Integration of relativistic equations of motion

CPO Ltd (through F H Read) has developed its own technique for integrating relativistic equations of motion. Here is a description.

Starting with

m = m0/(1-(v/c)2)1/2

we have

m/ = mvv//(c2 – v2)

and

vv/ = Σi vivi/

where x/ signifies dx/dt.

Then

fi = pi/ = m/vi + mvi/ = Σjaijvj/

where

aij = m(δij + vivj/(c2 – v2) )

The solution to this can be shown to be

vi/ = Σjbijfj

where

bij = (c2δij – vivj)/mc2

It is this equation for vi/ that is used in the part of the code that deals with trajectory integration.

The many tests of these equations include test2d05, test3d04 and test3d05.

For 2D non-meridional motion the integration is more difficult, see report 478.

Another useful equation, for the total relativistic velocity v, is

(v/u)2 = (1 + r/2)/(1 + r)2

where

r = K/m0c2

and where K is the kinetic energy, and

u = (2K/m0)1/2

is the nominal non-relativistic velocity.