Deformed Hamilton Mechanics in Noncommutative Phase Space

Based on the correspondence between operator commutative relations and Poisson brackets, we develop a framework of deformed Hamilton, Lagrange and Euler equations as well as their relations in the noncommutative phase space. We introduce a deformed factor and deformed matrix to measure the departure of the deformed symplectic structure from the canonical symplectic structure. We endow the noncommutative parameters with the Planck length, Planck constant and cosmological constant, which allows us to explore some puzzles from the Planck to cosmological scales. We find that there exist an observer-dependent effective force and moment of force break the translation and rotation symmetries. As a spacetime quantum fluctuation, these formulations and results provide some hints and insights into some unsolved phenomena such as intrinsic spacetime singularities, black hole radiation, dark matter and dark energy as well as anisotropic cosmic radiation background.