Классификация путей в геометрии Галилея

Let $X$ be $n$-dimensional linear space over field $\mathbb{R}$ of real numbers and let $GL(n,\mathbb{R})$ be the group of all invertible linear transformations of the space $X$. Two paths $x(t),y(t)\subset X,t\in (0,1),$ are called $G$-equivalent with respect to the action of the subgroup $G$ of the group $GL(n,\mathbb{R})$ if $g(x(t))=y(t)$ for some $g\in G$ and all $t\in (0,1)$. One of the important problems of differential geometry is finding necessary and sufficient conditions such that the paths $x(t), y(t)$ are $G$-equivalent. The solutions of this problem use methods of the theory of differential invariants, giving a description of finite rational bases of differential fields of $G$-invariant differential rational functions. These bases provide effective criteria for Gequivalence of paths. This approach was used in for solving the problem of the equivalence of paths with respect to the action of the symplectic, orthogonal and pseudo-orthogonal groups.
An important example of a non-Euclidean geometry is the Galileo geometry. The group $\Gamma (n,\mathbb{R})$ of all invertible linear transformations of the space $X$, preserving the Galilean metric, are called Galileo’s group. We give the following description of a finite rational basis in the differential field $\mathbb{R}\left \langle x_{1},....,x_{n} \right \rangle^{\Gamma (n,\mathbb{R})}$ of all $\Gamma (n,\mathbb{R})$-invariant differential rational functions:
In the field $\mathbb{R}\left \langle x_{1},...,x_{n} \right \rangle^{\Gamma (n,\mathbb{R})}$ the following differential polynomials form its a rational basis:\begin{align} \varphi _{k}(x_{1},...,x_{n})=\sum_{i=2}^{n}(x_{i}^{(k)})^{2},k=0,...,n-2;\\
\psi (x_{1},...,x_{n})=x_{1}.
\end{align} Using this rational basis, the following necessary and sufficient conditions for the $\Gamma (n,\mathbb{R})$-equivalence of two regular paths are established:
Two regular paths $x(t)=\left \{ x_{i}(t) \right \}_{i=1}^{n} $ and $y(t)=\left \{ y_{i}(t) \right \}_{i=1}^{n}$ are $\Gamma (n,\mathbb{R})$-equivalent if and only if $y_{1}(t)=\pm x_{1}(t)$ and $\sum_{i=2}^{n}(x_{i}^{m}(t))^{2}=\sum_{i=2}^{n}(y_{i}^{m}(t))^{2}$ for all $t\in (0,1)$ and $m=0,1,...,n-2.$
Keywords:Galileo space, a group of movements, differential invariant.