One-Term Groups and Ordinary Differential Equations

The flow $$x' = \exp ( tX) ( x)$$ x ′ = exp ( t X ) ( x ) of a single, arbitrary vector field $$X = \sum _{ i = 1}^n\, \xi _i ( x) \, \frac{ \partial }{ \partial x_i}$$ X = ∑ i = 1 n ξ i ( x ) ∂ ∂ x i with analytic coefficients $$\xi _i ( x)$$ ξ i ( x ) always generates a one-term (local) continuous transformation group satisfying: $$ \exp \big (t_1X\big ) \Big ( \exp \big (t_2X\big )(x) \Big ) = \exp \big ((t_1+t_2)X\big )(x), $$ exp ( t 1 X ) ( exp ( t 2 X ) ( x ) ) = exp ( ( t 1 + t 2 ) X ) ( x ) , and: $$ \left[ \exp (tX)(\cdot ) \right] ^{-1} = \exp (-tX)(\cdot ). $$ exp ( t X ) ( · ) - 1 = exp ( - t X ) ( · ) . In a neighborhood of any point at which $$X$$ X does not vanish, an appropriate local diffeomorphism $$x \mapsto y$$ x ↦ y may straighten $$X$$ X to just $$\frac{ \partial }{ \partial y_1}$$ ∂ ∂ y 1 , hence its flow becomes $$y_1' = y_1 + t$$ y 1 ′ = y 1 + t , $$y_2 ' = y_2, \dots , y_n' = y_n$$ y 2 ′ = y 2 , ⋯ , y n ′ = y n . In fact, in the analytic category (only), computing a general flow $$\exp ( tX) ( x)$$ exp ( t X ) ( x ) amounts to adding the differentiated terms appearing in the formal expansion of Lie’s exponential series: $$ \exp (tX)(x_i) = \sum _{k\geqslant 0}\, \frac{(tX)^k}{k!}(x_i) = x_i + t\,X(x_i) +\cdots + \frac{t^k}{k!}\, \underbrace{X\big (\cdots \big ( X\big (X}_{k\,\,\text {times}}(x_i)\big )\big )\cdots \big ) +\cdots , $$ exp ( t X ) ( x i ) = ∑ k ⩾ 0 ( t X ) k k ! ( x i ) = x i + t X ( x i ) + ⋯ + t k k ! X ( ⋯ ( X ( X ⏟ k times ( x i ) ) ) ⋯ ) + ⋯ , that have been studied extensively by Gröbner in [3]. The famous Lie bracket is introduced by looking at the way a vector field $$X = \sum _{ i = 1}^n \, \xi _i ( x) \frac{ \partial }{ \partial x_i}$$ X = ∑ i = 1 n ξ i ( x ) ∂ ∂ x i is perturbed, to first order, while introducing the new coordinates $$x' = \exp ( tY) ( x) =: \varphi ( x)$$ x ′ = exp ( t Y ) ( x ) = : φ ( x ) provided by the flow of another vector field $$Y$$ Y : $$ \varphi _*(X) = X' + t\,\left[ X',\,Y'\right] +\cdots , $$ φ ∗ ( X ) = X ′ + t X ′ , Y ′ + ⋯ , with $$X ' = \sum _{ i=1}^n \, \xi _i ( x') \, \frac{ \partial }{\partial x_i'}$$ X ′ = ∑ i = 1 n ξ i ( x ′ ) ∂ ∂ x i ′ and $$Y' = \sum _{ i = 1}^n \, \eta _i ( x') \, \frac{ \partial }{ \partial x_i'}$$ Y ′ = ∑ i = 1 n η i ( x ′ ) ∂ ∂ x i ′ denoting the two vector fields in the target space $$x'$$ x ′ having the same coefficients as $$X$$ X and $$Y$$ Y . Here, the analytical expression of the Lie bracket is: $$ \left[ X',\,Y'\right] = \sum _{i=1}^n\, \bigg ( \sum _{l=1}^n\, \xi _l(x')\,\frac{\partial \eta _i}{\partial x_l'}(x') - \eta _l(x')\,\frac{\partial \xi _i}{\partial x_l'}(x') \bigg )\, \frac{\partial }{\partial x_i'}. $$ X ′ , Y ′ = ∑ i = 1 n ( ∑ l = 1 n ξ l ( x ′ ) ∂ η i ∂ x l ′ ( x ′ ) - η l ( x ′ ) ∂ ξ i ∂ x l ′ ( x ′ ) ) ∂ ∂ x i ′ . An $$r$$ r -term group $$x' = f ( x; \, a)$$ x ′ = f ( x ; a ) satisfying his fundamental differential equations $$\frac{ \partial x_i'}{ \partial a_k} = \sum _{ j = 1}^r \, \psi _{ kj} ( a) \, \xi _{ ji} ( x')$$ ∂ x i ′ ∂ a k = ∑ j = 1 r ψ k j ( a ) ξ j i ( x ′ ) can, alternatively, be viewed as being generated by its infinitesimal transformations $$X_k = \sum _{ i = 1}^n \, \xi _{ ki} ( x) \, \frac{ \partial }{\partial x_i}$$ X k = ∑ i = 1 n ξ k i ( x ) ∂ ∂ x i in the sense that the totality of the transformations $$x' = f ( x; \, a)$$ x ′ = f ( x ; a ) is identical with the totality of all transformations: $$\begin{aligned} x_i'&= \exp \big ( \lambda _1\,X_1 +\cdots + \lambda _r\,X_r\big )(x_i) \\&= x_i + \sum _{k=1}^r\,\lambda _k\,\xi _{ki}(x) + \sum _{k,\,j}^{1\dots r}\, \frac{\lambda _k\,\lambda _j}{1\cdot 2}\, X_k(\xi _{ji}) + \cdots \ \ \ \ \ \ \ \ \ \ \ \ \ {\scriptstyle {(i\,=\,1\,\cdots \,n)}} \end{aligned}$$ x i ′ = exp ( λ 1 X 1 + ⋯ + λ r X r ) ( x i ) = x i + ∑ k = 1 r λ k ξ k i ( x ) + ∑ k , j 1 ⋯ r λ k λ j 1 · 2 X k ( ξ j i ) + ⋯ ( i = 1 ⋯ n ) obtained as the time-one map of the one-term group $$\exp \big ( t \sum \, \lambda _i X_i \big ) ( x)$$ exp ( t ∑ λ i X i ) ( x ) generated by the general linear combination of the infinitesimal transformations. A beautiful idea of analyzing the (diagonal) action $${x^{( \mu )}}' = f \big ( x^{ ( \mu )}; \, a\big )$$ x ( μ ) ′ = f ( x ( μ ) ; a ) induced on $$r$$ r -tuples of points $$\big ( x^{(1)}, \dots , x^{ ( r)} \big )$$ ( x ( 1 ) , ⋯ , x ( r ) ) in general position enables Lie to show that for every collection of $$r$$ r linearly independent vector fields $$X_k = \sum _{ i = 1}^n\, \xi _{ ki} ( x) \, \frac{ \partial }{ \partial x_i}$$ X k = ∑ i = 1 n ξ k i ( x ) ∂ ∂ x i , the parameters $$\lambda _1, \dots , \lambda _r$$ λ 1 , ⋯ , λ r in the finite transformation equations $$x' = \exp \big ( \lambda _1 \, X_1 + \cdots + \lambda _r \, X_r \big ) ( x)$$ x ′ = exp ( λ 1 X 1 + ⋯ + λ r X r ) ( x ) are all essential.