First integral by means of the uniformization over the torus

TL;DR

This paper generalizes the uniformization method over the torus to establish the existence of first integrals for plane vector fields, extending Finn's 1973 approach to classify differential operators.

Contribution

It introduces a generalized uniformization technique over the torus for analyzing the existence of first integrals in plane vector fields.

Findings

Unified approach to uniformization over the torus

Application to existence of first integrals

Extension of Finn's classification method

Abstract

The uniformization of a direction field was defined by Finn (in 1973) for the classification of certain differential operators. In the present note we recover the idea of uniformization in order to generalize it and apply it to the existence of first integrals. Roughly speaking, we say that a plane vector field $X$ on $D\subset\mathbb{R}^2$ is uniformizable over $T^2$ (the torus) if there is a constant vector field $Z$ on $\Delta\subset T^2$ and a diffeomorphism $\psi\colon D\to \Delta\subset T^2$ such that $d\psi\circ X = Z\circ\psi$.

Equations

$${\mathrm{ind}}_{\Gamma }(X):=\frac{1}{2\pi }\delta (X_{\Gamma }).$$

$${\mathrm{ind}}_{\Gamma }(X)=\frac{1}{2\pi }{\int }_0^{2\pi }\frac{\det ({X|}_{\Gamma },({X|}_{\Gamma }{)}^{\prime })}{\Vert {X|}_{\Gamma }{\Vert }^2}dt,$$

$$\left\{\begin{array}[]{l}h_{x}=b\,f_{x}-a\,f_{y}\\ h_{y}=b\,g_{x}-a\,g_{y}\end{array}\right.$$

$$d\psi \circ X\circ {\psi }^{-1}=(f_x\dot{x}+g_x\dot{y})\frac{\partial }{\partial u}+(f_y\dot{x}+g_y\dot{y})\frac{\partial }{\partial v}.$$

$$\begin{array}{rl}d\psi \circ X\circ {\psi }^{-1}(H)& =(f_x\dot{x}+g_x\dot{y})H_u+(f_y\dot{x}+g_y\dot{y})H_v\\ & =\dot{x}(H_u{f}_x+H_v{f}_y)+\dot{y}(H_u{g}_x+H_v{g}_y)\\ & =\dot{x}(b{f}_x-a{f}_y)+\dot{y}(b{g}_x-a{g}_y)=0.\end{array}$$

$$X(h)=\dot{x}{h}_x+\dot{y}{h}_y=\dot{x}(b{f}_x-a{f}_y)+\dot{y}(b{g}_x-a{g}_y)=0.$$

$$P:=\frac{a{g}_y-b{f}_y}{f_x{g}_y-f_y{g}_x}$$

$$Q:=\frac{b{f}_x-a{g}_x}{f_x{g}_y-f_y{g}_x},$$

$${\mathrm{ind}}_{\phi \circ \Gamma }(d\phi \circ Z\circ {\phi }^{-1})=1.$$

$$\phi :=\sigma \circ \tau \circ {\rho }_{p,q},$$

$$\rho_{p,q}:=\dfrac{1}{p^{2}+q^{2}}\Big{(}\begin{matrix}p&q\\ -q&p\end{matrix}\Big{)}$$

$$\tau (x,y)=(x,y+2\pi ),$$

$$\sigma (x,y)=(y\cos (x),y\sin (x)).$$

$$\varphi(x,y)=\dfrac{1}{p^{2}+q^{2}}\big{(}(py-qx+2\pi)\cos(px+qy)\,,\,(py-qx+2\pi)\sin(px+qy)\big{)}\,.$$

$$(py-qx){|}_{\Gamma (t)}=p(qt)-q(pt)=0\text{mbox}and(px+qy){|}_{\Gamma (t)}=(p^2+q^2)t.$$

$$\phi \circ \Gamma =\frac{2\pi }{H}(c,s),$$

$$[0,2\pi ]\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${(d\phi \circ Z\circ {\phi }^{-1})|}_{\phi \circ \Gamma }={d\phi \circ Z|}_{\Gamma }={d\phi (\partial /\partial x)|}_{\Gamma }={\frac{\partial \phi }{\partial x}|}_{\Gamma }.$$

