Dualidad de curvas, ecuaciones diferenciales binarias y la transformada de Legendre

Abstract

We consider binary differential equations of the form a(x, y)(dy)^2 + 2b(x, y)dxdy + c(x, y)(dx)^2 = 0 where a,b,c are smooth functions that vanish at (0,0). In particular, we consider points where (b(x, y))^2 − a(x, y)c(x, y))≥ 0. Under those conditions there exists a classification up to diffeomorphism with the jet space of the normal forms of the equations given by:

Lemon: y(dy)^2 + 2xdxdy − y(dx)^2 = 0 Star: y(dy)^2 − 2xdxdy − y(dx)^2 = 0 Monstar: y(dy)^2 + 1/2 xdxdy − y(dx)^2 = 0 Applying the Legendre transform to the models, we obtain their integral curves due to duality with the corresponding curves of the Legendre transform. The purpose of this work is to show the construction of the classification and the duality of those models under the Legendre transformation.