Algebraic approach to symmetries and invariants construction

This reports presents a new algebraic approach to approximating symmetries and invariants construction. Both invariants and symmetries are separated into dynamical and kinematical ones. Additionally each type of symmetries is separated into intrinsic and imposed symmetries. The intrinsic symmetries are generated by the dynamical system under study. The imposed symmetries ensure some desired properties of the dynamical system. This approach is very useful for optimal beam lines design problems. The kinematical invariants are used as a nonlinear theoretical probe. Such probes can be used for nonlinear effects investigation and control. Symmetries and invariants constructions procedure are based on the matrix formalism for the Lie algebraic methods. This formalism allows to create algebraic equations for determining block-matrices entering into the corresponding symmetries and invariants description. These equations can be solved in a symbolic mode, and the corresponding results are included in a special database. The algebraic approach is based on the Kronecker presentation of the Poincare-Witt basis for Lie algebras. All necessary statements are proved. Some practical applications for beam physics problems are discussed