Equations of motion in a non-integer-dimensional space

Abstract

Equations of motion are derived for a fractional dimensional system of n-spatial coordinates to be used as an effective description of anisotropic and confined systems. An existing measure theoretic approach is extended to multiple variables and different degrees of confinement in orthogonal directions and comparisons are made with the analytic continuation of Gaussian integrals. This is applied to the variational principle, and equations of motion for a field described by a Lagrange density are found. A specific example is looked at in Schrödinger wave mechanics, particularly in three-coordinate systems.