Singularity analysis and an approach to obtaining symmetry for a system of ordinary differential equations: Euler and Ramanujan equations

摘要

This paper discusses two different problems. The first is the classic system of the Euler equations for the rotation of a rigid body about a fixed point. We study the Euler system using singularity analysis for the zero torque and nonzero torque cases. Secondly, we implement an alternate approach to study a system consisting of three or more first-order odes. An existing dependent variable is considered to be independent and the old or original system is rewritten as a new system with a fewer number of first-order odes. We study the analysis of the new system using the Lie symmetry approach. The main reason behind following this approach is to overcome the tedious calculation of the Lie point symmetries for the original system. The method is explained using the Ramanujan system, which arises in modular theory, for which we introduce a new independent variable to reduce the system to two first-order equations, from which we obtain a single second-order equation with one Lie-Point symmetry. The reduced equation is an Abel's equation of the first kind which is the same result as can be obtained by the usual procedure.