Conservation of Motion

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The following derivation is based on the article The Principle of Conservation of the Center of Gravity and the Inertia of Energy, Albert Einstein, Annalen der Physik, 20 (1906): 626-633. There are two sections to this article. The first section is entitled A special case and is discussed in the web page Einstein’s Box. The second is entitled On the principle of the conservation of the motion of the center of gravity and is discussed below, in this web page.
    Consider a system of n charged particles (“discrete material points”) with masses m₁, m₂, … , m_n and “center of gravity” position vectors r₁, r₂, … , r_n, respectively. Assume that the following Maxwell equations hold 

The first set of three equations in Einstein’s Eq. (1) corresponds to Eq. (1a) above while the second set of three equations corresponds to Eq. (1b). The equation that follows Eq. (1) in Einstein’s paper corresponds to Eq. (1c). Einstein use’s the notation E = (X, Y, Z) and B = (L, M, N) and c = V (Note: Einstein uses a different system of units than used here).
   Einstein describes the next step in his article as follows

If one adds up equations (1) after they have successively been multiplied by

 

 

and integrates them over the entire space, one obtains, after a few integrations by parts, the following equation….

After dividing by m₀ and adding Eq. (2a) top (2b) we obtain 

This can be simplify using the following identities

After these substitutions are made the result is integrated of the volume dt to give

Revert back to charge density, r, and velocity, v, i.e. J = rv, and bring the time derivative outside the middle integral to obtain

Immediately after this Einstein writes

The first term of this equation represents the energy supplied by the electromagnetic field to the bodies m₁, m₂, … , m_n. According to our hypothesis on the dependence of the masses on energy, the first term of the sum should therefore be equated with the expression

This is a rather odd assertion because the first term is not the energy but the integral of the infinitesimal power times x.

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