$$\begin{split}\dfrac{\partial\varphi}{\partial x}(x,y)=\cfrac{1}{H}&\big{(}-q\cos(px+qy)-p(py-qx+2\pi)\sin(px+qy),\\ &-q\sin(px+qy)-p(py-qx+2\pi)\cos(px+qy)\big{)}\,.\end{split}$$

$${\frac{\partial \phi }{\partial x}|}_{\Gamma }=\frac{1}{H}(-qc-2\pi ps,-qs-2\pi pc).$$

$$({\frac{\partial \phi }{\partial x}|}_{\Gamma }{)}^{\prime }=(qs+2\pi pc,-qc-2\pi ps).$$

$${\big{\|}{\tfrac{\partial\varphi}{\partial x}|}_{\Gamma}\big{\|}}_{2}^{2}=\frac{1}{H}\big{(}(qc+2\pi ps)^{2}+(qs+2\pi pc)^{2}\big{)}\,,$$

$$\det({\tfrac{\partial\varphi}{\partial x}|}_{\Gamma},({\tfrac{\partial\varphi}{\partial x}|}_{\Gamma})^{\prime})=\cfrac{1}{H}\big{(}(qc+2p\pi s)^{2}+(qs+2p\pi c)^{2}\big{)}={\big{\|}{\tfrac{\partial\varphi}{\partial x}|}_{\Gamma}\big{\|}}_{2}^{2}\,.$$

$$\operatorname{ind}_{\varphi\circ\Gamma}(d\varphi\circ Z\circ\varphi^{-1})=\cfrac{1}{2\pi}\int_{0}^{2\pi}\frac{\det({\tfrac{\partial\varphi}{\partial x}|}_{\Gamma},({\tfrac{\partial\varphi}{\partial x}|}_{\Gamma})^{\prime})}{{\big{\|}{\tfrac{\partial\varphi}{\partial x}|}_{\Gamma}\big{\|}}_{2}^{2}}\,dt=1\,,$$

$${\mathrm{ind}}_{\phi \circ \Gamma }(d\psi \circ X\circ {\psi }^{-1})=1.$$

$$D\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1. Introduction and terminology

The existence of a first integral for a vector field is a classical problem in the theory of dynamical systems. One of the main tools to this aim is the study of the rectification or uniformization of a vector field. Roughly speaking, we understand as uniformization of a vector field with plane domain, as the existence of a diffeomorphic map that brings such a vector field to a constant vector field. Local uniformization is a well known property of any vector field with mild conditions (see e.g [4, Theorem 1.2]). Regarding global plane domains, there are several results over time that deal with several techniques in order to find a first integral or solve a related problem, see for instance [2, 6, 7, 9, 10, 11].

The technique we introduce in the present note is based in the ideas presented by Finn in [5] (who studied the behavior of the index of a plane direction field along a Jordan curve that represent the boundary of the domain) and the suggestion of Martínez-Alfaro for the generalization of this concept to other not plane manifolds. In [5], the author studied direction fields which have index [math] or $2$ along suitable curves. Thanks to the generalization we give we can extend this study to the index $1$. In brief, a vector field is uniformizable the sense of Finn, if it is conjugate with a constant one, both plane vector fields. In our case we assume that the constant vector field can be defined over the torus. We will see that this point of view will allows us to obtain an application to the existence of a first integral for certain vector fields.

We have organized the paper in a unique section where we develop the ideas described above. Let us introduce some terminology for the paper. Let $M$ be a manifold, we denote by $T(M)$ he bundle of tangent vector spaces of $M$. Let $D\subseteq\mathbb{R}^{2}$ be an closed plane domain, recall that a vector field $X\colon D\to T(D)$ is said to be uniformizable if it is conjugate with a constant vector field $U\colon E\to T(E)$ in the plane, for a certain plane domain $E$, i.e. if there exists a diffeomorphic map $\varphi\colon D\to E$ so that $d\varphi\circ X=U\circ\varphi$ (see [5, Definition 3] and [4, Section 1.3]). We denote the torus by $T^{2}$. The definition of the index of a direction field along a curve can be found in [5, Section 1]. Let us define it for a vector field. We assume that a Jordan curve is an smooth injective closed curve. As usual we sometimes identify the parametrization of a curve with its image. Let $D$ be a plane domain and let $\Gamma\colon[0,2\pi]\to D$ be a Jordan curve on $D$. Let $X_{\Gamma}$ be the vector field defined on $\Gamma$, and $\delta(X_{\Gamma})$ denotes the change of angle from the vector $X_{\Gamma}(0)$ to the vector $X_{\Gamma}(2\pi)$. Let $X$ be a vector field defined on $D$, we define the index of the vector field $X$ along the curve $\Gamma$ as

