# Relation between the minimal Lorentz surfaces in $\mathbb{R}^{4}_{2}$ and
$\mathbb{R}^{3}_{1}$

Krasimir Kanchev  ,  Ognian Kassabov  and  Velichka Milousheva  Department of
Mathematics and Informatics, Todor Kableshkov University of Transport, 158 Geo
Milev Str., 1574, Sofia, Bulgaria<EMAIL_ADDRESS>Institute of
Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev
Str. bl. 8, 1113, Sofia, Bulgaria<EMAIL_ADDRESS>Institute of
Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev
Str. bl. 8, 1113, Sofia, Bulgaria<EMAIL_ADDRESS>
###### Abstract.

In this paper we give Weierstrass-type representation formulas for the null
curves and for the minimal Lorentz surfaces in the Minkowski 3-space
$\mathbb{R}^{3}_{1}$ using real-valued functions. Applying the Weierstrass-
type representations for the null curves, we find a correspondence between the
null curves in $\mathbb{R}^{4}_{2}$ and the pairs of null curves in
$\mathbb{R}^{3}_{1}$. Based on this correspondence, we obtain a relation
between the minimal Lorentz surfaces in $\mathbb{R}^{4}_{2}$ and the pairs of
minimal Lorentz surfaces in $\mathbb{R}^{3}_{1}$.

###### Key words and phrases:

Minimal Lorentz surfaces, pseudo -Euclidean space, canonical coordinates,
Weierstrass-type representation

###### 2010 Mathematics Subject Classification:

Primary 53B30; Secondary 53A35; 53A10; 53B25

##  1\. Introduction

The study of minimal surfaces in different ambient spaces is one of the main
topics in both classical and modern differential geometry and has been
attracting the attention of many mathematicians for more than two centuries.
In the last years, the geometry of Lorentz surfaces in pseudo-Euclidean spaces
has been of wide interest, since pseudo-Riemannian geometry has many
interesting applications in Physics.

One of the instruments in the study of minimal surfaces in the Euclidean space
is the generalized Gauss map [15]. In [10], Gálvez et al. obtained a
representation for spacelike surfaces in the Minkowski three-dimensional space
$\mathbb{R}^{3}_{1}$ using the Gauss map and the conformal structure given by
the second fundamental form. Timelike surfaces in $\mathbb{R}^{3}_{1}$ with
prescribed Gauss curvature and Gauss map were studied in [1], where a
Kenmotsu-type representation for these surfaces was given. In [20], M. Magid
proved that the Gauss map and the mean curvature of a timelike surface satisfy
a system of partial differential equations and found a Weierstrass
representation formula for timelike surfaces in $\mathbb{R}^{3}_{1}$.

A representation of a minimal Lorentz surface was given by M. Dussan and M.
Magid in [8], where they solved the Björling problem for timelike surfaces in
$\mathbb{R}^{4}_{2}$. The Björling problem for timelike surfaces in the
Lorentz-Minkowski spaces $\mathbb{R}^{3}_{1}$ and $\mathbb{R}^{4}_{1}$ was
solved in [4] and [9], respectively. Lorentz surfaces in $\mathbb{R}^{4}_{2}$
were studied in [3] and [21] by use of spinors and Lorentz numbers (also known
as para-complex, split-complex, double or hyperbolic numbers), and a
generalized Weierstrass representation of a minimal Lorentz surface in
$\mathbb{R}^{4}_{2}$ was given in [21]. Many other researches have studied
spacelike and timelike surfaces in a 3-dimensional Lorentz space via
Weierstrass-type formulas, see e.g. [6], [19], [22].

On the other hand, the class of minimal surfaces in the Euclidean space
$\mathbb{R}^{4}$ and the classes of spacelike or timelike surfaces with
vanishing mean curvature in the pseudo-Euclidean spaces $\mathbb{R}^{4}_{1}$
and $\mathbb{R}^{4}_{2}$ are characterized by special systems of PDEs (called
natural systems) for the Gauss curvature $K$ and the normal curvature
$\varkappa$. Explicit solutions to these systems of PDEs can be found by use
of a special type of Weierstrass representation (in terms of canonical
coordinates), which is called canonical Weierstrass-type representation. This
problem is solved for the class of minimal spacelike surfaces in
$\mathbb{R}^{4}_{1}$ in [12]. In the case of a minimal Lorentz surface in
$\mathbb{R}^{4}_{2}$, the problem is solved in [17].

A correspondence between the solutions of the system of natural PDEs
describing the minimal surfaces in $\mathbb{R}^{4}$ and the pairs of solutions
to the natural PDEs of the minimal surfaces in $\mathbb{R}^{3}$ is obtained in
[11]. Using this correspondence one can find also a correspondence between the
minimal surfaces in $\mathbb{R}^{4}$ and the pairs of minimal surfaces in
$\mathbb{R}^{3}$. Applying a similar approach, a relation between the maximal
spacelike surfaces in $\mathbb{R}^{4}_{2}$ and the pairs of maximal spacelike
surfaces in $\mathbb{R}^{3}_{1}$ is found in [13].

The main purpose of the present paper is to find a correspondence of this kind
between the minimal Lorentz surfaces in $\mathbb{R}^{4}_{2}$ and the pairs of
minimal Lorentz surfaces in $\mathbb{R}^{3}_{1}$. As the case of Lorentz
surfaces is much more complicated than the case of spacelike surfaces, we use
a different method, applying the theory of the Weierstrass-type
representations for the null curves in $\mathbb{R}^{4}_{2}$, which was
developed in [16]. First, we use the main definitions and results from [16]
and obtain a correspondence between the null curves in $\mathbb{R}^{4}_{2}$
and the pairs of null curves in $\mathbb{R}^{3}_{1}$.

It is known, that there exists a Weierstrass-type representation for the
minimal Lorentz surfaces in $\mathbb{R}^{3}_{1}$, analogous to the classical
Weierstrass representation in $\mathbb{R}^{3}$, see [18]. An analogous natural
representation for the surfaces in $\mathbb{R}^{4}_{2}$ exists. In this paper,
we apply the method developed in [17] and [16] which is based on the
Weierstrass-type representation for the null curves. This method has the
obvious advantage that the surfaces and their invariants are expressed in
terms of real-valued functions.

