|
|
International youth conference "Geometry & Control"
April 15, 2014 17:00, Poster session, Moscow, Steklov Mathematical Institute of RAS
|
|
|
|
|
|
Polynomial Integrals of the Geodesics Equations in Two-Dimensional Case
Yulia Bagderina Institute of Mathematics with Computer Center of RAS, Ufa, Russia
|
Number of views: |
This page: | 151 | Materials: | 54 |
|
Abstract:
Let $M^2$ be two-dimensional surface with the Riemannian metric
$$
\tag{1}
ds^2=g_{11}(x,y)dx^2+2g_{12}(x,y)dxdy+g_{22}(x,y)dy^2.
$$
Geodesics equations of a given metric can be treated as a system of Euler-Lagrange equations
$$
\tag{2}
\frac{d}{dt}L_{\dot{x}}-L_x=0,\qquad \frac{d}{dt}L_{\dot{y}}-L_y=0
$$
with the Lagrangian
$$
\tag{3}
L(x,y,\dot{x},\dot{y})=\frac 12g_{11}(x,y)\dot{x}^2
+g_{12}(x,y)\dot{x}\dot{y}+\frac 12g_{22}(x,y)\dot{y}^2.
$$
Geodesic flow of the metric (1) is Liouville integrable, if it possesses a smooth
first integral $F$ functionally independent of the Lagrangian (3). In the present work we consider
the problem of the existence of the polynomial integral of the system (2), (3) of the first degree
$$
\tag{4}
F_1=b_0(x,y)\dot{x}+b_1(x,y)\dot{y},
$$
the second degree
$$
\tag{5}
F_2=b_0(x,y)\dot{x}^2+2b_1(x,y)\dot{x}\dot{y}+b_2(x,y)\dot{y}^2
$$
and the third degree
$$
\tag{6}
F_3=b_0(x,y)\dot{x}^3+3b_1(x,y)\dot{x}^2\dot{y}+3b_2(x,y)\dot{x}\dot{y}^2+b_3(x,y)\dot{y}^3.
$$
For an integral (4), (5) or (6) the existence conditions are obtained as the compatibility conditions
of an overdetermined system of linear homogeneous first-order equations in the functions $b_i(x,y)$.
Here these conditions are expressed in terms of the invariants
$$
\tag{7}
I_1(x,y)=\frac{J_1}{j_0J_0^3},\qquad I_2(x,y)=\frac{J_2}{j_0J_0^2}
$$
of the equivalence transformations of the family of equations (2), (3) defined by
$$
\tilde t=k(t+t_0),\qquad \tilde x=\varphi(x,y),\qquad
\tilde y=\psi(x,y),\qquad k,t_0={\rm const}.
$$
In (7) the value $J_0$ up to a constant multiplier coincides with the main (scalar)
curvature $K$ of the surface $M^2$,
$$
j_0=g_{11}g_{22}-g_{12}^2,\qquad
J_1=g_{22}J_{0x}^2-2g_{12}J_{0x}J_{0y}+g_{11}J_{0y}^2.
$$
All results on the integrals (4)–(6) are obtained in assumption of the non-degeneracy
of the surface $M^2$. It means, when the conditions
$$
\tag{8}
j_0\neq 0,\qquad J_0\neq 0,\qquad J_1\neq 0
$$
hold. The geometrical sense of the first two conditions (8) is obvious (non-degeneracy of the
matrix $g_{ij}(x,y)$ and nonzero curvature of the surface). The sense of the third condition (8)
is not so evident. The question is what properties has the degenerate surface $M^2$ with the
curvature $K$, which satisfies the relation
$$
\tag{9}
g_{22}(x,y)\left(\frac{\partial K}{\partial x}\right)^2
-2g_{12}(x,y)\frac{\partial K}{\partial x}\frac{\partial K}{\partial y}
+g_{11}(x,y)\left(\frac{\partial K}{\partial y}\right)^2=0.
$$
Supplementary materials:
abstract.pdf (57.3 Kb)
Language: English
|
|