[TABLE]

A well known formula to compute this index is:

[TABLE]

see for instance [3]. A parametrization of the vector field can be written as $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$, where $P,Q\colon D\to\mathbb{R}$ and $x,y$ are suitable coordinates of $D$. Let $h\colon D\to\mathbb{R}$ be a differentiable function. We say that $h$ is a first integral of $X$ if $X(h)=Ph_{x}+Qh_{y}=0$, where $h_{x}$ and $h_{y}$ denotes the respective derivatives. We will mention a suitable domain called corona, which is a set of the form $\{(x,y)\in\mathbb{R}^{2}:0<r<\|(x,y)\|_{2}<R<\infty\}$. We denote by $S^{n}$ the sphere of dimension $n\in\mathbb{N}$. We will identify $T^{2}$ with $S^{1}\times S^{1}$ and with $\mathbb{R}^{2}/[0,2\pi]^{2}$. The type $(p,q)$ of a Jordan curve on the torus ($S^{1}\times S^{1}$) is the number of laps around each circumference $S^{1}$. We can see the type as an element of the homotopy group $\pi_{1}(T^{2})$. We refer to [1, 4] for notation, definitions and basic results that not appear here.

2. First integrals and uniformization over the torus

In this unique section we introduce the concept of uniformization over the torus and illustrates by means of an example that such a concept is useful for the existence of a first integral of a given vector field. Then, we give a condition for the uniformization over the torus of certain vector fields that involves the index along the boundary of the domain of these vector fields. We will assume that a domain is a bounded and connected region delimited by Jordan curves. We also assume that every function and vector field will be sufficiently differentiable.

Definition 2.1**.**

Let $r\geq 1$ and $D\subseteq\mathbb{R}^{2}$ be a bounded and connected domain. Let $X\colon D\to T(D)$ be a vector field. Let $\Delta\subseteq T^{2}$ be a domain. We say that $X$ is uniformizable over the torus if there exists $\Delta\subseteq T^{2}$ such that $X$ is conjugate to a constant vector field defined on $\Delta$, i.e. there exists a constant vector field $Z\colon\Delta\to T(\Delta)$ and a diffeomorphism $\psi\colon D\to\Delta$ such that $d\psi\circ X=Z\circ\psi$.

In the present paper we just deal with the definition above, over the torus, but one can easily extend it to any other manifold.

Example 2.2**.**

The constant vector field $X=a\;\partial/\partial x+b\;\partial/\partial y$, where $a,b\in\mathbb{R}$ so that $a/b\in\mathbb{Q}\setminus\{0\}$, defined in a domain of $\mathbb{R}^{2}$, is a clear example of uniformizable vector field over the torus and over the cylinder.

Let us give an application to the existence of a first integral. Observe that we also can apply the following to the Finn uniformization.

Proposition 2.3**.**

Let $D\subseteq\mathbb{R}^{2}$ be a compact connected domain and $X$ be a vector field on $D$ with Hausdorff flow. Assume that $X$ is uniformizable over the torus, then there is a first integral for $X$.

Proof.

By hypothesis, there is a diffeomorphism $\psi=(f,g)$ through which $X$ is conjugate to a constant vector field $Z:=a\frac{\partial}{\partial u}+b\frac{\partial}{\partial v}$ on the torus, where $a,b\in\mathbb{R}^{r}\setminus\{0\}$ satisfy that $a/b$ (or $b/a$) is a rational number. Moreover, the partial differential equation

[TABLE]

has solution (see for instance [8]). Let $h$ be such a solution. We claim that $h$ is a first integral of $X$.

