Vol.2, No 8, 1998 pp. 629 - 634
UDC: 531.011
ONE APROACH TO THE PROBLEM OF TWO BODIES
Veljko A. Vujičić
Mathematical Institute of Serbian Academy of Science and Arts,11001 Beograd, P.O. 367, Yugoslavia
Abstract. The motion of two bodies, viewed as material points, is known in the celestial mechanics as "the two bodies' problem". Kepler's Laws, Newton's Law of Gravity as well as Galilee's theorems about the free fall of the body are exactly related to this problem. In this paper, however, it can be taken as an example of two material points, but its relation to the above-mentioned laws makes it a problem of great importance. The formula for the gravitational force is obtained in the form:   F=chi*m1*m2/p which obviously differs from the Newton's one. For assumptions which are not in agreement with the first of Kepler's laws, the formula for F simply reduces to the Newton's law of gravitation. Under these assumptions an analytic expression for gravitational "constant" is obtained. It is shown that the numerical value of k, for all planets of the Sun-system, is very close to the widely accepted value of gravitational constant.
Two bodies as material points with masses m1 and m2 are moving towards each other in the plane z = 0 so that the distance between their centers of inertia is a known function of time p(t).
Key words: Two body problem, gravitational force, gravitational constant.
JEDAN PRISTUP PROBLEMU DVA TELA
Razmatra se zadatak određivanja sile uzajamnog dejstva dva tela - materijalne tačke, čije je me-đusobno rastojanje funkcija vremena. Pomoću Lagranžovih diferencijalnih jednačina kretanja prve vrste lako se određuje da je tražena sila obrnuto proporcionalna rastojanju materijalnih tačaka. U slucaju da posmatrane materijalne tačke pretstavljaju nebeska tela Sunčevog sistema ovaj zadatak dobija značaj problema dva tela. Nađena sila F=chi*m1*m2/p se razlikuje od Njutnove sile gravitacije. Tek za pretpostavku da je pomenuto rastojanje konstatno, što nije u saglasnosti sa Keplerovim zapažanjima dobijaju se formule za Njutnovu silu gravitacije i odgovarajuću gravitacionu konstantu.
Ključne reči: problem dva tela, gravitaciona sila, gravitaciona konstanta