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The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. (English) Zbl 0623.34058
The Campbell-Baker-Hausdorff-Dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems. Specifically, suppose u(t) takes values in a \(C^{\infty}\) manifold and satisfies the initial value problem \(u'(t)=A(t)(u(t))\), \(u(0)=a\), where A(t) is a vector field on the manifold depending continuously on t. Then \(u(t)=\exp z(t)(a)\) (here exp z(a) means the solution at \(s=1\) to \(v'(s)=z(v(s))\), \(v(0)=a)\) for \[ z(t)\sim \sum^{\infty}_{r=1}\sum_{\sigma \in P_ r}((- 1)^{e(\sigma)}/r^ 2\left[ \begin{matrix} r-1\\ e(\sigma)\end{matrix} \right]) \] \[ \int_{T_ r(t)}[\cdot \cdot \cdot [A(s_{\sigma (1)})A(s_{\sigma (2)})]...]A(s_{\sigma (r)})]ds \] as \(t\to 0\), where \(T_ r(t)=\{s\in {\mathbb{R}}^ r: 0<s_ 1<s_ 2<...<s_ r<t\}\), \(P_ r\) is the set of permutations on \(\{\) 1,...,r\(\}\), e(\(\sigma)\) is the number of errors in ordering consecutive terms in \(\{\sigma\) (1),...,\(\sigma\) (r)\(\}\), and [ ] is the usual commutator of vector fields. Under appropriate analyticity assumptions the series for z(t) is convergent for small t. This formula gives an explicit formulation of results of K.-T. Chen published in 1957. Applications are given to problems in sub- Riemannian geometry, and to improving convergence estimates for the Campbell-Baker-Hausdorff-Dynkin formula in the context of Banach algebras. Our formula can be thought of as a noncommutative generalization of the familiar formula \(u(t)=a \exp (\int^{t}_{0}A(s)ds)\) in the scalar linear case in the same way that the Campbell-Baker-Hausdorff-Dynkin formula is a noncommutative generalization of the familiar formula \(e^ xe^ y=e^{x+y}\).

34E05 Asymptotic expansions of solutions to ordinary differential equations
22E60 Lie algebras of Lie groups
53B99 Local differential geometry
Full Text: DOI
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