ABSTRACT
Here the primes denote the time derivatives. Introduce the proper time
τ = ∫ t t0
√ 1− β2dt, (3)
and write down the action integral
S = ∫ t t0
L0dt. (4)
The variation of this action gives the equations of motion in the form
S = ∫ τ 0
L0 dt
dτ dτ. (5)
Since the value of the upper integration limit depends on the shape of the trajectory, this upper limit should also be varied. Therefore, the proper time cannot be taken for an independent (i.e., not being a subject of variation) variable for the Lagrange function L0 dtdτ .
