Abstract
In the present work the geodesic equation represents the equations of motion of the particles along the geodesics is derived. The deviation of the curved space-time metric tensor from that of the Minkowski tensor is considered as a perturbation. The quantities is expanded in powers of c-2. The equations of motion of the relativistic three body problem in the PN formalism are obtained.
Highlights
The Three body problem concerns with the motion of a small particle of negligible mass moving under the gravitational influence of two massive objects m1 and m2
In 1772, Euler [5] first introduced a synodic coordinate system, the use of which led to an integral of the equations of motion, known today as the Jacobian integral
Euler himself did not discover the Jacobia integral which was first given by Jacobi [1] who, as Wintner remarks, “rediscovered” the synodic system
Summary
The Three body problem concerns with the motion of a small particle of negligible mass moving under the gravitational influence of two massive objects m1 and m2. It’s history began with Euler and Lagrange continues with Jacobi [1], Hill [2], Poincaré [3], and Birkhoff [4]. In 1772, Euler [5] first introduced a synodic (rotating) coordinate system, the use of which led to an integral of the equations of motion, known today as the Jacobian integral. Euler himself did not discover the Jacobia integral which was first given by Jacobi [1] who, as Wintner remarks, “rediscovered” the synodic system. The actual situation is somewhat complex since Jacobia published his integral in a sideral (fixed) system in which its significance is definitely less than in the synodic system.
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