Tham khảo tài liệu 'classical mechanics - 3rd ed. - goldstein, poole & safk episode 3 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Noether s Theorem 595 Equations form the main conclusion of Noether s theorem which thus says that if the system or the Lagrangian density has symmetry properties such that conditions 2 and 3 above hold for transformations of the type of Eqs. then there exist r conserved quantities. The conservation of the stress-energy tensor is easily recovered as a special case of Eq. . If c does not contain any of the XM then it and therefore the action integral will be invariant under transformations such as Eq. where À takes on all the values ỊI. Equation then reduces to d d dxv P P which is identical with Eqs. with TpV given by Eq. . A large number of other symmetries are covered by transformations of the form of Eq. . One of the most interesting is a family of transformations of the field variables only called gauge transformations of the first kind such that ỖX 0 Sĩỉp ecptip no summation on p where the Cp are constants. If the Lagrangian density and therefore the action integral is invariant under this transformation then there is a conservation equation of the form 0 dxv where Cp -rip. Equation is in the form of an equation of continuity with in the role of a current density jv. Hence invariance under a gauge transformation of the first kind leads to the identification of a conserved current that would be appropriate for an electric charge and cunent density to be associated with the field. As an illustration let us consider the first example of Section the complex scalar field. A transformation of the type ị ộeie Ộ ộ e ie corresponds in infinitesimal form to a gauge transformation of the first type Eq. with c Ỉ c i. The familiar gauge transformation of the electromagnetic field which adds a 4-gradient A P to Ap. is part of a gauge transformation of the second kind and is not considered here. 596 Chapter 13 Formulations for Continuous Systems and Fields It