In Sections 3 and 4, we recall the Weierstrass-type representation for the
null curves and for the minimal Lorentz surfaces in $\mathbb{R}^{4}_{2}$ and
give similar Weierstrass-type representation formulas for the null curves and
for the minimal Lorentz surfaces in $\mathbb{R}^{3}_{1}$ using real-valued
functions. We also define canonical coordinates and give canonical
Weierstrass-type representation formulas for minimal Lorentz surfaces in
$\mathbb{R}^{3}_{1}$. Using the Weierstrass-type representation for the null
curves, in Section 5, we give a relation between the non-degenerate null
curves in $\mathbb{R}^{4}_{2}$ and $\mathbb{R}^{3}_{1}$. In Section 6, we use
the canonical isotropic coordinates for the minimal Lorentz surfaces and
obtain a relation between the set of minimal Lorentz surfaces in
$\mathbb{R}^{4}_{2}$ and the set of ordered pairs of minimal Lorentz surfaces
in $\mathbb{R}^{3}_{1}$. Finally, we give formulas expressing the Gauss
curvature $K$ and the normal curvature $\varkappa$ of a minimal Lorenz surface
in $\mathbb{R}^{4}_{2}$ in terms of the Gauss curvatures of the corresponding
minimal Lorentz surfaces in $\mathbb{R}^{3}_{1}$.

##  2\. Preliminaries

We denote by $\mathbb{R}^{3}_{1}$, resp. $\mathbb{R}^{4}_{2}$, the standard
three-dimensional, resp. four-dimensional pseudo-Euclidean space with the
indefinite scalar product

$\mathrm{a}\cdot\mathrm{b}=-a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3},\,$

resp.

$\mathrm{a}\cdot\mathrm{b}=-a_{1}b_{1}+a_{2}b_{2}-a_{3}b_{3}+a_{4}b_{4}\,.$

All considered functions, curves and surfaces in this paper are supposed to be
real and smooth.

Let $\mathcal{M}=(\mathcal{D},\mathrm{x})$ be a Lorentz surface in
$\mathbb{R}^{4}_{2}$, where $\mathcal{D}\subset\mathbb{R}^{2}$ , and
$\mathrm{x}:\mathcal{D}\to\mathbb{R}^{4}_{2}$ is an immersion. It is known
(see e.g. [5]) that for every minimal Lorentz surface in $\mathbb{R}^{4}_{2}$
there exists a pair $(\alpha_{1},\alpha_{2})$ of null curves in
$\mathbb{R}^{4}_{2}$, such that $\alpha^{\prime 2}_{1}=0$, $\alpha^{\prime
2}_{2}=0$, ${\alpha^{\prime}_{1}\cdot\alpha^{\prime}_{2}\neq 0}$ and

(2.1)
$\mathrm{x}(t_{1},t_{2})=\frac{\alpha_{1}(t_{1})+\alpha_{2}(t_{2})}{2}\,,$

where $(t_{1},t_{2})$ is a pair of isotropic coordinates for $\mathcal{M}$.
This implies that with respect to such parametrization the coefficients $E$,
$F$, $G$ of the first fundamental form of $\mathcal{M}$ are expressed as
follows:

$E=G=0\,;\qquad F=\frac{\alpha^{\prime}_{1}\cdot\alpha^{\prime}_{2}}{4}\,.$

Hence, the change of the sign of one of the isotropic coordinates
$(t_{1},t_{2})$ changes the sign of $F$. Consequently, without loss of
generality we may suppose that $F>0$ (or $F<0$) at each point of
$\mathcal{M}$.

On the other hand, if $\mathrm{x}$ is parametrized by isothermal coordinates,
then $\mathcal{M}$ is minimal if and only if $\mathrm{x}$ is hyperbolic
harmonic (see [16]). Then, we can introduce a function $\mathrm{y}$, which is
hyperbolic harmonic conjugate to $\mathrm{x}$. The induced minimal Lorentz
surface is called conjugate to the given surface. In isotropic coordinates
$(t_{1},t_{2})$, we have:

(2.2)
$\mathrm{y}(t_{1},t_{2})=\frac{\alpha_{1}(t_{1})-\alpha_{2}(t_{2})}{2}\,.$

The null curve $\alpha$ is called _nondegenerate_ if
${\alpha^{\prime\prime}}^{2}\neq 0$ everywhere. Note that if we make the
parameter change $t=t(s),\,t^{\prime}\neq 0$ , in the null curve $\alpha(t)$
then
${\alpha^{\prime\prime}_{s}}^{2}={\alpha^{\prime\prime}_{t}}^{2}{t^{\prime}}^{4}$.
This implies that the degeneracity of a null curve does not depend on the
parametrization. A minimal Lorentz surface $\mathcal{M}$ in
$\mathbb{R}^{4}_{2}$ is said to be of _general type_ , if its corresponding
null curves $\alpha_{1}$ and $\alpha_{2}$ are both nondegenerate [14], [16].
This definition does not depend on the local coordinates and is invariant
under a motion in $\mathbb{R}^{4}_{2}$.

######  Definition 2.1.

[14], [16] Let $\mathcal{M}$ be a minimal Lorentz surface of general type in
$\mathbb{R}^{4}_{2}$ with corresponding null curves ${\alpha_{1}}$,
${\alpha_{2}}$. We say that the surface $\mathcal{M}$ is of

  * •

_first type_ , if both vectors ${\alpha^{\prime\prime}_{1}}$,
${\alpha^{\prime\prime}_{2}}$ are spacelike;

  * •

_second type_ , if both vectors ${\alpha^{\prime\prime}_{1}}$,
${\alpha^{\prime\prime}_{2}}$ are timelike;

  * •

_third type_ , if one of the vectors ${\alpha^{\prime\prime}_{1}}$,
${\alpha^{\prime\prime}_{2}}$ is spacelike and the other one is timelike.

Note that this definition does not depend on the choice of the local
coordinates and is invariant under a motion in $\mathbb{R}^{4}_{2}$. Moreover,
for a surface ot third type we may suppose ${\alpha^{\prime\prime}_{1}}^{2}>0$
and ${\alpha^{\prime\prime}_{2}}^{2}<0$ (otherwise we may renumerate the
curves and the parameters).

A non-degenerate curve $\alpha$ is said to be parametrized by a _natural
parameter_ , if ${\alpha^{\prime\prime}}^{2}=\pm 1$ . Since this parameter
plays a role similar to the role of an arc-length parameter for non-null
curves, it is known in the literature as pseudo arc-length parameter [24, 7].
If $t$ is an arbitrary parameter of $\alpha$, then a natural parameter $s$ is
given by

$s=\int\sqrt[4]{\big{|}{\alpha^{\prime\prime}}^{2}(t)\big{|}}\>dt\,.$

######  Definition 2.2.

[16] If $t_{1}$, resp. $t_{2}$ is a natural parameter of $\alpha_{1}$, resp.
$\alpha_{2}$, then, we say that $(t_{1},t_{2})$ are _canonical coordinates_ of
the corresponding minimal Lorentz surface of general type $\mathcal{M}$ in
$\mathbb{R}^{4}_{2}$, (if $\mathcal{M}$ is of third type we suppose
${\alpha^{\prime\prime}_{1}}^{2}=1$ and ${\alpha^{\prime\prime}_{2}}^{2}=-1$
).

