Motion transformation in a higher order dimensional space

Abstract

The objective of the work presented in this paper is an attempt at solving and transforming of the known from the classical mechanics two dimensional-plane single mass mechanical and mathematical vibration models in a higher order dimensional space with any virtual sectional curvature-positive or negative, constant or variable. A characterization of the Riemannian manifolds is performed by means of curvature operators. The computer codes Mathematica and MATLAB are used in the numerical simulation. The objects of the investigation are a sphere – with a positive constant sectional curvature, a cylinder-with a zero constant sectional curvature, helicoid-with a negative variable sectional curvature, a torus-with a variable (±) sectional curvature, any virtual surface of second order-with a variable (±) sectional curvature, pseudo-sphere – with a negative constant sectional curvature and a saddle-with a negative variable sectional curvature. The system motion is investigated in a qualitative aspect in time and frequency domain on the cited surfaces. The common algorithm derived in the paper can transform any motion from 3D space to curved manifold. We can derive the trajectory in an explicit form on the curved manifold. We can change the trajectory by a suitable variation of the curved manifold. The objective of the work presented in this paper with respect to the analogical work reported before is to widen the range and to verify and to illustrate the performance of the common algorithm on a wider scale of surfaces and a comparison is derived on the behaviour of the common algorithm depending on the surface and on the surface properties qualification. The results derived are supposed to form a basis for a series of future investigations for simulation and solution of the dynamic behaviour of 3D-spatial single and double mass dynamic models in a higher order dimensional space with any virtual sectional curvature.