Ramsey Approach to Dynamics: Ramsey Theory and Conservation Laws

We propose the Ramsey approach for the analysis of behavior of isolated mechanical systems containing interacting particles. The total momentum of the system in the frame of the center of masses is zero. The mechanical system is described by the Ramsey, bi-colored, complete graph. Vectors of momenta of the particles pi serve as the vertices of the graph. We start from the graph representing the system in the frame of the center of masses, the momenta of the particles in this system are pcmi. If (pcmi(t)∙pcmj(t))≥0 is true, vectors of momenta of the particles numbered i and j are connected with the red link; if (pcmi(t)∙pcmj(t))<0 takes place, the vectors of momenta are connected with the green link. Thus, the complete, bi-colored graph emerges. Consider the isolated system built of six interacting particles. According to the Ramsey Theorem, the graph inevitably comprises at least one monochromatic triangle. The coloring procedure is invariant relatively rotations/translations of frames; thus, the graph representing the system contains at least one monochromatic triangle in any of frames emerging by rotation/translation of the original frame. This gives rise to the novel kind of a mechanical invariant. Similar coloring is introduced for the angular momenta of the particles. However, the coloring procedure is sensitive to Galilean/Lorenz transformations. Extensions of the suggested approach are discussed.