The discussion focuses on the implications of Simple Harmonic Motion (SHM) when velocities approach relativistic speeds. It establishes that the traditional equations for SHM, such as a = -ω²y, do not apply under these conditions, leading to a different set of differential equations that are more complex due to their non-linearity. The relativistic extension of SHM is explored in the paper "Relativistic (an)harmonic oscillator" by Moreau, Easther, and Neutze, which details the derivation of relativistic equations of motion and their analytical solutions, highlighting the effects of time dilation and the anharmonic nature of motion at high velocities.
PREREQUISITESPhysicists, students of advanced mechanics, and anyone interested in the intersection of classical mechanics and relativistic physics will benefit from this discussion.
Washer101 said:On The Student Room I saw a couple of guys looking at whether the rules for Simple Harmonic Motion at non-relativistic speeds (e.g. a = -w2y and so on) would work if the maximum velocity of the particle hits relativistic speeds.
Many Thanks
Have a look at
Relativistic (an)harmonic oscillator
Moreau, William; Easther, Richard; Neutze, Richard
American Journal of Physics, Volume 62, Issue 6, pp. 531-535 (1994).
The relativistic extension of one-dimensional simple harmonic motion is developed in the Lagrangian formalism. The relativistic equations of motion are derived and solved analytically. The motion with respect to proper time is analyzed in terms of an effective potential energy. While the motion remains bounded and periodic, the effect of time dilation along the world line is to cause it to become anharmonic with the period increasing with amplitude and the curvature concentrated at the turning points.