Variation of the light-like particle energy and its critical curve equations

Abstract

We consider variation of energy of the light-like particle in Riemann space-time, find lagrangian, canonical momenta and forces. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is shown to not lead to violation of conformity of varied curve to the null path in contradistinction of the interval variation. Though found equations are differ from standard form of geodesics equations, for the Schwarzschild space-time their solutions coincide with each other to within parameter of differentiation.