Appendix A: Indicial Notation

Abstract

Lagrangian dynamics and the various variants in use (Hamilton’s equations, Kane’s equations, the method of quasicoordinates and the Kane-Hamilton synthesis introduced in this text) use abstract vector spaces (for example configuration space and state space). These are K dimensional vector spaces with length defined as the square root of the dot product of a vector with itself. Two vectors are perpendicular (or orthogonal) if their dot product is zero. The dot product is defined as one would expect by analogy to the dot product for ordinary vectors: the product of the first pair of components, plus the product of the second pair of components and so on out until all K pairs of components have been multiplied and added. Ordinary vector notation, either classical or in the context of linear algebra does not suffice for everything one wants to do, so I will introduce an indicial notation closely related to that of tensor analysis. It will not be tensor analysis, but the reader familiar with tensors will find much that is familiar in this appendix.