One-Term Groups and Ordinary Differential Equations

Abstract

The flow $x' = \exp ( tX) ( x)$ of a single, arbitrary vector field $X = \sum _{ i = 1}^n\, \xi _i ( x) \, \frac{ \partial }{ \partial x_i}$ with analytic coefficients $\xi _i ( x)$ always generates a one-term (local) continuous transformation group satisfying:

$$ \exp \big (t_1X\big ) \Big ( \exp \big (t_2X\big )(x) \Big ) = \exp \big ((t_1+t_2)X\big )(x), $$

and:

$$ \left[ \exp (tX)(\cdot ) \right] ^{-1} = \exp (-tX)(\cdot ). $$

In a neighborhood of any point at which $X$ does not vanish, an appropriate local diffeomorphism $x \mapsto y$ may straighten $X$ to just $\frac{ \partial }{ \partial y_1}$, hence its flow becomes $y_1' = y_1 + t$, $y_2 ' = y_2, \dots , y_n' = y_n$. In fact, in the analytic category (only), computing a general flow $\exp ( tX) ( x)$ amounts to adding the differentiated terms appearing in the formal expansion of Lie’s exponential series:

$$ \exp (tX)(x_i) = \sum _{k\geqslant 0}\, \frac{(tX)^k}{k!}(x_i) = x_i + t\,X(x_i) +\cdots + \frac{t^k}{k!}\, \underbrace{X\big (\cdots \big ( X\big (X}_{k\,\,\text {times}}(x_i)\big )\big )\cdots \big ) +\cdots , $$

that have been studied extensively by Gröbner in [3]. The famous Lie bracket is introduced by looking at the way a vector field $X = \sum _{ i = 1}^n \, \xi _i ( x) \frac{ \partial }{ \partial x_i}$ is perturbed, to first order, while introducing the new coordinates $x' = \exp ( tY) ( x) =: \varphi ( x)$ provided by the flow of another vector field $Y$:

$$ \varphi _*(X) = X' + t\,\left[ X',\,Y'\right] +\cdots , $$

with $X ' = \sum _{ i=1}^n \, \xi _i ( x') \, \frac{ \partial }{\partial x_i'}$ and $Y' = \sum _{ i = 1}^n \, \eta _i ( x') \, \frac{ \partial }{ \partial x_i'}$ denoting the two vector fields in the target space $x'$ having the same coefficients as $X$ and $Y$. Here, the analytical expression of the Lie bracket is:

$$ \left[ X',\,Y'\right] = \sum _{i=1}^n\, \bigg ( \sum _{l=1}^n\, \xi _l(x')\,\frac{\partial \eta _i}{\partial x_l'}(x') - \eta _l(x')\,\frac{\partial \xi _i}{\partial x_l'}(x') \bigg )\, \frac{\partial }{\partial x_i'}. $$

An $r$-term group $x' = f ( x; \, a)$ satisfying his fundamental differential equations $\frac{ \partial x_i'}{ \partial a_k} = \sum _{ j = 1}^r \, \psi _{ kj} ( a) \, \xi _{ ji} ( x')$ can, alternatively, be viewed as being generated by its infinitesimal transformations $X_k = \sum _{ i = 1}^n \, \xi _{ ki} ( x) \, \frac{ \partial }{\partial x_i}$ in the sense that the totality of the transformations $x' = f ( x; \, a)$ is identical with the totality of all transformations:

$$\begin{aligned} x_i'&= \exp \big ( \lambda _1\,X_1 +\cdots + \lambda _r\,X_r\big )(x_i) \\&= x_i + \sum _{k=1}^r\,\lambda _k\,\xi _{ki}(x) + \sum _{k,\,j}^{1\dots r}\, \frac{\lambda _k\,\lambda _j}{1\cdot 2}\, X_k(\xi _{ji}) + \cdots \ \ \ \ \ \ \ \ \ \ \ \ \ {\scriptstyle {(i\,=\,1\,\cdots \,n)}} \end{aligned}$$

obtained as the time-one map of the one-term group $\exp \big ( t \sum \, \lambda _i X_i \big ) ( x)$ generated by the general linear combination of the infinitesimal transformations. A beautiful idea of analyzing the (diagonal) action ${x^{( \mu )}}' = f \big ( x^{ ( \mu )}; \, a\big )$ induced on $r$-tuples of points $\big ( x^{(1)}, \dots , x^{ ( r)} \big )$ in general position enables Lie to show that for every collection of $r$ linearly independent vector fields $X_k = \sum _{ i = 1}^n\, \xi _{ ki} ( x) \, \frac{ \partial }{ \partial x_i}$, the parameters $\lambda _1, \dots , \lambda _r$ in the finite transformation equations $x' = \exp \big ( \lambda _1 \, X_1 + \cdots + \lambda _r \, X_r \big ) ( x)$ are all essential.