Relativistic mechanics of a particle

As measured in some inertial reference frame, the instantaneous particle velocity is v = (dx/dt, dy/dt, dz/dt). It is customary to define noninvariant parameters yd = v/c and y = (1 — yd2)-A The definition of the invariant interval ds2 implies that dr2 = dt2(l - yd2), where dt is the measured time interval. Hence = y, as 

Unlike classical mechanics, the 4-acceleration is not independent of the 4-velocity. Because ulxulx is invariant, its r-derivative must vanish, so that 

This implies a relationship between classical velocity and acceleration, 

The variational formalism makes it possible to postulate a relativistic Lagrangian that is Lorentz invariant and reduces to Newtonian mechanics in the classical limit. Introducing a parameter m, the proper mass of a particle, or mass as measured in its own instantaneous rest frame, the Lagrangian for a free particle can be postulated to have the invariant form A = mulxiilx = — mc2. The canonical momentum is pf, = iiiuj, and the Lagrangian equation of motion is 

The free-particle Lagrangian A is a space-time constant — mc2. If terms are added that are invariant functions of x/2, the equations of motion become 

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