The Noether Symmetries and Invariants of Some Partial Differential Equations

Abstract

A connection is obtained between isometries and Noether symmetries for the area-minimizing La- grangian. It is shown that the Lie algebra of Noether symmetries for the Lagrangian minimizing an (n − 1)-area enclosing a constant n-volume in a Euclidean space is so(n) ⊕s Rn and in a space of constant curvature the Lie algebra is so(n). Here for the non-compact space this has to be taken in the sense of being cut at a fixed boundary that respects the symmetry of the space and is not a volume enclosing hypersurface otherwise. Further if the space has one section of constant cur- vature of dimension n1 , another of n2 , etc. to nk and one of zero curvature of dimension m, with n≥ k j=1 nj + m (as some of the sections may have no symmetry), then the Lie algebra of Noether symmetries is ⊕k so(nj + 1) ⊕ (so(m) ⊕s Rm ). j=1 For a subclass of the general class of linear hyperbolic systems, obtainable from complex base hy- perbolic equation, semi-invariant and joint invariants are investigate by complex and real symmetry analysis. A comparison of all the invariants derived by complex and real methods is presented here which shows that the complex procedure provides a few invariants different from those extracted by real symmetry analysis for a linear hyperbolic system. The equations for the classification of symmetries of the scalar linear elliptic equation are obtained in terms of Cotton’s invariants. New joint differential invariants of the scalar linear elliptic equations in two independent variables are derived, in terms of Cotton’s invariants by application of the infinitesimal method. Joint differential invariants of the scalar linear elliptic equation are also derived from the bases of the joint differential invariants of the scalar linear hyperbolic equation under the application of the complex linear transformation. We also find a basis of joint differential invariants for such equations by utilization of the operators of invariant differentiation. The other invariants are functions of the bases elements and their invariant derivatives. Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by splitting the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear complex transformations of the independent variables. It is shown that Cotton-type invariants derived from these two approaches are identical. Furthermore, Cotton-type and joint invariants for a general system of two linear elliptic equations are also obtained from the Laplace-type and joint invariants for a system of two linear hyperbolic equations equivalent to the system of linear elliptic equations by complex changes of the independent variables.