Editing One-parameter group (section)

==Physics==
In [[physics]], one-parameter groups describe [[dynamical systems]].<ref>Zeidler, E. (1995) ''Applied Functional Analysis: Main Principles and Their Applications'' Springer-Verlag</ref> Furthermore, whenever a system of physical laws admits a one-parameter group of [[derivative|differentiable]] [[symmetry group|symmetries]], then there is a [[Conservation law (physics)|conserved quantity]], by [[Noether's theorem]].

In the study of [[spacetime]] the use of the [[unit hyperbola]] to calibrate spatio-temporal measurements has become common since [[Hermann Minkowski]] discussed it in 1908. The [[principle of relativity]] was reduced to arbitrariness of which diameter of the unit hyperbola was used to determine a [[world-line]]. Using the parametrization of the hyperbola with [[hyperbolic angle]], the theory of [[special relativity]] provided a calculus of relative motion with the one-parameter group indexed by [[rapidity]]. The ''rapidity'' replaces the ''velocity'' in kinematics and dynamics of relativity theory. Since rapidity is unbounded, the one-parameter group it stands upon is non-compact. The rapidity concept was introduced by [[E.T. Whittaker]] in 1910, and named by [[Alfred Robb]] the next year. The rapidity parameter amounts to the length of a [[versor#Hyperbolic versor|hyperbolic versor]], a concept of the nineteenth century. Mathematical physicists [[James Cockle (lawyer)|James Cockle]], [[William Kingdon Clifford]], and [[Alexander Macfarlane]] had all employed in their writings an equivalent mapping of the Cartesian plane by operator <math>(\cosh{a} + r\sinh{a})</math>, where <math>a</math> is the hyperbolic angle and <math>r^2 = +1</math>.