With this paper, a consistent and comprehensive treatise on the foundations of the extended Hamilton-Lagrange formalism will be presented. In this formalism, the system's dynamics is parametrized along a system evolution parameter $s$, and the physical time $t$ is treated as a dependent variable $t(s)$ on equal footing with all other configuration space variables $q^{i}(s)$. In the action principle, the conventional classical action $L dt$ is then replaced by the generalized action $L_{e}ds$, with $L$ and $L_{e}$ denoting the conventional and the extended Lagrangian, respectively. It is shown that a class of extended Lagrangians $L_{e}$ exists that are correlated to corresponding conventional Lagrangians $L$ without being homogeneous functions in the velocities. Then the Legendre transformation of $L_{e}$ to an extended Hamiltonian $H_{e}$ exists. With this class of extended Hamiltonians, an extended canonical formalism is presented that is completely analogous to the conventional Hamiltonian formalism. The physical time $t$ and the negative value of the conventional Hamiltonian then constitute and an additional pair of conjugate canonical variables. The extended formalism also includes a theory of extended canonical transformations, where the time variable $t(s)$ is also subject to transformation. In the extended formalism, the system's dynamics is described as a motion on a hypersurface within an extended phase space of even dimension. With the extended Lagrangian $L_{e}$, it is shown that the generalized path integral approach yields the Klein-Gordon equation as the corresponding quantum description. Moreover, the space-time propagator for a free relativistic particle will be derived. These results can be regarded as the proof of principle of the relativistic generalization of Feynman's path integral approach to quantum physics.
Time evolution operator for a $\left\{ h(1) \oplus h(1) \right\} \uplus u(2)$ time-dependent quantum Hamiltonian; a self-consistent resolution method based on Feynman's disentangling rules
The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the Hermitian quantum theory, i.e., the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the open system (mass or energy exchange systems) and nonconservative force systems or dissipative systems. On this basis, we have given the generalized Lagrange function, it has to do with the kinetic energy, potential…
The Dirac's formalism for constrained systems is applied to the analysis of time-dependent Hamiltonians in the extended phase space. We show that the Lewis invariant is a reparametrization invariant and we calculate the Feynman propagator using the extended phase description. We show that the quantum phase of the Feynman propagator is given by the boundary term of the canonical transformation of the extended phase space.
