The Harmonic Oscillator in Extended Relativistic Dynamics
Extended relativistic dynamics is a speculative modification of special relativity that proposes a maximum possible value of the acceleration. This Demonstration shows the velocity and acceleration of the extended relativistic harmonic oscillator compared to the classical harmonic oscillator as functions of the natural frequency and amplitude .
This Demonstration shows how the motion of a harmonic oscillator behaves according to "extended relativity" [1, 2], as compared to the classical model. In extended relativity, it is assumed that both the velocity and acceleration are bounded: the velocity by , and the acceleration by , a value which, by some indications, approximates .
For the classical model of the harmonic oscillator, the Hamiltonian is
For the extended relativistic model of the harmonic oscillator, the Hamiltonian is generalized to [1, 2, 3]
By solving the Hamiltonian system
we find the motion of the harmonic oscillator.
In the classical model, the effective frequency (, where is the period of the motion) and the natural frequency are equal. In extended relativity, however, the effective frequency is considerably smaller. It can be shown [2, 3] that as the natural frequency goes to infinity, the effective frequency approaches
 Y. Friedman, "The Maximal Acceleration, Extended Relativistic Dynamics and Doppler Type Shift for an Accelerated Source," Annalen der Physik (Berlin), 523(5), 2011 pp. 408–416. arxiv.org/abs/0910.5629.
 Y. Friedman, "Extending the Relativity of Time," Proceedings of IARD 2012, 8th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields, Florence, Italy. Journal of Physics: Conference Series, 437, 2013 #012017. doi:10.1088/1742-6596/437/1/012017.
 Y. Friedman, "Digitization of the Harmonic Oscillator in Extended Relativity," Physica Scripta 87(6), 2013 #065702. doi:10.1088/0031-8949/87/06/065702.