## 375(10): Lagrangians for the General Two Body Problem in Gravitation

This note gives the lagrangians and Euler Lagrange equations for the non relativistic and relativistic orbits of any mass m1 around any mass m2 interacting with an assumed central potential. The three equations of motion are (19) to (21) and can be solved in any coordinate system, planar or non planar. Eq. (19) gives an ellipse, and Eqs. (20) and (21) must be solved simultaneously with Maxima. The orbits can become intricate even in the two body problem. This is almost always solved using m2 >> m1, e.g. the earth sun system. In a binary pulsar however m1 and m2 are about the same. The relativistic lagrangian is Eq. (24), and can be reduced to Eq. (40), which can be solved numerically using the Euler Lagrange equation (41). The relations (38) and (39) can be used to construct two more relativistic orbital equations. Eqs. (42) and (43) give the masses of the stars of the Hulse Taylor binary pulsar in kilograms, S. I. units. The initial condition can be taken to be the periastron, which is the minimum separation of m1 and m2 for the Hulse Taylor pulsar. This is 7.6527 ten power eight metres. The initial velocity relative to the centre of mass at the periastron is 4.50 ten power five metres per second. Various precessions can be calculated from the lagrangian. The initial velocity could be used as an input parameter to see how the orbits vary. The precession is observed to be 4.2 degrees per earth year. The non relativistic orbit of m1 about m2 is given by Eq. (8) and the relativistic orbit is given by Eqs. (40) and (41). These must be be solved numerically. So I will proceed to writing up UFT375, Sections 1 and 2 as usual. The main theme is to find precessions with the ECE2 lagrangian, Eq. (40). However during the course of writing the notes, a wild error of eight orders of magnitude has been found in the Einstein theory in the S2 star system. The weak field Einstein theory also fails completely, by orders of magnitude, in the Hulse Taylor binary pulsar. The theory of this note can also be applied to the S2 star system with m2 >> m1. The most general n body problem of gravitation is exceedingly intricate, and cannot be solved analytically. Even the two body problem, when correctly solved as in this note, gives very intricate orbits when two approximately equal masses orbit each other.

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