2 then the result is a quadratic equation for the mass, m - 2m(H1 + L2)/v2 + p1/v2 = 0, whose solution factors are (m - p1/ v2) = 0 iff p1v2 = H1 + L2. These derivations imply that the quantum of mass is equal to the ratio of one particle momentum over the velocity of the other particle in the system. In deriving the differential form, one starts by noting that p2 = ∂L2/∂v2 and v1 = ∂H1/∂p1, equivalently, p2 ∂v2 = ∂L2 and v1∂p1 = ∂H1. If p1 = p2 and v1 = v2 then ∂(pv) = ∂(H1 + L2) and the spacetime integral is pv = ∫∫∫∫∂(H+ L).


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