If $t$ and $s$ are two natural parameters of $\alpha$, then it follows that
$t^{\prime}(s)=\pm 1$ . Consequently,

(2.3) $t=\pm s+c\,,\qquad c=\textnormal{const}\,.$

This shows that the canonical coordinates of a minimal Lorentz surface of
general type in $\mathbb{R}^{4}_{2}$ are defined up to numeration, sign and
additive constants. For the surfaces of third type the numeration is fixed,
since ${\alpha^{\prime\prime}_{1}}^{2}=1$ and
${\alpha^{\prime\prime}_{2}}^{2}=-1$ .

Note that we can consider $\mathbb{R}^{3}_{1}$ as a subspace of
$\mathbb{R}^{4}_{2}$, defined by $x_{3}=0$. So, every object in
$\mathbb{R}^{3}_{1}$ can be considered as an object in $\mathbb{R}^{4}_{2}$.
Hence, the above remarks about the existence and uniqueness of the natural
parameters of curves and the canonical coordinates of the surfaces in
$\mathbb{R}^{4}_{2}$ hold also for $\mathbb{R}^{3}_{1}$. Moreover, the minimal
Lorentz surfaces in $\mathbb{R}^{3}_{1}$ considered as surfaces in
$\mathbb{R}^{4}_{2}$ are always of first type, since
${\alpha^{\prime\prime}}^{2}\geq 0$ for the null curves in
$\mathbb{R}^{3}_{1}$.

The main invariants of a minimal Lorentz surface $\mathcal{M}$ in
$\mathbb{R}^{4}_{2}$ are the Gauss curvature $K$ and the curvature of the
normal connection (normal curvature) $\varkappa$. The following statement
holds true.

######  Proposition 2.1.

[14] A minimal Lorentz surface $\mathcal{M}$ in $\mathbb{R}^{4}_{2}$ is of
general type if and only if $K^{2}-\varkappa^{2}\neq 0$ . Moreover,
$\mathcal{M}$ is of first or second type (resp. of third type) if and only if
$K^{2}-\varkappa^{2}>0$ (resp. $K^{2}-\varkappa^{2}<0$ ).

##  3\. Weierstrass-type representation for null curves in
$\mathbb{R}^{4}_{2}$ and $\mathbb{R}^{3}_{1}$

The null curves in $\mathbb{R}^{4}_{2}$ can be determined by a triple of real
functions as follows.

######  Theorem 3.1.

[16] Let $\alpha$ be a null curve in $\mathbb{R}^{4}_{2}$ such that
$\alpha^{\prime}=(\xi_{1},\xi_{2},\xi_{3},\xi_{4})$ satisfies
$\xi_{1}-\xi_{2}\neq 0$ . Then, $\alpha^{\prime}$ has the form

(3.1) $\alpha^{\prime}=f\big{(}\,gh+1\,,\,gh-1\,,\,h-g\,,\,h+g\,\big{)},$

where the functions $f\neq 0$, $g$, $h$ are uniquely determined by

$f=\displaystyle\frac{1}{2}(\xi_{1}-\xi_{2})\,;\qquad
g=\displaystyle\frac{\xi_{4}-\xi_{3}}{\xi_{1}-\xi_{2}}\;;\qquad
h=\displaystyle\frac{\xi_{4}+\xi_{3}}{\xi_{1}-\xi_{2}}\;.$

Conversely, let $(f,g,h)$ be a triple of functions, $f\neq 0$ . Then there
exists a null curve $\alpha$ in $\mathbb{R}^{4}_{2}$ such that (3.1) holds and
$\xi_{1}-\xi_{2}\neq 0$ .

######  Remark 3.1.

The restriction $\xi_{1}-\xi_{2}\neq 0$ in the last theorem is not essential
for the geometric properties of the curve, because every null curve in
$\mathbb{R}^{4}_{2}$ with $\xi_{1}-\xi_{2}=0$ can be transformed by a proper
motion to a curve satisfying the condition $\xi_{1}-\xi_{2}\neq 0$ .

Now, consider the null curves in $\mathbb{R}^{3}_{1}$. Note that (3.1) implies
that $x_{3}=0$ is equivelent to $g=h$. So, we obtain the following analogue of
Theorem 3.1:

######  Theorem 3.2.

Let $\alpha$ be a null curve in $\mathbb{R}^{3}_{1}$, such that
$\alpha^{\prime}=(\xi_{1},\xi_{2},\xi_{3})$ satisfies $\xi_{1}-\xi_{2}\neq 0$
. Then, $\alpha^{\prime}$ has the form:

(3.2) $\alpha^{\prime}=f\big{(}\,g^{2}+1\,,\,g^{2}-1\,,\,2g\,\big{)},$

where $f\neq 0$ and $g$ are functions, given uniquely by

(3.3) $f=\displaystyle\frac{1}{2}(\xi_{1}-\xi_{2})\,;\qquad
g=\displaystyle\frac{\xi_{3}}{\xi_{1}-\xi_{2}}\;.$

Conversely, let $(f,g)$ be a pair of functions, $f\neq 0$ . Then, there exists
a null curve $\alpha$ in $\mathbb{R}^{3}_{1}$, such that $\alpha^{\prime}$ has
the form (3.2) and $\xi_{1}-\xi_{2}\neq 0$ .

Formulas (3.2) and (3.1) are analogous to the classical Weierstrass formula
for minimal surfaces in $\mathbb{R}^{3}$. So, we call them the _Weierstrass-
type representation_ of a null curve in $\mathbb{R}^{3}_{1}$, resp.
$\mathbb{R}^{4}_{2}$, _gererated by the pair of functions $(f,g)$_, resp. _the
triple of functions $(f,g,h)$_.

In what follows, we find the transformation formulas under proper motions in
$\mathbb{R}^{3}_{1}$ for the functions that participate in (3.2). First, we
consider a proper orthochronous motion.

