Classical Dynamics Based on the Minimal Length Uncertainty Principle

Abstract

In this paper we consider the quadratic modification of the Heisenberg algebra and its classical limit version which we call the β-deformed Poisson bracket for corresponding classical variables. We use the β-deformed Poisson bracket to discuss some physical problems in the β-deformed classical dynamics. Finally, we consider the (α,β)- deformed classical dynamics in which minimal length uncertainty principle is given by $ [ \hat {x} , \hat {p}] = i \hbar (1 + \alpha \hat {x}^{2} + \beta \hat {p}^{2} ) $ . For two small parameters α,β, we discuss the free fall of particle and a composite system in a uniform gravitational field.