分形空间中力学规律的适应性原理与标度不变性原理

The Adaptability Principle of Mechanical Law and the Scale-invariant Principle of Mechanical Law in Fractal Space

本文应重整化群的参数空间变换和标度空间变换的方法证明了分形空间的力学规律的适应性原理和标度不变性原理。从空间变换的角度讲,从分形空间的物理规律到欧式空间的物理规律变换就是标度不变性的变换,从欧氏空间的物理规律到分形空间的物理规律变换就是一个适应性变换。其推论是分形空间的能量守恒定律、力的合成及其位移的合成在分形空间上保持形式不变。同时,Carpinteri的分形空间的空间维数关系利用重整化方法获得证明,即分形空间的体积的维数溢出量等于分形空间的截线和截面的维数溢出量之和。

The adaptability principle of mechanical law and the scale-invariant principle of mechanical law in fractal space are proved by using parameter - space and scale-space transforms in renormalization groups. From the space-transform angle, the transform of mechanical law from fraetal space to European space is the scale-invariant transform while the transform of mechanical law from European space to fractal space is the adaptability transform. Their deductions are that law of conservation of energy and vectorial resultant of force and displacement in fractal space hold the line in form and Carpinteri＇s dimensional formula of fractal space is also proved. Namely, the spilling dimension of volume in fractal space equals to the sum of the spilling dimension of transversal and section.