######  Theorem 3.3.

Let $\hat{\alpha}$ and $\alpha$ be null curves in $\mathbb{R}^{3}_{1}$ having
Weierstrass-type representations of the form (3.2). Then, the following
conditions are equivalent:

  1. (1)

$\hat{\alpha}$ and $\alpha$ are related by a proper orthochronous motion in
$\mathbb{R}^{3}_{1}$ of the type
$\hat{\alpha}(t)=A\alpha(t)+\mathrm{b}$, where
$A\in\mathbf{SO}^{+}(2,1,\mathbb{R})$ and $\mathrm{b}\in\mathbb{R}^{3}_{1}$.

  2. (2)

The functions in the Weierstrass-type representation formulas for
$\hat{\alpha}$ and $\alpha$ are related by

(3.4)
$\hat{f}=f(cg+d)^{2}\,;\qquad\hat{g}=\displaystyle\frac{ag+b}{cg+d}\,;\qquad
a,b,c,d\in\mathbb{R}\,;\;\;ad-bc=1\,.$

###### Proof.

For any vector $\mathrm{x}\in\mathbb{R}^{3}_{1}$ we define a $2\times 2$
matrix $S$ by:

$\mathrm{x}=(x_{1},x_{2},x_{3})\ \leftrightarrow\
S=\left(\begin{array}[]{cc}-x_{3}&x_{2}+x_{1}\\\
x_{2}-x_{1}&x_{3}\end{array}\right).$

It is easy to see that the above correspondence gives a linear isomorphism
between $\mathbb{R}^{3}_{1}$ and the linear space of all $2\times 2$ real
matrices with vanishing trace. Moreover, this isomorphism satisfies the
condition $\det S=-\mathrm{x}^{2}$. So, every linear operator, acting in the
space of $2\times 2$ real matrices, which preserves the determinant and the
trace, determines an orthogonal operator in $\mathbb{R}^{3}_{1}$. If
$B\in\mathbf{SL}(2,\mathbb{R})$, then $\det BSB^{-1}=\det S$ and
$\mathop{\mathrm{trace}}\nolimits BSB^{-1}=\mathop{\mathrm{trace}}\nolimits
S$. For each such matrix $B$ there is an orthogonal matrix
$A\in\mathbf{O}(2,1,\mathbb{R})$ corresponding to $B$. Consequently, there
exists a group homomorphism $B\rightarrow A$:

(3.5) $\hat{S}=BSB^{-1}\ \rightarrow\ \hat{\mathrm{x}}=A\mathrm{x}\,.$

This homomorphism from $\mathbf{SL}(2,\mathbb{R})$ in
$\mathbf{O}(2,1,\mathbb{R})$ is a spinor map. The kernel of this map has two
elements (see [23]): $I$ and $-I$, $I$ being the unit matrix, and the image
coincides with the connected component of the unit element in
$\mathbf{O}(2,1,\mathbb{R})$. This is the group
$\mathbf{SO}^{+}(2,1,\mathbb{R})$ of proper orthochronous motions in
$\mathbb{R}^{3}_{1}$. The form of the kernel and the image of (3.5) imply that
the map (3.5) gives rise to a group isomorphism:

$\mathbf{SL}(2,\mathbb{R})/\\{I,-I\\}\ \cong\
\mathbf{SO}^{+}(2,1,\mathbb{R})\,.$

Applying this to the null curves in $\mathbb{R}^{3}_{1}$, we derive the
assertion. ∎

Every proper non-orthochronous motion in $\mathbb{R}^{3}_{1}$ is a composition
of a proper orthochronous motion with a proper non-orthochronous motion,
obtained with changing the signs of the first two coordinates, i.e. with
changing the signs of $f$ and $g$. Hence, the transformation formulas for a
proper non-orthochronous motion in $\mathbb{R}^{3}_{1}$ are:

(3.6)
$\hat{f}=-f(cg+d)^{2}\,;\qquad\hat{g}=\displaystyle\frac{ag+b}{cg+d}\,;\qquad
a,b,c,d\in\mathbb{R}\,;\;\;ad-bc=-1\,.$

Analogously, we can see that the transformation formulas for a non-proper
orthochronous motion in $\mathbb{R}^{3}_{1}$ are

(3.7)
$\hat{f}=f(cg+d)^{2}\,;\qquad\hat{g}=\displaystyle\frac{ag+b}{cg+d}\,;\qquad
a,b,c,d\in\mathbb{R}\,;\;\;ad-bc=-1\,,$

and the transformation formulas for a non-proper non-orthochronous motion in
$\mathbb{R}^{3}_{1}$ are

(3.8)
$\hat{f}=-f(cg+d)^{2}\,;\qquad\hat{g}=\displaystyle\frac{ag+b}{cg+d}\,;\qquad
a,b,c,d\in\mathbb{R}\,;\;\;ad-bc=1\,.$

Let $\alpha$ be a null curve in $\mathbb{R}^{4}_{2}$ parametrized by an
arbitrary parameter.

The following statement holds true.

######  Proposition 3.4.

[16] Let $\alpha$ be a null curve in $\mathbb{R}^{4}_{2}$ with Weierstrass-
type representation (3.1). Then, $\alpha$ is non-degenerate if and only if
$g^{\prime}h^{\prime}\neq 0$ at each point.

Now, let $\alpha$ be parametrized by a natural parameter, i.e.
${\alpha^{\prime\prime}}^{2}=\pm 1$. Using (3.1) we get
${\alpha^{\prime\prime}}^{2}=4f^{2}g^{\prime}h^{\prime}$ and hence,
$4f^{2}g^{\prime}h^{\prime}=\pm 1$ , or equivalently
$4f^{2}|g^{\prime}h^{\prime}|=1$ . So, we obtain:

(3.9)
$f=\displaystyle\frac{\omega}{2\sqrt{|g^{\prime}h^{\prime}|}}\,;\qquad\omega=\pm
1\,,$

and the sign of $\omega$ coinsides with the sign of $f$. Using (3.1) and
(3.9), we obtain:

######  Theorem 3.5.

[16] Let $\alpha$ be a null curve in $\mathbb{R}^{4}_{2}$, parametrized by a
natural parameter, i.e. ${\alpha^{\prime\prime}}^{2}=\pm 1$. Then, $\alpha$
has the following Weierstrass-type representation:

(3.10)
$\alpha^{\prime}=\displaystyle\frac{\omega}{2\sqrt{|g^{\prime}h^{\prime}|}}\left(gh+1,gh-1,h-g,h+g\right),$

where $g$ and $h$ are smooth real functions satisfying
$g^{\prime}h^{\prime}\neq 0$, and $\omega=\pm 1$. The functions $g$ and $h$ as
well as $\omega$ are determined uniquely by $\alpha$ in accordance with (3.1).

