Conservation integrals in the sense of Noether’s theorem for an analytic function on a physical plane and application

Conservation integrals in the sense of Noether’s theorem for an analytic function on a physical plane are investigated. For any physical system described by a continuum field theory, when its general solution under the consideration of plane problem can be expressed in terms of analytic functions, there always exist these kinds of conservation integrals. Especially, it is found that any conformal transformation satisfying Cauchy–Riemann equations is the symmetry-transformation for obtaining these kinds of conservation integrals. Since conformal transformations include the translation, rotation and scale change of planner coordinates, the obtained conservation integrals possess universality and diversity. By adjusting the conformal transformation, not only there are countless conserved quantities and conservation integrals indicated by Noether’s theorem in physics, but also these kinds of conservation integrals accord with the imaginary part of Cauchy’s integral theorem mathematically. In order to show some insight into application, the breakdown at the sharp point of a conductor, the steady streaming flow past a spinning cylinder and a crack in plane elasticity are considered. Some conserved quantities and parameters with a kind of physical invariance are presented. Especially, the energy release rate for crack extension and T stress for path selection can be expressed.

► Conservation integrals of analytic functions are established by Noether’s theorem. ► There exist countless conservation integrals when changing conformal mapping. ► A finite value can always be obtained from the obtained conservation integral. ► A model to predict the breakdown at the sharp point of a conductor is presented. ► Expressions for crack extension force and T stress are presented.