Let us denote by $(u,v)$ the coordinates in $T^{2}$ and by $(x,y)$ the coordinates in $D$. By hypothesis we know that $d\psi\circ X=Z\circ\psi$. Let us denote $X=P(x,y)\frac{\partial}{\partial x}+Q(x,y)\frac{\partial}{\partial y}$, where $\dot{x}=P(x,y)$ and $\dot{y}=Q(x,y)$. On the one hand, since $Z$ is constant, there exists $H$ a first integral of $Z$. We can choose, for instance, $H(u,v)=bu-av$, since $H_{u}=b$ and $H_{v}=-a$ and $Z(H)=aH_{u}+bH_{v}=0$. On the other hand we can choose the variables $u$ and $v$ so that $\psi^{-1}(u,v)=(x,y)$, hence

[TABLE]

Therefore

[TABLE]

We know that $h$ is solution of (2.1), hence by (2.2)

[TABLE]

That means that $h$ is a first integral for $X$. ∎

Example 2.4**.**

Let $D\subseteq\mathbb{R}^{2}$ be a plane domain and $X\colon D\to T(D)$ be vector field. Let $(a,b)\in\mathbb{R}^{2}\setminus 0$ such that $a/b$ (or $b/a$) is rational. Assume that there is a diffeomorphism $\psi\colon D\to\Delta\subseteq T^{2}$ as the one defined above $\psi:=(f,g)$ and assume that $X:=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ is such that

[TABLE]

and

[TABLE]

where the numerators does not vanishes at the same time in $D$, so the vector field will be regular. Observe that the denominator is the Jacobian of $\psi$, hence it not vanishes in $D$. This vector field is uniformizable by means of the constant vector field $a\frac{\partial}{\partial u}+b\frac{\partial}{\partial v}$, where $u,v$ denotes the coordinates in the torus.

Proposition and example above illustrates that uniformization over the torus could be a tool in order to find out the existence of a first integral, in consequence it is interesting to find conditions related to such a property. In the following theorem and corollary we show that there exist certain uniformizables vector fields such that the index along suitable Jordan curves, that we can take closer to the the boundary of its domain, is equal to $1$. Later, we easily deduce a corollary for a more general class of vector fields.

Let $Z\colon T^{2}\to T(T^{2})$ be a constant vector field and $\Gamma$ be a Jordan curve of type $(p,q)\in\mathbb{Z}^{2}\setminus\{0\}$ on the torus. Clearly $\operatorname{ind}_{\Gamma}Z=0$. Then we consider the set $T^{2}\setminus\Gamma_{0}$ for a suitable Jordan curve with the same type as $\Gamma$ and transform it diffeomorphically in a corona on $\mathbb{R}^{2}$. Next result states that the index of the image of such a constant vector field along the image of $\Gamma$ is $1$.

Theorem 2.5**.**

Let $\Gamma$ be a Jordan curve on the torus $T^{2}$ with type $(p,q)\in\mathbb{Z}^{2}\setminus\{0\}$. Let $Z\colon T^{2}\to T(T^{2})$ be a constant vector field. Then, there exist a Jordan curve $\Gamma_{0}$ such that $\Gamma\cap\Gamma_{0}=\emptyset$, a domain $D\in\mathbb{R}^{2}$ and a diffeomorphism $\varphi\colon T^{2}\setminus\Gamma_{0}\to D$ such that

[TABLE]

Proof.

We will use the representation of the torus by means of the usual equivalence relation in $[0,2\pi]^{2}$. Since the type of a curve is invariant by diffeomorphism, without loss of generality, we can assume that $\Gamma(t)=(pt,qt)$ for $t\in[0,2\pi]$. Again, without loss of generality we take the constant vector field $Z:=\partial/\partial x$. Thus, $\Gamma$ has type $(p,q)$ and the representation of $\Gamma$ in $\mathbb{R}^{2}/[0,2\pi]^{2}$ is the segment between $(0,0)$ and $(2p\pi,2q\pi)$ in $\mathbb{R}^{2}$. The angle of this segment with respect to the positive abscissa is $\arctan(\frac{q}{p})$.