Conversely, if $(g,h)$ is a pair of smooth real functions satisfying
$g^{\prime}h^{\prime}\neq 0$ and $\omega=\pm 1$, then there exists a null
curve $\alpha$ in $\mathbb{R}^{4}_{2}$ parametrized by a natural parameter,
such that $\alpha^{\prime}$ is expressed by the given functions in the form
(3.10).

######  Remark 3.2.

If we apply (3.10) to the pair of functions $(g,h)$, we obtain two null curves
for $\omega=1$ and $\omega=-1$. These curves are related by a proper
orthochronous motion in $\mathbb{R}^{4}_{2}$.

Note also that, if $\alpha$ is a null curve in $\mathbb{R}^{3}_{1}$, then
${\alpha^{\prime\prime}}^{2}=4f^{2}g^{\prime 2}$ implies the following
statement:

######  Proposition 3.6.

If $\alpha$ is a null curve in $\mathbb{R}^{3}_{1}$ with Weierstrass-type
representation (3.2), then $\alpha$ is non-degenerate if and only if
$g^{\prime}\neq 0$ at each point.

If the parameter in (3.2) is natural, then
${\alpha^{\prime\prime}}^{2}=4f^{2}g^{\prime 2}$ implies $2|f||g^{\prime}|=1$
. Hence, we find

(3.11) $f=\displaystyle\frac{\omega}{2|g^{\prime}|}\,;\qquad\omega=\pm 1,\,$

where the sign of $\omega$ coincides with the sign of $f$. Now, from (3.2) and
(3.11) we derive

######  Theorem 3.7.

Let $\alpha$ be a null curve in $\mathbb{R}^{3}_{1}$, parametrized by a
natural parameter, i.e. ${\alpha^{\prime\prime}}^{2}=1$. Then, $\alpha$ has
the following Weierstrass-type representation:

(3.12)
$\alpha^{\prime}=\displaystyle\frac{\omega}{2|g^{\prime}|}\left(\,g^{2}+1\,,\,g^{2}-1\,,\,2g\,\right),$

where $g^{\prime}\neq 0$ and $\omega=\pm 1$ . The function $g$ and the sign of
$\omega$ are uniquely determined by $\alpha$.

Conversely, if $g$ is a function, $g^{\prime}\neq 0$, $\omega=\pm 1$, then
there exists a null curve $\alpha$ in $\mathbb{R}^{3}_{1}$ parametrized by a
natural parameter such that (3.12) holds.

######  Remark 3.3.

Using (3.12) for a given function $g$ we obtain two null curves in
$\mathbb{R}^{3}_{1}$ corresponding to $\omega=1$ and $\omega=-1$,
respectively. These two curves are related by a non-proper motion in
$\mathbb{R}^{3}_{1}$.

##  4\. Weierstrass-type representation for minimal Lorentz surfaces in
$\mathbb{R}^{4}_{2}$ and $\mathbb{R}^{3}_{1}$

Let us recall the following result given in [16].

######  Theorem 4.1.

Let $\mathcal{M}$ be a minimal Lorentz surface in $\mathbb{R}^{4}_{2}$ and
$(\alpha_{1},\alpha_{2})$ be its corresponding pair of null curves. Then,

(4.1)
$\alpha^{\prime}_{i}=f_{i}\big{(}g_{i}h_{i}+1,g_{i}h_{i}-1,h_{i}-g_{i},h_{i}+g_{i}\big{)};\qquad
i=1,2,$

where $(f_{i},g_{i},h_{i})$; $i=1,2$ are two triples of smooth real functions
such that

(4.2) $f_{1}(t_{1})\neq 0;\qquad f_{2}(t_{2})\neq 0;\qquad g_{1}(t_{1})\neq
g_{2}(t_{2});\qquad h_{1}(t_{1})\neq h_{2}(t_{2}).$

Conversely, if $(f_{i},g_{i},h_{i})$; $i=1,2$ are two triples of smooth real
functions satisfying conditions (4.2), then, there exists a minimal Lorentz
surface in $\mathbb{R}^{4}_{2}$ such that its corresponding null curves have a
Weierstrass-type representation of the form (4.1) expressed by the given
functions.

The Gauss curvature $K$ and the normal curvature $\varkappa$ are expressed as
follows [16]:

(4.3)
$\begin{array}[]{llr}K&=&\displaystyle\frac{2}{f_{1}f_{2}(g_{1}-g_{2})(h_{1}-h_{2})}\left(\displaystyle\frac{g^{\prime}_{1}g^{\prime}_{2}}{(g_{1}-g_{2})^{2}}+\displaystyle\frac{h^{\prime}_{1}h^{\prime}_{2}}{(h_{1}-h_{2})^{2}}\right);\\\\[12.91663pt]
\varkappa&=&\displaystyle\frac{2}{f_{1}f_{2}(g_{1}-g_{2})(h_{1}-h_{2})}\left(\displaystyle\frac{g^{\prime}_{1}g^{\prime}_{2}}{(g_{1}-g_{2})^{2}}-\displaystyle\frac{h^{\prime}_{1}h^{\prime}_{2}}{(h_{1}-h_{2})^{2}}\right).\end{array}$

We can obtain similar results for a minimal Lorentz surface in
$\mathbb{R}^{3}_{1}$, by putting $g_{i}=h_{i}$ in the corresponding statements
for a minimal surface in $\mathbb{R}^{4}_{2}$. Namely, we have

######  Theorem 4.2.

Let $\mathcal{M}$ be a minimal Lorentz surface in $\mathbb{R}^{3}_{1}$ and
$(\alpha_{1},\alpha_{2})$ be the corresponding pair of null curves. Then,
$\alpha^{\prime}_{1}$ and $\alpha^{\prime}_{2}$ have the following form:

(4.4)
$\alpha^{\prime}_{i}=f_{i}\big{(}\,g_{i}^{2}+1\,,\,g_{i}^{2}-1\,,\,2g_{i}\,\big{)};\qquad
i=1;2\,,$

where $(f_{i},g_{i})$; $i=1;2$ are two pairs of functions, such that:

(4.5) $f_{1}(t_{1})\neq 0\,;\qquad f_{2}(t_{2})\neq 0\,;\qquad
g_{1}(t_{1})\neq g_{2}(t_{2})\,.$

Conversely, let $(f_{i},g_{i})$; $i=1;2$ be two pairs of functions, satisfying
(4.5). Then, there exists a minimal Lorentz surface in $\mathbb{R}^{3}_{1}$,
such that its corresponding curves have the form (4.4).

