Equivalence of paths in galileean-symplectic geometry

K. K. Muminov, V. I. Chilin

National University of Uzbekistan named after M. Ulugbek, Tashkent

Abstract: Let $V$ be a $2n$-dimensional linear space over the field of real numbers $R$, let $GL\left( 2n,\,R \right)$ be the group of all invertible linear transformations of the space $V$, and let $G$ be a subgroup of the group $ GL\left( 2n,\,R \right)$. A path in $V$ is a vector function $x:(0,1)\to V, x(t)=\left\{ {{x}_{i}}(t) \right\}_{i= 1}^{2n},$ for which all coordinate maps ${{x}_{i}}:\left( 0,1 \right)\to R$ are infinitely differentiable functions. Two paths $x(t)$ and $y(t)$ are said to be $\,G$-equivalent if there is an element $\,g\in \,\,G$ such that $y(t) \,=gx(t)$ for all $t\in \,(0,1).$ We consider the Galilean-symplectic group $\Gamma Sp\left( 2n,\,R \right)$ of all such linear transformations $g=(g_{i,j})_{i,j=1}^{2n} \in GL\left( 2n,\,R \right) $, for which $g_{11}=\pm 1, g_{2n,2n}=\pm 1,\, (g_{i,j})_{i ,j=2}^{2n-1}\in Sp(2n-2,R)$, where $Sp(2n-2,R)$ is the symplectic group of invertible linear transformations in the space $R^{2n-2} $. To solve this problem, the differential field of all $\Gamma Sp(2n,R)$-invariant $d$-rational functions is considered and a description of the finite system of generators of this field is given. We consider the class of $\Gamma Sp(2n,R)$-regular paths lying in $R^{2n}$, i.e. such paths $x(t)=\left\{x_{i}(t) \right\}_{i=1}^{2n}\subset R^{2n}$, $t\in (0,1)$, for which $\det M_{2n-2}(x(t))$ is not equal to zero for all $t\in (0,1)$, where
$$M_{2n-2}(x(t))=\left( x_{i}^{(j)}(t) \right)_{j=0,1,...,2n-3,\,\,\,\,i=2,...,2n-1}, \ x_{i}^{(0)}(t) = x_{i}(t),$$
$x_{i}^{(j)}(t)$ is the $j$-th derivative of the coordinate function ${x}_{i}\left( t \right)\,$, $i=2,...;2n-1, \ j=1,...,2n-3$. For this class of paths, necessary and sufficient conditions for their equivalence under the action of the group $\Gamma Sp(2n,R)$ are obtained. The article gives an explicit description of a finite system of generators in a differential field of differential rational functions invariant with respect to the action of the Galileo-symplectic group $\Gamma Sp(2n,R)$. With the help of these generators, necessary and sufficient conditions are established for $\Gamma Sp(2n,R)$-equivalence of paths.

Keywords: Galilean-symplectic group of motions, differential invariants, regular paths in a finite-dimensional space.

UDC: 514.74

MSC: 53A15, 53A35, 53A55