Generator (mathematics) - Differential Equations

Differential Equations

In the study of differential equations, and commonly those occurring in in physics, one has the idea of a set infinitesimal displacements that can be extended to obtain a manifold, or at least, a local part of it, by means of integration. The general concept is of using the exponential map to take the vectors in the tangent space and extend them, as geodesics, to an open set surrounding the tangent point. In this case, it is not unusual to call the elements of the tangent space the generators of the manifold. When the manifold possesses some sort of symmetry, there is also the related notion of a charge or current, which is sometimes also called the generator, although, strictly speaking, charges are not elements of the tangent space.

• Elements of the Lie algebra to a Lie group are sometimes referred to as "generators of the group," especially by physicists. The Lie algebra can be thought of as the infinitesimal vectors generating the group, at least locally, by means of the exponential map, but the Lie algebra does not form a generating set in the strict sense.

• In stochastic analysis, an Itō diffusion or more general Itō process has an infinitesimal generator.
• The generator of any continuous symmetry implied by Noether's theorem, the generators of a Lie group being a special case. In this case, a generator is sometimes called a charge or Noether charge, examples include:
  • angular momentum as the generator of rotations,
  • linear momentum as the generator of translations,
  • electric charge being the generator of the U(1) symmetry group of electromagnetism,
  • the color charges of quarks are the generators of the SU(3) color symmetry in quantum chromodynamics,
• More precisely, "charge" should apply only to the root system of a Lie group.