For the curvatures of the surface we find

$K=\displaystyle\frac{4g^{\prime}_{1}g^{\prime}_{2}}{f_{1}f_{2}(g_{1}-g_{2})^{4}}\,,\qquad\varkappa=0\,.$

Let $\mathcal{M}$ be a minimal Lorentz surface of general type in
$\mathbb{R}^{4}_{2}$, parametrized by canonical isotropic coordinates, i.e.
the corresponding null curves are parametrized by natural parameters. Then,
using Theorem 3.5 , we obtain the next result:

######  Theorem 4.3.

[16] Let $\mathcal{M}$ be a minimal Lorentz surface of general type in
$\mathbb{R}^{4}_{2}$ parametrized by canonical isotropic coordinates. Then,
$\mathcal{M}$ has the following Weierstrass-type representation:

(4.6)
$\alpha^{\prime}_{i}=\displaystyle{\frac{\omega_{i}}{2\sqrt{|g^{\prime}_{i}h^{\prime}_{i}|}}}\left(g_{i}h_{i}+1,g_{i}h_{i}-1,h_{i}-g_{i},h_{i}+g_{i}\right);\qquad\omega_{i}=\pm
1,$

where $(g_{i},h_{i})$; $i=1,2$ are two pairs of smooth real functions such
that:

(4.7) $g^{\prime}_{1}(t_{1})h^{\prime}_{1}(t_{1})\neq 0;\qquad
g^{\prime}_{2}(t_{2})h^{\prime}_{2}(t_{2})\neq 0;\qquad g_{1}(t_{1})\neq
g_{2}(t_{2});\qquad h_{1}(t_{1})\neq h_{2}(t_{2}),$

and, in addition, if $\mathcal{M}$ is of third type, then
$g^{\prime}_{1}h^{\prime}_{1}>0$ and $g^{\prime}_{2}h^{\prime}_{2}<0$.

Conversely, if $(g_{i},h_{i})$; $i=1,2$ are two pairs of smooth real functions
satisfying (4.7), then there exists a minimal Lorentz surface of general type
in $\mathbb{R}^{4}_{2}$ parametrized by canonical isotropic coordinates and
having Weierstrass-type representation (4.6).

Formula (4.6) is called _canonical Weierstrass-type representation_ for a
minimal Lorentz surface of general type in $\mathbb{R}^{4}_{2}$ [16].

Note that using (4.6) we can calculate that

(4.8)
$F=-\frac{\omega_{1}\omega_{2}(g_{1}-g_{2})(h_{1}-h_{2})}{8\sqrt{|g^{\prime}_{1}h^{\prime}_{1}g^{\prime}_{2}h^{\prime}_{2}|}}\,.$

If the canonical isotropic coordinates are chosen in such a way that $F<0$,
then from (4.3) and (3.9) we obtain that the curvatures $K$ and $\varkappa$
are expressed in terms of canonical coordinates as follows:

(4.9)
$\begin{array}[]{llr}K&=&\displaystyle\frac{8\sqrt{|g^{\prime}_{1}h^{\prime}_{1}g^{\prime}_{2}h^{\prime}_{2}|}}{|(g_{1}-g_{2})(h_{1}-h_{2})|}\left(\displaystyle\frac{g^{\prime}_{1}g^{\prime}_{2}}{(g_{1}-g_{2})^{2}}+\displaystyle\frac{h^{\prime}_{1}h^{\prime}_{2}}{(h_{1}-h_{2})^{2}}\right);\\\\[12.91663pt]
\varkappa&=&\displaystyle\frac{8\sqrt{|g^{\prime}_{1}h^{\prime}_{1}g^{\prime}_{2}h^{\prime}_{2}|}}{|(g_{1}-g_{2})(h_{1}-h_{2})|}\left(\displaystyle\frac{g^{\prime}_{1}g^{\prime}_{2}}{(g_{1}-g_{2})^{2}}-\displaystyle\frac{h^{\prime}_{1}h^{\prime}_{2}}{(h_{1}-h_{2})^{2}}\right).\end{array}$

Analogously, we have the following theorem giving the canonical Weierstrass-
type representation of a minimal Lorentz surface in $\mathbb{R}^{3}_{1}$.

######  Theorem 4.4.

Let $\mathcal{M}$ be a minimal Lorentz surface of general type in
$\mathbb{R}^{3}_{1}$, parametrized by canonical isotropic coordinates. Then,

(4.10)
$\alpha^{\prime}_{i}=\displaystyle\frac{\omega_{i}}{2|g^{\prime}_{i}|}\left(\,g_{i}^{2}+1\,,\,g_{i}^{2}-1\,,\,2g_{i}\,\right);\qquad\omega_{i}=\pm
1\,,$

where $g_{i}$; $i=1;2$ are two functions, such that:

(4.11) $g^{\prime}_{1}(t_{1})\neq 0\,;\qquad g^{\prime}_{2}(t_{2})\neq
0\,;\qquad g_{1}(t_{1})\neq g_{2}(t_{2})\,.$

Conversely, let $g_{i}$; $i=1;2$ be functions, satisfying (4.11) and
$\omega_{i}=\pm 1$ . Then, there exists a minimal Lorentz surface of general
type in $\mathbb{R}^{3}_{1}$ parametrized by canonical isotropic coordinates
and having Weierstrass-type representation (4.10).

The Gauss curvature of $\mathcal{M}$ is expressed in terms of canonical
coordinates as follows:

(4.12)
$K=\displaystyle\frac{16\omega_{1}\omega_{2}|g^{\prime}_{1}g^{\prime}_{2}|g^{\prime}_{1}g^{\prime}_{2}}{(g_{1}-g_{2})^{4}}\,.$

##  5\. Relation between the null curves in $\mathbb{R}^{4}_{2}$ and
$\mathbb{R}^{3}_{1}$

Let $\alpha$ be a non-degenerate null curve in $\mathbb{R}^{4}_{2}$,
parametrized by a natural parameter and having a Weierstrass-type
representation (3.10). Using the function $g$, we obtain a non-degenerate null
curve $\alpha_{g}$ in $\mathbb{R}^{3}_{1}$ parametrized by a natural parameter
and determined by (3.12). This curve $\alpha_{g}$ is determined up to a non-
proper motion in $\mathbb{R}^{3}_{1}$, since $\omega_{g}=\pm 1$. Analogously,
the function $h$ generates a null curve $\alpha_{h}$ in $\mathbb{R}^{3}_{1}$
parametrized by a natural parameter and determined up to a non-proper motion
in $\mathbb{R}^{3}_{1}$. So, we obtain a map from the set of non-degenerate
null curves in $\mathbb{R}^{4}_{2}$ to the set of the ordered pairs of non-
degenerate null curves in $\mathbb{R}^{3}_{1}$:

(5.1) $\alpha\rightarrow(\alpha_{g}\,,\alpha_{h})\,.$

We will study the properties of this map under some transformations of the
curves in $\mathbb{R}^{4}_{2}$.

