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

Relativistic potential energy of a bound particle (Non-dissipative relativistic harmonic oscillator)