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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 $$e^ xe^ y=e^{x+y}$$.

##### MSC:
 34E05 Asymptotic expansions of solutions to ordinary differential equations 22E60 Lie algebras of Lie groups 53B99 Local differential geometry
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##### References:
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