######  Theorem 5.1.

The map (5.1) is invariant under a proper motion in $\mathbb{R}^{4}_{2}$ and
under a change of the natural parameter.

###### Proof.

First, let us suppose that the null curve $\hat{\alpha}$ is obtained from
$\alpha$ by a proper motion in $\mathbb{R}^{4}_{2}$. According to Theorem 5.2
in [16], the functions $\hat{g}$ and $\hat{h}$ are obtained from $g$ and $h$,
respectively, by a linear-fractional transformation of determinant $\pm 1$ ,
and both determinants have the same sign. Now, using Theorem 3.3 we derive
that $\hat{\alpha}_{g}$ is obtained from $\alpha_{g}$ by a motion (possibly
improper) in $\mathbb{R}^{3}_{1}$ and analogously, $\hat{\alpha}_{h}$ is
obtained from $\alpha_{h}$.

Now, we suppose that the null curve $\tilde{\alpha}$ is obtained by a change
of the parameter $t=t(s)$ ($t^{\prime}\neq 0$) of the curve $\alpha$ in
$\mathbb{R}^{4}_{2}$. Then, we have
$\tilde{\alpha}^{\prime}_{s}=\alpha^{\prime}_{t}t^{\prime}$. Using (3.1) we
find

$\tilde{f}(s)=f(t(s))t^{\prime}(s)\,;\qquad\tilde{g}(s)=g(t(s))\,;\qquad\tilde{h}(s)=h(t(s))\,.$

In particular, change (2.3) transforms the functions $g$ and $h$ in the
canonical Weierstrass-type representation (3.10) as follows:

$\tilde{g}(s)=g(\pm s+c)\,;\qquad\tilde{h}(s)=h(\pm s+c)\,.$

Using (3.12) we can see that the functions $g$ and $h$ are transformed in the
same way if we consider the null curves in $\mathbb{R}^{3}_{1}$ parametrized
by a natural parameter. This proves the statement. ∎

Let the null curve $\hat{\alpha}$ be obtained from $\alpha$ with a non-proper
motion in $\mathbb{R}^{4}_{2}$. Then, the corresponding functions $\hat{g}$
and $\hat{h}$ are obtained by changing the places of $g$ and $h$.
Consequently, (5.1) implies:

$\hat{\alpha}\rightarrow(\alpha_{h}\,,\alpha_{g})\,.$

Now, consider the case of a null curve $\hat{\alpha}$, obtained by applying an
orientation-preserving anti-isometry in $\mathbb{R}^{4}_{2}$ on $\alpha$. The
corresponding functions $\hat{g}$ and $\hat{h}$ are obtained from $g$ and $h$
using a linear-fractional transformations with determinants of different sign.
Hence, (5.1) gives:

(5.2) $\hat{\alpha}\rightarrow(\alpha_{g}\,,\alpha_{-h})\,.$

Let us consider two non-degenerate null curves $\alpha_{g}$ and $\alpha_{h}$
in $\mathbb{R}^{3}_{1}$ parametrized by natural parameters and generated by
some functions $g$ and $h$, respectively, in accordance with Theorem 3.7. Let
$\alpha$ be the null curve in $\mathbb{R}^{4}_{2}$ obtained by the
Weierstrass-type representation (3.10) . Then, according to the map (5.1),
$\alpha$ corresponds exactly to the initial pair $(\alpha_{g}\,,\alpha_{h})$ .
Consequently, the map (5.1) is surjective.

##  6\. Relation between the minimal Lorentz surfaces in $\mathbb{R}^{4}_{2}$
and $\mathbb{R}^{3}_{1}$

In the previous section, we considered a relation between the non-degenerate
null curves in $\mathbb{R}^{4}_{2}$ and $\mathbb{R}^{3}_{1}$. In this section,
we will find a relation between the minimal Lorentz surfaces in
$\mathbb{R}^{4}_{2}$ and the pairs of minimal Lorentz surfaces in
$\mathbb{R}^{3}_{1}$. We will use the canonical isotropic coordinates for the
minimal Lorentz surfaces. In general, the canonical coordinates exist only in
a neighbourhood of a point, so our considerations will be local. To formalize
the correspondence, we will consider the pairs $(\mathcal{M},p)$, where
$\mathcal{M}$ is a minimal Lorentz surface of general type in
$\mathbb{R}^{4}_{2}$ and $p\in\mathcal{M}$ is a fixed point. We will identify
two pairs $(\mathcal{M},p)$ and $(\hat{\mathcal{M}},\hat{p})$ in
$\mathbb{R}^{4}_{2}$, if $(\hat{\mathcal{M}},\hat{p})$ is obtained from
$(\mathcal{M},p)$ by a change of the coordinates and a proper motion,
$\hat{p}$ and $p$ being the corresponding points. Analogously, we identify two
similar pairs in $\mathbb{R}^{3}_{1}$ if one of them is obtained from the
other by a change of the coordinates and a motion (possibly non-proper) in
$\mathbb{R}^{3}_{1}$.

Let $\mathcal{M}$ be a minimal Lorentz surface of general type in
$\mathbb{R}^{4}_{2}$ parametrized by canonical isotropic coordinates in a
neighbourhood of a point $p\in\mathcal{M}$. Suppose $p$ has vanishing
coordinates and $(\alpha_{1},\alpha_{2})$ is the pair of null curves in
$\mathbb{R}^{4}_{2}$ that corresponds to $\mathcal{M}$. Then, $\alpha_{1}$ and
$\alpha_{2}$ are non-degenerate and parametrized by natural parameters.
Applying (5.1) to $\alpha_{1}$, we obtain two non-degenerate null curves
$\alpha_{1g}$ and $\alpha_{1h}$ in $\mathbb{R}^{3}_{1}$ parametrized by
natural parameters. Analogously, $\alpha_{2}$ generates two non-degenerate
null curves $\alpha_{2g}$ and $\alpha_{2h}$ in $\mathbb{R}^{3}_{1}$,
parametrized by natural parameters. Then, using (2.1) we obtain two minimal
Lorentz surfaces $(\mathcal{M}_{g},p_{g})$ and $(\mathcal{M}_{h},p_{h})$ of
general type in $\mathbb{R}^{3}_{1}$, $p_{g}$ and $p_{h}$ being the points in
$\mathcal{M}_{g}$ and $\mathcal{M}_{h}$ with vanishing coordinates. So, we
obtain a map from the set of minimal Lorentz surfaces in $\mathbb{R}^{4}_{2}$
into the set of the ordered pairs of minimal Lorentz surfaces in
$\mathbb{R}^{3}_{1}$:

