Almost all real linear second order ordinary differential equations are solved by geodesic curves in two dimensional Riemannian hyperbolic geometry
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915211142758400 |
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| author | Rudnicki, Łukasz |
| author_facet | Rudnicki, Łukasz |
| contents | I show that a real linear second order ordinary differential equation $u''\left(x\right)+h\left(x\right)u\left(x\right)=0$, with differentiable $h(x)$, locally admits two linearly independent solutions which exist on an open interval around any $x_0\in\mathbb{R}$: \[
u_\mathtt{top}(x)=\exp\left[\int_{x_0}^{x}\!\!dξ\,Φ\left(ξ\right)\frac{Φ'\left(ξ\right)-\sqrt{\left[h\left(ξ\right)-Φ^{2}\left(ξ\right)\right]^{2}+\left[Φ'\left(ξ\right)\right]^{2}}}{h\left(ξ\right)-Φ^{2}\left(ξ\right)}\right], \] \[
u_\mathtt{bot}(x)=\exp\left[\int_{x_0}^{x}\!\!dξ\,Φ\left(ξ\right)\frac{Φ'\left(ξ\right)+\sqrt{\left[h\left(ξ\right)-Φ^{2}\left(ξ\right)\right]^{2}+\left[Φ'\left(ξ\right)\right]^{2}}}{h\left(ξ\right)-Φ^{2}\left(ξ\right)}\right], \] where $Φ(x)$ is any geodesic curve in a two dimensional hyperbolic geometry of a Riemannian manifold $\mathbb{M}_h$, which is non-vertical at $x_0$. I define $\mathbb{M}_h$ to be an upper half plane $\{\left(x,\varPhi\right)\in\mathbb{R}^2\,|\,\varPhi>0\}$, with points in which $\varPhi^2=h(x)$ being removed, equipped with metric $g_h=\left[\left(h(x)-\varPhi^2\right)^2dx^2+d\varPhi^2\right]/\varPhi^2$. A non-trivial character of the presented result stems from the fact that $g_h$ is solely defined in terms of the function $h(x)$.
I also show that a local diffeomorphism between $\mathbb{M}_h$ and Poincaré upper half plane $\mathbb{H}$ is induced by any pair of linearly independent solutions of $u''\!\left(x\right)+h\left(x\right)u\left(x\right)=0$. If this pair is selected to be $u_\mathtt{top}(x)$ and $u_\mathtt{bot}(x)$, the associate geodesic curve $Φ(x)$ is mapped to a vertical geodesic curve on $\mathbb{H}$.
Thus, I establish a link between linear second order ordinary differential equations and two dimensional hyperbolic geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17816 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost all real linear second order ordinary differential equations are solved by geodesic curves in two dimensional Riemannian hyperbolic geometry Rudnicki, Łukasz Classical Analysis and ODEs Differential Geometry I show that a real linear second order ordinary differential equation $u''\left(x\right)+h\left(x\right)u\left(x\right)=0$, with differentiable $h(x)$, locally admits two linearly independent solutions which exist on an open interval around any $x_0\in\mathbb{R}$: \[ u_\mathtt{top}(x)=\exp\left[\int_{x_0}^{x}\!\!dξ\,Φ\left(ξ\right)\frac{Φ'\left(ξ\right)-\sqrt{\left[h\left(ξ\right)-Φ^{2}\left(ξ\right)\right]^{2}+\left[Φ'\left(ξ\right)\right]^{2}}}{h\left(ξ\right)-Φ^{2}\left(ξ\right)}\right], \] \[ u_\mathtt{bot}(x)=\exp\left[\int_{x_0}^{x}\!\!dξ\,Φ\left(ξ\right)\frac{Φ'\left(ξ\right)+\sqrt{\left[h\left(ξ\right)-Φ^{2}\left(ξ\right)\right]^{2}+\left[Φ'\left(ξ\right)\right]^{2}}}{h\left(ξ\right)-Φ^{2}\left(ξ\right)}\right], \] where $Φ(x)$ is any geodesic curve in a two dimensional hyperbolic geometry of a Riemannian manifold $\mathbb{M}_h$, which is non-vertical at $x_0$. I define $\mathbb{M}_h$ to be an upper half plane $\{\left(x,\varPhi\right)\in\mathbb{R}^2\,|\,\varPhi>0\}$, with points in which $\varPhi^2=h(x)$ being removed, equipped with metric $g_h=\left[\left(h(x)-\varPhi^2\right)^2dx^2+d\varPhi^2\right]/\varPhi^2$. A non-trivial character of the presented result stems from the fact that $g_h$ is solely defined in terms of the function $h(x)$. I also show that a local diffeomorphism between $\mathbb{M}_h$ and Poincaré upper half plane $\mathbb{H}$ is induced by any pair of linearly independent solutions of $u''\!\left(x\right)+h\left(x\right)u\left(x\right)=0$. If this pair is selected to be $u_\mathtt{top}(x)$ and $u_\mathtt{bot}(x)$, the associate geodesic curve $Φ(x)$ is mapped to a vertical geodesic curve on $\mathbb{H}$. Thus, I establish a link between linear second order ordinary differential equations and two dimensional hyperbolic geometry. |
| title | Almost all real linear second order ordinary differential equations are solved by geodesic curves in two dimensional Riemannian hyperbolic geometry |
| topic | Classical Analysis and ODEs Differential Geometry |
| url | https://arxiv.org/abs/2503.17816 |