Probabilistic Evolutions in Classical Dynamics: Conicalization and Block Triangularization of Lennard-Jones Systems

This work aims at the solution of two-body problem with Lennard-Jones potential via recently developed Probabilistic Evolution Approach (PEA) proposed by M. Demiralp for explicit ODEs, as final target. PEA has been quite well developed and it is well-known that the conicality in descriptive functions facilitates the construction of the truncation approximants. Conicality, if does not exist, can be obtained by using the space extension method at the expense of an increase in the number of the unknowns and may bring the block triangularity which is another important facility to investigate and control the properties of PEA. In space extension we construct the Hamilton equations of the motion for the system first and then define new appropriate unknown functions depending on the existing unknowns such that the resulting ODEs to be used in PEA become conical. This attempt provides us with the block triangularity at the same time.