Let the Jordan curve $\Gamma_{0}(t):=(pt+1/2,qt+1/2)$, clearly $\Gamma\cap\Gamma_{0}=\emptyset$. We define the diffeomorphism $\varphi\colon T^{2}\setminus\Gamma_{0}\to\mathbb{R}^{2}$ given by

[TABLE]

where

[TABLE]

is the clockwise rotation centered at the origin and angle $\arctan(\frac{q}{p})$, which move $\Gamma$ to the segment $[0,2\pi\sqrt{p^{2}+q^{2}}]\times\{0\}$; $\tau$ is the translation

[TABLE]

which translate the segment in the plane $[0,2\pi\sqrt{p^{2}+q^{2}}]\times\{0\}$ to the segment $[0,2\pi\sqrt{p^{2}+q^{2}}]\times\{2\pi\}$; and $\sigma$ is the change to polar variables, that is

[TABLE]

Therefore

[TABLE]

We now use the formula (1.1) in order to obtain the index of the vector field $d\varphi\circ Z\circ\varphi^{-1}$ along $\varphi\circ\Gamma$ by a direct computation. On the one hand we have that

[TABLE]

For the aim of brevity we define $H:=p^{2}+q^{2}$, $c:=\cos(Ht)$ and $s:=\sin(Ht)$. Hence

[TABLE]

defines a curve on a certain domain $D\subset\mathbb{R}^{2}$. On the other hand we get the following commutative diagram

[TABLE]

Therefore,

[TABLE]

The partial derivative of $\varphi$ with respect to $x$ is

[TABLE]

Thus, taking into account (2.4) and the given notation, we obtain that

[TABLE]

Now, having in mind that $c^{\prime}=-Hs$ and $s^{\prime}=Hc$, we can compute the derivative

[TABLE]

Finally, we have that the square norm is

[TABLE]

in consequence

[TABLE]

This equality together with (2.5), implies that

[TABLE]

which is the claim of the theorem. ∎

Remark 2.6*.*

Observe that the domain $D$ given in the proof of theorem above can be taken diffeomorphic to the interior of the corona $\{(x,y)\in\mathbb{R}^{2}:1\leq\sqrt{x^{2}+y^{2}}\leq 2\}$. Thus any vector field defined on a compact domain dipheomorphic to $C$, defined by two contours $\Gamma_{1}$ and $\Gamma_{2}$, so that it is uniformizable over the torus by means of the inverse of the map $\varphi$ defined in (2.3) is a good candidate that could satisfies that $\operatorname{ind}_{\Gamma_{i}}X=1$, for $i=1,2$.

Remark 2.7*.*

One can see that our approach to the study of the index is quite different to the one given by Finn in [5]. There are two main reasons that bring us to change the original point of view: First, the notion of index along a curve has evolved, in fact, sometimes it is called winding number of a vector field (see e.g. [3]). Second, some results given in the original paper, although they were correct, they were based (a little) on the intuition.

In our last result we provide a necessary condition for the uniformization over the torus for suitable vector fields.

Corollary 2.8**.**

Let $X$ be a vector field on a bounded and connected domain. Assume that $X$ is uniformizable over the torus through a domain in $T^{2}$ which is diffeomorphic to $T^{2}\setminus\Gamma_{0}$ for a Jordan curve $\Gamma_{0}$. Then, for each Jordan curve $\Gamma$ with the same type as $\Gamma_{0}$ satisfying that $\Gamma\cap\Gamma_{0}=\emptyset$, there exists a diffeomorphism $\psi$ such that

[TABLE]

Proof.

The proof is easy to deduce taking into account the following diagram

[TABLE]

where $Z$, $\widehat{Z}$, $\varphi$ and $\alpha$ are the maps defined in the hypothesis and $U$ and $\beta$ are defined in Theorem 2.5. Observe that we can choose $\widehat{Z}$ as a constant vector field, since $Z$ is constant. Now, having in mind that the vector field $\widehat{Z}=d\alpha\circ d\varphi\circ X\circ\varphi^{-1}\circ\alpha^{-1}$ is constant and applying theorem above for each $\Gamma$ as in the hypothesis, we obtain the result. ∎

Despite we cannot give a sufficient condition, the condition for the index is meaningful and provides a good clue for the wanted uniformization.

Acknowledgment. The original idea of a generalization of the uniformization in the sense of Finn was suggested by Professor José A. Martínez Alfaro around the year 2010. I am very grateful for his support and encouragement along those years.