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The physical process of a field depends on twin sets of equation system, one is the basic equations satisfied by the physical variables, and the other is the equations satisfied by independent functions included in general solution. The conservation laws for the independent functions included in general solution are called non-classical or generalized Noether's conservation laws. This kind of conservation laws has not been focused on and is a theoretical lack of many physical fields.
The scientific research focuses on the discovery of physical innatequality, namely conservation laws. The original Noether's theorem focuses on the classical conservation laws from an action integral of Lagrangian of basic physical variables. However, as already discussed, Noether's theorem represents a generalized mathematical identity because the quantity's invariance of functional is equivalent to the mathematical form's invariance of its Euler-Lagrange equations (PDEs). That is, for any nondegenerate functional, the generalized Noether's identity includes two parts: one is a possible conservation quantity Pj, and the other is Euler-Lagrange expression Eα
where Dj is the total derivative and Qα is the characteristics.
The generalized Noether's identity (1) can be used for the following two aspects: (a)By replacing Eα in Eqs. (1) by PDEs, their functionals can be constructed; (b)When this kind of functionals is known, their conservation laws follow by Noether's identity (1).
With the help of the above two aspects, a functional can be constructed from the PDEs satisfied by functions included in the general solutions. Also, a functional can be constructed from the PDEs satisfied by some special physical variables, such as displacement, strain, stress, Airy stress function, the first stress and strain invariants. All the conservation laws from this kind of functionals are called non-classical or generalized Noether's conservation laws. The status of this kind of conservation laws is shown in Fig. 1.