A global regularisation for integrating the Caledonian symmetric four-body problem

Abstract

International audienceSeveral papers in the last decade have studied the Caledonian symmetric four-body problem (CSFBP), a restricted four-body system with a symmetrically reduced phase space. During these studies, difficulties have arisen when the system approaches a close encounter. These are due to collision singularities causing numerical integration algorithms to fail. In this paper, we give the full details of a regularisation approach that now enables us to study these close encounters and collision events. The resulting equations of motion can be efficiently integrated by a high-order integrator. The results from numerical testing of the algorithm verify that the regularisation is advantageous in preserving numerical stability. The effectiveness of the approach is illustrated for a range of CSFBP orbits. Numerical experiments show that the newly developed regularisation algorithm has excellent energy conservation properties

On the C8/3-regularisation of simultaneous binary collisions in the collinear 4-body problem

“…For details on these sub-problems see, for instance, [1]. This observation agrees with the regularisation results of [2,33,35].…”

The rhomboidal symmetric four-body problem

“…A technique for regularizing the n-body problem by doubling the dimension of the phase space was suggested by Heggie (1974). This technique was applied by Sivasankaran et al (2010) to regularize the symmetric four-body problem. A potential difficulty are simultaneous binary collisions possible in this system.…”

The Eight-Body Cubic Collision-Based Periodic Orbit

“…Orbits with four bodies in one spatial dimension are featured in [29], [15], [12] and [33]. Orbits in two spatial dimensions featuring collisions were studied as early as 1979 in [7] and as recently as 2021 in [30], with other notable works including [24], [4], [31], [3], [2], [22], [34], and [1]. Additionally, in [29] and [15], large families of highly-symmetric orbits are given in one, two and three dimensions, all of which can be expressed in two degrees of freedom.…”