  In this paper, we argue in favor of first-order homogeneous Lagrangians in
the velocities. The relevant form of such Lagrangians is discussed and
justified physically and geometrically. Such Lagrangian systems possess
Reparametrization Invariance (RI) and explain the observed common Arrow of Time
as related to the non-negative mass for physical particles. The extended
Hamiltonian formulation, which is generally covariant and applicable to
reparametrization-invariant systems, is emphasized. The connection between the
explicit form of the extended Hamiltonian $\boldsymbol{H}$ and the meaning of
the process parameter $\lambda$ is illustrated. The corresponding extended
Hamiltonian $\boldsymbol{H}$ defines the classical phase space-time of the
system via the Hamiltonian constraint $\boldsymbol{H}=0$ and guarantees that
the Classical Hamiltonian $H$ corresponds to $p_{0}$ -- the energy of the
particle when the coordinate time parametrization is chosen. The
Schr\"odinger's equation and the principle of superposition of quantum states
emerge naturally. A connection is demonstrated between the positivity of the
energy $E=cp_{0}>0$ and the normalizability of the wave function by using the
extended Hamiltonian that is relevant for the proper-time parametrization
