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Finslerian geometrodynamics

We construct a unified framework of geometrodynamics based on the Finsler geometry to reveal the relationship between spacetime and dynamics.The Lagrangian of electron in electromagnetic field as the Finsler function gives the Finslerian metric, which modifies spacetime metric in the Finsler-Randers space. The geodesic equation gives the effective mass, forces, and effective (or geometric) fields. Using the Chern connection, we construct the generalized Einstein-Maxwell equations. In the local natural basis, we give generalized Maxwell equations and wave equations. We find that the geometric field couples with electromagnetic field and gives effective charges and currents. We analyze several typical cases, such as flat spacetime, vacuum and Berwald structure. We find that the electromagnetic field vanishes, but there still exists the magnetic potential in the Berwald space. These results provide some hints to understand some puzzles, such as axion and dark energy. These formulations stimulate some clues to explore deeper geometric structures behind physical phenomena.