In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism from the real line
One-parameter groups were introduced by Sophus Lie in 1893 to define infinitesimal transformations.
The action of a one-parameter group on a set is known as a flow.
A smooth vector field on a manifold, at a point, induces a local flow - a one parameter group of local diffeomorphisms, sending points along integral curves of the vector field.
being the group of unitary operators on a Hilbert space.
See Stone's theorem on one-parameter unitary groups.
In his monograph Lie Groups, P. M. Cohn gave the following theorem: In physics, one-parameter groups describe dynamical systems.
[4] Furthermore, whenever a system of physical laws admits a one-parameter group of differentiable symmetries, then there is a 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.
Whittaker in 1910, and named by Alfred Robb the next year.
The rapidity parameter amounts to the length of a hyperbolic versor, a concept of the nineteenth century.
Mathematical physicists James Cockle, William Kingdon Clifford, and Alexander Macfarlane had all employed in their writings an equivalent mapping of the Cartesian plane by operator
An important example in the theory of Lie groups arises when
as This result can be used, for example, to show that any continuous homomorphism between matrix Lie groups is smooth.
is constructed by winding a straight line round
In that case the induced topology may not be the standard one of the real line.