| Abstract |
Conservation laws of systems of partial differential equations (PDE) arise in a wide variety of applications and contexts. They often serve as mathematical expressions for fundamental physical principles, such as the conservation of mass, momentum, charge, and energy. Conservation laws of ordinary differential equations (ODE) yield first integrals. Noether celebrated a theorem now known as Noether's Theorem that establishes a relation between conserved vectors and symmetries of the Euler-Lagrange equations (variational problems). Noether's theorem works only for the variational domain and still has some limitations like it can't undergo non-variational, odd order, fractional, evolution problems, etc. We will discuss here the generalization of Noether's theorem that is more convenient and less conditional. This theorem is applicable to all domains including variational, non-variational, fractional, and evolution problems as well as odd order equations. This theorem has a vast number of applications and we will discuss a few of them.
|
