Variational principles of physics and the infinite Ramsey theory

Abstract Application of the Ramsey Infinite Theorem to the fundamental variational principles of physics is addressed. The Hamilton Least Action Principle states that, for a true/actual trajectory of a system, Hamilton’s Action is stationary for the paths, which evolve from the preset initial space-time point to the preset final space-time point. The Hamilton Principle distinguishes between the actual and trial/test trajectories of the system in the configurational space. This enables the transformation of the infinite set of points of the configurational space (available for the system) into the bi-colored, infinite, complete, graph, when the points of the configurational space are seen as the vertices, actual paths connecting the vertices/ points of the configurational space are colored with red; whereas, the trial links/paths are colored with green. Following the Ramsey Infinite Theorem, there exists the infinite, monochromatic sequence of the pathways/clique, which is completely made up of actual or virtual paths, linking the interim states of the system. The same procedure is applicable to the Maupertuis’s principle (classical and quantum), Hilbert-Einstein relativistic variational principle and reciprocal variational principles. Exemplifications of the Infinite Ramsey Theorem are addressed.