Symbolic computation of approximate symmetries for ordinary differential equations

This report presents a method of constructing approximate symmetries and invariants in dynamical systems. At first, a similar approach was developed for linear systems (for example, the time-dependent invariants by Leach and Lewis). Beam line systems are considered as an example of such dynamical systems. It is known that such concept as emittance invariants has been used in beam physics for a long time (for example, the so called Courant–Snyder invariant). There are some recently works devoted to this problem. Among them we must mention the works by Dragt and his colleagues. Now we consider the problem of construction of approximate invariants and symmetries based on the matrix formalism for Lie algebraic tools. The knowledge of symmetries of the dynamical system is also useful for problems of synthesis of a beamline with desired characteristics.