Exactly Solvable Dynamical Models with a Minimal Length Uncertainty

We present exact analytical solutions to the classical equations of motion and analyze the dynamical consequences of the existence of a minimal length for the free particle, particle in a linear potential, anti-symmetric constant force oscillator, harmonic oscillator, vertical harmonic oscillator, linear diatomic chain, and linear triatomic chain. It turns out that the speed of a free particle and the magnitude of the acceleration of a particle in a linear potential have larger values compared to the non-minimal length counterparts - the increase in magnitudes come from multiplicative factors proportional to what is known as the generalized uncertainty principle parameter. Our analysis of oscillator systems suggests that the characteristic frequencies of systems also have larger values than the non-minimal length counterparts. In connection with this, we discuss a kind of experimental test with which the existence of a minimal length may be detected on a classical level.