On the General Covariant Formalism of the Analytical Mechanics

This paper examines a mathematical reformulation of classical analytical mechanics based on the principle of general covariance in general relativity. While traditional classical mechanics treats time as an independent and absolute variable, general relativity assigns time the same status as other physical quantities—an essential shift in perspective for formulating a theory of quantum gravity. Following the framework of Carlo Rovelli, this study introduces a generally covariant formulation of analytical mechanics that eliminates the privileged role of time and treats all observables equally. Beginning with a review of classical formulations such as Lagrangian, Hamiltonian, and Hamilton– Jacobi mechanics, the paper develops a relativistic formulation grounded in presymplectic geometry that removes time parametrization. The structure of phase space with constraints and the gauge invariance of the equations of motion are also addressed, demonstrating that physical motion is not defined by the flow of time, but rather by the interrelations among observables. This approach offers profound implications for the philosophical and mathematical foundations of physics and provides a theoretical basis for quantum gravity.