The present authors have given a mathematical model of Mach's principle and of the Mach-Einstein doctrine about the complete induction of the inertial masses by the gravitation of the universe. The analytical formulation of the Mach-Einstein doctrine is based on Riemann's generalization of the Lagrangian analytical mechanics (with a generalization of the Galilei transformation) on Mach's definition of the inertial mass and on Einstein's principle of equivalence. All local and cosmological effects, - which are postulated as consequences of Mach's principle by C Neumann, Mach, Friedländer and Einstein - result from the Riemannian dynamics with the Mach-Einstein doctrine. In celestial mechanics it follows, in addition, Einstein's formula for the perihelion motion, too. In cosmology, the Riemannian mechanics yields two models of an evolutionary universe with the expansion laws R ~ t or R ~ t2. In this paper secular consequences of the Mach-Einstein doctrine are examined concerning palaeogeophysics and celestial mechanics. The research resulted in a secular decrease of the Earth's flattening and in the secular acceleration of the motion of the Moon and of the planets. The numerical values of this secular effect agree very well with the empirical facts. In all cases, the secular variation a of the parameter a is of the order of magnitude a=-H0a, where H0 means the instantaneous value of the Hubble constant: H0=(R/R0)˜(0.5-1.0)·10-10a-1. The relation of the secular consequences of the Mach-Einstein doctrine to those of Dirac's hypothesis on the expanding Earth, and to Darwin's theory of tidal friction are also discussed.
The general problem of solar variability (including the solar constant and various minima) is discussed in detail. We have no theoretical arguments about the amplitudes of the solar cycle, today. A new point is the statement of White et al. (1992) on the temperature of solar-type non-cycling stars. The theorems of the creation and annihilation of vorticity and magnetical fields prove that these processes essentially have a thermodynamical component.
Euler's results concerning the theory of gravitation are discussed in this paper in order to give an insight into the scientific problems of Euler's age and his relation to Newton's work. Geophysical aspects are emphasized in his work.
The general relativistic and covariant differential form of Helmholtz's first vorticity theorem is presented. We prove in relation with it an invariant kinematic identity which is the generalisation of the Helmholtz theorem for general continua.