Some physical applications of systems of differential equations admitting a superposition rule
The authors refer to old results of S. Lie and G. Scheffers concerning systems of ordinary differential equations admitting a superposition rule. Such systems are associated with the action of a Lie group \(G\) on a manifold \(M\) being in general a homogeneous space \(G/H\) and are called Lie-Scheffers systems. The basic point is (following \textit{J. Wei} and \textit{E. Norman} [J. Math. Phys. 4, 575-581 (1963; Zbl 0133.34202)]) to regard equations of the type \[ \dot g(t)\cdot g(t)^{-1}= a(t)= \sum^r_{\alpha=1} A_\alpha(t)\cdot a_\alpha\in T_e(G), \] in a Lie group \(G\), where the r.-h.s. is formed in a certain subalgebra of the Lie algebra of \(G\) and \(g(t)\) is locally written as the product \(g(t)= \prod^r_{\alpha=1} \exp(u_\alpha(t)\cdot a_\alpha)\). Several examples, also with physical relevance, are given and demonstrate the general character of the construction.
scientific article; zbMATH DE number 1799969