1 “Sir Alan gave but never broke Like steel he studied and like oak, He brought up science by his hand The royal smith kept on the brand!” The invariant integral: some news Genady P. Cherepanov Honorary Life Member, the New York Academy of Sciences, USA (Elected on Dec. 8, 1976 together with Linus C. Pauling and George Polya) genacherepanov@hotmail.com Abstract. Earlier, this author introduced the invariant integral as a general mathematical tool for solving the physical problems based on conservation laws, without using partial differential equations, similarly to the calculus of variations. In this paper, the invariant integral was introduced for cosmic, gravitational, electromagnetic, and elastic fields combined. In a particular case of the united cosmicgravitational field, from the corresponding invariant integral the force F acting upon point mass m from point mass M and from the cosmic field was derived : . Here: G is the gravitational constant, Λ is the cosmological constant, and R is the distance between the masses. The first term provides Newton’s gravitation and the second term the cosmic repulsion. This force was used to build an elementary non-relativistic cosmological model of Universe and estimate the size of Universe as well as explain the accelerated expansion of Universe recently observed by astrophysicists. The orbital speed of stars in galaxies was found out to be constant and equal to about 250 km/s. Keywords: invariant integral, cosmic and gravitational field, interaction force, expansion of Universe 1. Introduction The integrals which are invariant with respect to the integration contour or surface provide a way to write down the laws of conservation of energy, mass, momentum and so on. From them, one can derive the differential equations as the local representation of the same conservation laws. However, the invariant integral approach is more powerful because it allows one to also deal with the field singularities where the differential equations have no meaning. In 1967, using the energy conservation law, this author derived the main invariant integral for elastic and inelastic materials and introduced it into fracture science [1]. In this approach, the invariance of the integral with respect to any integration paths followed from the energy conservation law, so that this fact seemed to be trivial and was not discussed. Particularly, the characterizing index of power-law hardening materials and the similarity were found out, which constituted the basis of the later HRR approach. In 1968, while studying strain concentration by notches and cracks, Jim Rice
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