Superposition formulas based on nonprimitive group action (Q2781689)

The authors regard the action of an \(r\)-dimensional matrix Lie group \(G\) on some manifold \(M\subset \mathbb{R}^n\), \(X^1,\dots, X^r\) denote then some base for the Lie algebra \(L\) of \(G\) in terms of generators corresponding to the left regular representation of \(G\) on a space \(F\subset C^\infty(M)\). Then for given \(z_1(t),\dots z_r(t)\) one gets to the path NEWLINE\[NEWLINEX(t)= z_1(t) X^1+\cdots z_r(t) X^rNEWLINE\]NEWLINE in the Lie algebra \(L\) a corresponding path \(g(t)\) with \(g(0)= e\) in \(G\) and by the action of \(G\) on \(M\) for a given initial value \(x(0)\) a path \(x(t)\) in \(\mathbb{R}^n\), which solves the equation NEWLINE\[NEWLINE\mathring x(t)= z_1(t) A^1(x(t))+\cdots+ z_r(t) A^r(x(t)),NEWLINE\]NEWLINE where the \(A^j\) are determined by the coefficient functions of the generators \(X^1,\dots, X^r\).NEWLINENEWLINENEWLINESince the matrix coefficients of \(g(t)\) may be expressed by a sufficient number of particular solutions, one gets a superposition formula for the above nonlinear differential equation, which expresses the general solution by particular solutions and initial values.NEWLINENEWLINENEWLINEExamples are given for the \(\text{SL}(2, \mathbb{R})\)-action on \(\mathbb{R}\) and \(\mathbb{R}^2\), and the general case of \(\text{SL}(u+ 1,\mathbb{R})\) is discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00040].