Center of mass references

 Classical Electrodynamics 2^nd Ed., Jackson, John Wiley & Sons, (1975). From page 617, Problem 12.16

(b) Show that when b = 0 the conservation law is

                       

where X is the coordinate of the center of mass of the electromagnetic fields, defined by

                       

where u is the electromagnetic energy density and E_em and P_em are the total energy and momentum of the fields.

Relativity: Special, General and Cosmological, Rindler, Oxford Univ., Press, (2001). From page 126, Problem 6.5 

6.5. The position vector of the center of mass (CM) of a system of particles in any inertial frame is defined by , the ms being the relativistic masses. By considering two equal particles traveling in the opposite directions along parallel lines, show that the CM of the system in one IF does not necessarily coincide with its CM in another IF. Prove that, nevertheless, if the particles of the system suffer collision forces only, the CM in every IF moves with the velocity of the ZM frame [Hint:  are constant;  is zero between collisions and at any collision we can factor out the r of the participating particles .]

 The Variational Principles of Mechanics – 4^th Ed., Lanczos, Dover Pub., (1970).  From page 393

If k takes the values 1, 2, 3 and j = 4, we have obtained the last three conservation laws in the form

                            (23.6)

If we defined the center of energy of the system by putting

                  (23.7)

 then (23.6) becomes

                                                (23.8)

 or

                                                     (23.9)

 Thus we have obtained a purely kinematic definition of the total momentum, in analogy with Newton’s definition: “The total momentum of a body is equal to the total energy times the velocity of the centre of energy.” Newton speaks of the mass of the body instead of its energy. The replacement of the word “mass” by “energy” is in complete harmony with Einstein’s fundamental discovery � derived from the principle of relativity (1906; cf. p. 292) � that mass and energy are identical (…).

Classical Field Theory: Electromagnetism and Gravitation, Low, John Wiley & Sons,(1997), From page 277-278, 

The importance of symmetry is that it makes possible the construction of six more global constants by defining a third-rank tensor

                    (6.6.24)

M^mnl is conserved with respect to the m index:

provided that T^mn = T^nm.

     Therefore, we have six global constants constructed as usual 

     (6.6.25) 

The space components Lij define an angular momentum 

                                                          (6.6.26)

where pⁱ = T⁰ⁱ is the momentum density. The components of L_0i are 

        (6.6.27)         

where P0 is the total energy: 

                                             (6.6.28) 

 is the center of energy 

                            (6.6.29) 

And Pi and x0 are, as usual, total momentum and time. We thus learn that for a relativistic system, dL_ij /dx⁰ =0, or 

                                         (6.6.30) 

This is as close to a center of mass theorem as one can come in a relativistic theory: The center of energy moves with constant velocity, v_c = P/W. None of the other center-of-mass theorems of nonrelativistic mechanics hold.