On the invariant integration of a vector in some problems in mechanics

TL;DR

This paper discusses the concept of invariant integration of vectors and tensors in mechanics, emphasizing its importance in solving problems without relying on Cartesian coordinates, especially in polar coordinate systems.

Contribution

It highlights the significance of invariant integration in mechanics and illustrates its application in two-dimensional Euclidean space using polar coordinates.

Findings

Invariant integration is essential for solving mechanics problems without Cartesian coordinates.

The concept, introduced by V.N. Folomeshkin, remains underexplored in research.

Application examples demonstrate the practical importance of invariant integration.

Abstract

Invariant integration of vectors and tensors over manifolds was introduced around fifty years ago by V.N. Folomeshkin, though the concept has not attracted much attention among researchers. Although it is a sophisticated concept, the operation of the invariant integration of vectors is actually required to correctly solve some problems in mechanics. Two such problems are discussed in the present exposition, in the context of a two-dimensional Euclidean space covered by a polar coordinate system. The notion of invariant integration becomes necessary when the space is described without any reference to a Cartesian coordinate system.