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Further, in the new system of coordinates the figure formed by the world-lines differs from that figure in the old system by the variation which is a function of only. Therefore according to (73) the second variation of is

By putting equal to zero the sum of this expression and the preceding one we obtain (76).

§ 46. We have thus deduced for some cases the equations of the gravitation field from the variation theorem. Probably this can also be done for thermodynamic systems, if the Lagrangian function is properly chosen in connexion with the thermodynamic functions, entropy and free energy. But as soon as we are concerned with irreversible phenomena, when e.g. the energy-current consists in a conduction of heat, the variation principle cannot be applied. We shall then be obliged to take Einstein's field-equations as our point of departure, unless, considering the motions of the individual atoms or molecules, we succeed in treating these by means of the generalized principle of Hamilton.

§ 47. Finally we shall consider the stresses, the energy etc. which belong to the gravitation field itself. The results will be the same for all the systems treated above, but we shall confine ourselves to the case of §§ 44 and 45. We suppose certain external forces to act on the material points, though we shall see that strictly speaking this is not allowed.

For any displacements of the matter and variations of the gravitation field we first have the equation which summarizes what we found above

In virtue of the equations of motion of the matter, the terms with cancel each other on the right hand side and similarly, on account of the equations of the gravitation field, the terms with and . Thus we can write[1]

  1. To make the notation agree with that of § 38 has been replaced by .