Symplectic structure, Lagrangian, and involutiveness of first integrals of the principal chiral field equation (Q795996)

The matrix equation (1) \(\bar U_ t-\bar V_ x=[\bar U,\bar V]\) is considered where \(\bar U=\zeta^{-1}U_ 0+U_ 1+\zeta A\), \(\bar V=- \zeta^{-1}U_ 0+U_ 1+\zeta A;\quad U_ 1,A,U_ 0\) are \(N\times N\) matrices and A is a constant diagonal matrix with distinct diagonal elements. (1) must be satisfied identically with respect to the parameter \(\zeta\). These equations could be transformed into the form \(M_{\eta}+N_{\xi}=0\), \(M_{\eta}-N_{\xi}=[N,M]\), \(\eta =x-t\), \(\xi =x+t\), which is referred to as chiral equations and M, N are called chiral fields. The main objective is to established the connection between the symplectic structure of chiral fields and the first integrals. Two symplectic structures are defined. The Lagrangian and Hamiltonian are constructed, corresponding to one of the symplectic structures. Next, the first integrals are constructed and shown to be in involution.