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Differential Algebraic Formulation of Normal Form Theory

A differential algebraic (DA) formulation of a normal form theory for repetitive systems is presented. Contrary to previous approaches, no Lie algebraic tools are used. The resulting algorithm is very transparent and not restricted to the treatment of symplectic systems. In the case of symplectic systems, the normal form algorithm provides a nonlinear coordinate transformation in which the motion is confined to circles. The transformation exists if the tunes are not on a resonance; in this case, it can be used to compute tune shifts in a similar way as in the Lie algebraic picture. In the case of nonsymplectic systems, the motions in the new coordinates are growing or shrinking exponential spirals. In the case all spirals are shrinking, which occurs in electron rings, all amplitude dependent tune shifts vanish and in a formal sense tune resonances do not occur. The algorithm has been implemented in the code COSY INFINITY. For symplectic systems, which can also be studied with the DA-Lie algorithm also implemented in COSY INFINITY, identical results are obtained at a reduced computational expense.