Orbital energy in the context of Vis-viva equation

In the gravitational two-body problem, the specific orbital energy ${\displaystyle \varepsilon }$ (or specific vis-viva energy) of two orbiting bodies is the constant quotient of their mechanical energy (the sum of their mutual potential energy, ${\displaystyle \varepsilon _{p}}$, and their kinetic energy, ${\displaystyle \varepsilon _{k}}$) to their reduced mass.

According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time:$${\displaystyle {\begin{aligned}\varepsilon &=\varepsilon _{k}+\varepsilon _{p}\\&={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=-{\frac {1}{2}}{\frac {\mu ^{2}}{h^{2}}}\left(1-e^{2}\right)=-{\frac {\mu }{2a}}\end{aligned}}}$$where