We obtain the expressions for the energy and momentum of a relativistic particle by incorporating the equivalence of mass and energy into Newtonian mechanics.
With the relativistic energy established we find that an infinite amount of energy is required to accelerate a particle to light speed. The existence of this ultimate speed calls us to question Galilean relativity, and with it the structure of Newtonian space and time.
All integrals will be taken from 0 to v.
This can also be seen from the recursion formula, Eq. (16), which indicates that En is of order times .
This is similar to the Frobenius method of solving differential equations by means of power series.
Just as with a physical system, the same equations started with different initial conditions will generate different results.
The analogy with Frobenius theory can be made exact at this point since we have a differential equation for E, which we could expand as a series in v.