  In the extended Lagrange formalism of classical point dynamics, the system's
dynamics is parametrized along a system evolution parameter $s$, and the
physical time $t$ is treated as a \emph{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 $L_{\e}ds$. Supposing that both Lagrangians describe the
same physical system then provides the correlation of $L$ and $L_{\e}$. In the
existing literature, the discussion is restricted to only those extended
Lagrangians $L_{\e}$ that are homogeneous forms of first order in the
velocities. As a new result, it is shown that a class of extended Lagrangians
$L_{\e}$ exists that are correlated to corresponding conventional Lagrangians
$L$ \emph{without being homogeneous functions in the velocities}. With these
extended Lagrangians, the system's dynamics is described as a motion on a
hypersurface within a \emph{symplectic extended} phase space of even dimension.
As a consequence of the formal similarity of conventional and extended Lagrange
formalisms, Feynman's non-relativistic path integral approach can be converted
into a form appropriate for \emph{relativistic} quantum physics. To provide an
example, the non-homogeneous extended Lagrangian $L_{\e}$ of a classical
relativistic point particle in an external electromagnetic field will be
presented. This extended Lagrangian has the remarkable property to be a
quadratic function in the velocities. With this $L_{\e}$, it is shown that the
generalized path integral approach yields the Klein-Gordon equation as the
corresponding quantum description. This result can be regarded as the proof of
principle of the \emph{relativistic generalization} of Feynman's path integral
approach to quantum physics.
