The Invariance of Oscillatory Mechanical Systems

Abstract

The authors of this paper begin by stating that oscillations are natural and objective phenomena that do not depend on mathematical description. Some authors base their descriptions on Newton's axioms or laws, other most frequently on Hamilton's canonic differential equations. Therefore, the very foundations of rational mechanics contain inconsistency. Newton's mechanics is concerned with solving two main problems. The first is to determine the motion with all forces being known, and the other, to determine the forces with motions being known. However, Hamilton's mechanics has only one problem integration of 2n differential (Hamilton's) equations, that are not invariant with respect to linear differential transformations. It is a widely accepted understanding that acceleration is not invariant over Riemann's multiplicities with respect to different coordinate systems, which leads to inaccurate conclusions regarding the motion itself. A more precise interpretation of these problems is stated here. It should not be concluded, that the equilibrium state, that is stable with respect to one coordinate system, is simultaneously indeterminate or instable with respect to other coordinate system. The inconsistency is especially obvious in conception of metric tensor of Riemman geometry, opposed to corresponding Lagrange's basic tensor of n dimensional configurational multiplicities.