202302121102

Type :Note Tags : Differential Equations

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One Parameter Group Of Diffeomorphisms

Let $U\subseteq {\mathbb{R}}^n$ be the Domain.

Note

A One Parameter Group Of Diffeomorphisms on $U$ is $\begin{array}{ccccc}g& :& \mathbb{R}\times U& \to & U\\ & & (t,\overline{x})& \mapsto & g^t(\overline{x})\end{array}$ Such that

1. $g$ is differentiable.
2. $g^t$ is a diffeomorphism.
3. $\forall t,s\in \mathbb{R},g^{t+s}(\overline{x})=g^t(g^s(\overline{x}));g^0(\overline{x})=\overline{x}$

And the pair $(U,\{g^t\})$ is called a Phase Flow.

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References