  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. 
Extended Hamilton-Lagrange formalism and its application to Feynman's   path integral for relativistic quantum physics