  The well known relation of Einstein relativistic energy for a free particle
is extended to cover the total relativistic energy of a bound particle by
calculating the relativistic potential energy. A non dissipative harmonic
oscillator (NDHO) is a fundamental bound system. Therefore, the potential
energy of an NDHO is analytically extended from the non relativistic to the
relativistic regime for the first time. This study is essentially concerned
with the relativistic mass, where the Lorentz factor transforms the state of
the second-order differential equation of an NDHO from linear into nonlinear.
The results are finally confirmed by demonstrating energy conservation since
the sum of kinetic and potential energies remains constant throughout the non
relativistic and relativistic regimes.
