Classical Dynamics Based on the Minimal Length Uncertainty Principle

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.