The geometry of a second order differential equation, part 2

Posted on 4 July, 2020 by jesperg

We start over with a general second order ordinary differential equation

\displaystyle y''=\phi(x,y,y').

and this time we won’t impose any restrictions on the right-hand side. We express y'=dy/dx and y''=(d^2ydx-d^2xdy)/dx^3 which gives that

\displaystyle d^2ydx-d^2xdy=\phi\left(x,y,\frac{dy}{dx}\right)dx^3.

The variables dx and dy denote velocity and are tangent space coordinates. The variables d^2x and d^2y denote acceleration and are jet space (of curves) coordinates. Since the right-hand side is simultaneous homogeneous of degree 3 in dx and dy we instead consider equations of the form

\displaystyle d^2ydx-d^2xdy=f(x,y,dx,dy),

where f is homogeneous of degree 3 in dx and dy, that is f(x,y,\lambda dx,\lambda dy)=\lambda^3f(x,y,dx,dy). Euler’s theorem then gives that we can express

\displaystyle f(x,y,dx,dy)=Adx^3+3Bdx^2dy+3Cdxdy^2+Ddy^3

with

\displaystyle \begin{aligned} A&=\tfrac{1}{6}f_{dxdxdx}(x,y,dx,dy)\\B&=\tfrac{1}{6}f_{dxdxdy}(x,y,dx,dy)\\C&=\tfrac{1}{6}f_{dxdydy}(x,y,dx,dy)\\D&=\tfrac{1}{6}f_{dydydy}(x,y,dx,dy)\end{aligned}

and A,B,C,D are homogeneous of degree 0 in dx and dy. We put p=dy/dx and consider A,B,C,D to be functions in x,y and p:

\displaystyle \begin{aligned} A(x,y,p)&=\tfrac{1}{6}f_{dxdxdx}(x,y,1,p)\\B(x,y,p)&=\tfrac{1}{6}f_{dxdxdy}(x,y,1,p)\\C(x,y,p)&=\tfrac{1}{6}f_{dxdydy}(x,y,1,p)\\D(x,y,p)&=\tfrac{1}{6}f_{dydydy}(x,y,1,p)\end{aligned}

We have that

\displaystyle d^2ydx-d^2xdy=Adx^3+3Bdx^2dy+3Cdxdy^2+Ddy^3

Polarisation gives that

\displaystyle\begin{aligned} d^2x&=-Bdx^2-2Cdxdy-Ddy^2\\d^2y&=\quad Adx^2+2Bdxdy+Cdy^2,\end{aligned}

which are geodesic equations of a connection of the two-dimensional tangent spaces in the three-dimensional projectivised tangent bundle (also called the manifold of elements). Each tangent space has coordinates dx and dy and the projectivised tangent bundle has coordinates x, y and p. We can read off the Christoffel symbols of this connection as

\displaystyle \begin{aligned}\Gamma^1_{11}=B\quad\Gamma^1_{12}=\Gamma^1_{21}=C\quad\Gamma^1_{22}=D\\\Gamma^2_{11}=-A\quad\Gamma^2_{12}=\Gamma^2_{21}=-B\quad\Gamma^2_{22}=-C.\end{aligned}

We collect the Christoffel symbols in a connection one-form

\displaystyle \mathbf{\Gamma}=\begin{pmatrix}B&C\\-A&-B\end{pmatrix}dx+\begin{pmatrix}C&D\\-B&-C\end{pmatrix}dy.

We calculate the curvature two-form \mathbf{\Omega} by applying the exterior covariant derivative D to the connection one-form

\displaystyle \begin{aligned}\mathbf{\Omega}&=D\mathbf{\Gamma}=\delta\mathbf{\Gamma}+\mathbf{\Gamma}\wedge\mathbf{\Gamma}\\&=\begin{pmatrix}b&c\\-a&-b\end{pmatrix}dx\wedge dy+\begin{pmatrix}kp&-k\\kp^2&-kp\end{pmatrix}(dy-pdx)\wedge dp\end{aligned}

where \delta means ordinary exterior derivative and

\displaystyle \begin{aligned}a&=B_x-A_y+2AC-2B^2\\b&=C_x-B_y+AD-BC\\c&=D_x-C_y+2BD-2C^2\\k&=-\frac{A_p}{p^3}=\frac{B_p}{p^2}=-\frac{C_p}{p}=D_p=\tfrac{1}{6}\phi_{y'y'y'y'}(x,y,p).\end{aligned}

We extend the tangent spaces of the base manifold to projectives planes by adding a line at infinity in each tangent space. The points in the affine part of the projective tangent spaces have homogenous coordinates (dx:dy:1) and points on the line at infinity have homogenous coordinates (dx:dy:0). We also extend the connection into two projective connections \mathbf{\Pi}_1 and \mathbf{\Pi}_2:

\displaystyle  \mathbf{\Pi}_1=\begin{pmatrix}B&C&1\\-A&-B&0\\0&0&0\end{pmatrix}dx+\begin{pmatrix}C&D&0\\-B&-C&1\\0&0&0\end{pmatrix}dy,

\displaystyle  \mathbf{\Pi}_2=\begin{pmatrix}B&C&1\\-A&-B&0\\-a&-b&0\end{pmatrix}dx+\begin{pmatrix}C&D&0\\-B&-C&1\\-b&-c&0\end{pmatrix}dy.

At an element (x_0,y_0,p_0) a connection in the bundle defines an osculating connection in the base manifold by regarding p=p_0 as constant. Also an osculating connection is defined in the dual manifold by regarding (x,y)=(x_0,y_0) as constant. The curvature of the osculating connection of \mathbf{\Pi}_2 in the base manifold is

\displaystyle D\mathbf{\Pi}_2|_{\text{primal}} = \begin{pmatrix}0&0&0\\0&0&0\\ \alpha&\beta&0\end{pmatrix}dx\wedge dy

with

\displaystyle \begin{aligned}\alpha&=A_{yy}-2B_{xy}+C_{xx}-3AC_y+2AD_x+6BB_y-3BC_x-3CA_y+DA_x\\\beta&=B_{yy}-2C_{xy}+D_{xx}-AD_y+3BD_x+3CB_y-6CC_x-2DA_y+3DB_x.\end{aligned}

The curvature of the osculating connection of \mathbf{\Pi}_1 in the dual manifold is

\displaystyle D\mathbf{\Pi}_1|_{\text{dual}} =k\begin{pmatrix}p&-1&0\\p^2&-p&0\\0&0&0\end{pmatrix}(dy-pdx)\wedge dp

The curvature form of the primal osculating connection is characterised by that all lines through the origin are invariant. The curvature form of the dual osculating connection is characterised by that all points on the line dy-pdx=0 are invariant. These two relations are dual and it follows that the two osculating connections are dual.

From the curvatures we get two invariants in the bundle:

\displaystyle \begin{aligned} k(x,y,p) &= \tfrac{1}{6}\phi_{y'y'y'y'}(x,y,p)\\ \gamma(x,y,p)&=\alpha(x,y,p)+\beta(x,y,p) p\end{aligned}

which are dual to each other.

 

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