# Vector quantities and their components

# Vector quantities and their components

Mechanics is not only about geometric displacements, but also about concepts of kinematics involving time, such as velocity, momentum, and acceleration. In order to deal with such quantities, Heinrich Hertz introduced the general concept of a vector quantity with respect to a mechanical system, as well as the covariant components of a vector to use the modern language of tensor analysis. Hertz's geometric interpretation of the generalised momenta reveals the real strength and intuitive appeal of his geometry of systems of points. This chapter argues that Hertz's introduction of the reduced components of a displacement was partly or entirely a result of the role his geometry of systems of points was intended to play in the presentation of mechanics, in particular, in connection with the Lagrangian and Hamiltonian formalisms. Before turning to the origin of Hertz's concept of vector quantities and their components, the chapter summarises how their definitions and most important properties appear in his book *Principles of Mechanics*. The interaction between physical content and mathematical form is also discussed.

*Keywords:*
vector quantities, tensor analysis, displacement, kinematics, mechanics, geometry, covariant components, physical content, mathematical form

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