(6.1)
$\big{(}\mathcal{M},p\big{)}\rightarrow\big{(}(\mathcal{M}_{g},p_{g}),(\mathcal{M}_{h},p_{h})\big{)}\,.$

According to Theorem 5.1, the map (5.1) is invariant under a proper motion in
$\mathbb{R}^{4}_{2}$ and a change of the natural parameters. The same is true
for (2.1). Hence, we obtain:

######  Theorem 6.1.

The map (6.1) from the set of minimal Lorentz surfaces of general type in
$\mathbb{R}^{4}_{2}$ to the set of ordered pairs of minimal Lorentz surfaces
of general type in $\mathbb{R}^{3}_{1}$ is invariant under a proper motion in
$\mathbb{R}^{4}_{2}$ and a change of the canonical coordinates.

Note that the null curves, given with (5.1) are defined up to a sign. If we
change the signs of both $\alpha_{1g}$ and $\alpha_{2g}$ in the definition of
$\mathcal{M}_{g}$, then the new surface will be obtained from
$\mathcal{M}_{g}$ by a non-proper motion. On the other hand, if we change only
the sign of one of the curves, then according to (2.2) we obtain a surface
conjugate to $\mathcal{M}_{g}$. The same is true for $\mathcal{M}_{h}$.
Consequently, the induced minimal Lorentz surfaces are defined up to a non-
proper motion in $\mathbb{R}^{3}_{1}$ and conjugate surface. So, we can give
the following definitions:

######  Definition 6.1.

Two minimal Lorentz surfaces in $\mathbb{R}^{3}_{1}$ are called _equivalent_ ,
if they coincide up to a motion (possibly non-proper) or conjugation.

######  Definition 6.2.

Two minimal Lorentz surfaces in $\mathbb{R}^{4}_{2}$ are called _equivalent_
if one of them coincides with the other (or with the conjugate of the other)
up to a proper motion and an orientation-preserving anti-isometry.

######  Theorem 6.2.

Let $(\hat{\mathcal{M}},\hat{p})$ and $(\mathcal{M},p)$ be equivalent minimal
Lorentz surfaces in $\mathbb{R}^{4}_{2}$ and
$\big{(}(\hat{\mathcal{M}}_{g},\hat{p}_{g}),\
(\hat{\mathcal{M}}_{h},\hat{p}_{h})\big{)}$ and
$\big{(}(\mathcal{M}_{g},p_{g}),\ (\mathcal{M}_{h},p_{h})\big{)}$ be the
corresponding pairs of minimal Lorentz surfaces in $\mathbb{R}^{3}_{1}$. Then,
$(\hat{\mathcal{M}}_{g},\hat{p}_{g})$ and $(\mathcal{M}_{g},p_{g})$ (resp.
$(\hat{\mathcal{M}}_{h},\hat{p}_{h})$) and $(\mathcal{M}_{h},p_{h})$) are
equivalent.

###### Proof.

Let $(\hat{\mathcal{M}},\hat{p})$ and $(\mathcal{M},p)$ be minimal Lorentz
surface in $\mathbb{R}^{4}_{2}$ which are equivalent. Theorem 6.1 implies that
if $(\hat{\mathcal{M}},\hat{p})$ and $(\mathcal{M},p)$ are related by a proper
motion, then they give rise to equivalent surfaces in $\mathbb{R}^{3}_{1}$.

Let $(\hat{\mathcal{M}},\hat{p})$ and $(\mathcal{M},p)$ be related by an
orientation-preserving anti-isometry in $\mathbb{R}^{4}_{2}$ . According to
(5.2), the corresponding functions $\hat{h}_{1}$ and $\hat{h}_{2}$ are
obtained by changing the sign in $h_{1}$ and $h_{2}$. Consequently, (6.1)
imp[lies

(6.2)
$\big{(}\hat{\mathcal{M}},\hat{p}\big{)}\rightarrow\big{(}(\mathcal{M}_{g},p_{g}),(\mathcal{M}_{-h},p_{-h})\big{)}\,.$

Note that, according to (3.6) and (3.7), if we change the signs of $h_{1}$ and
$h_{2}$, then the corresponding curves are obtained by a motion (possibly non-
proper) in $\mathbb{R}^{3}_{1}$ . Hence, the surface
$(\mathcal{M}_{-h},p_{-h})$ is obtained from $(\mathcal{M}_{h},p_{h})$ by the
same motion in $\mathbb{R}^{3}_{1}$ . Consequently, every orientation-
preserving anti-isometry in $\mathbb{R}^{4}_{2}$ , gives a pair of surfaces in
$\mathbb{R}^{3}_{1}$, which is equivalent to
$\big{(}(\mathcal{M}_{g},p_{g}),(\mathcal{M}_{h},p_{h})\big{)}$.

Finally, we suppose that $(\hat{\mathcal{M}},\hat{p})$ is conjugate to
$(\mathcal{M},p)$. Then, using (2.2), we get that the corresponding curves
satisfy $\hat{\alpha}_{1}=\alpha_{1}$, $\hat{\alpha}_{2}=-\alpha_{2}$ . Using
(4.6), we see that under a change of the sign of $\alpha_{2}$, the functions
$g_{2},h_{2}$ do not change. Consequently, $(\hat{\mathcal{M}},\hat{p})$ and
$(\mathcal{M},p)$ produce two equivalent pairs of surfaces in
$\mathbb{R}^{3}_{1}$ . ∎

######  Theorem 6.3.

The map (6.1) gives a bijection between the classes of equivalent minimal
Lorentz surfaces in $\mathbb{R}^{4}_{2}$ and the pairs of classes of
equivalent minimal Lorentz surfaces in $\mathbb{R}^{3}_{1}$ .

###### Proof.

Let us consider two minimal Lorentz surfaces of general type
$(\mathcal{M}_{g},p_{g})$ and $(\mathcal{M}_{h},p_{h})$ in
$\mathbb{R}^{3}_{1}$ parametrized by canonical coordinates. Applying Theorem
4.4, we obtain two pairs of functions $(g_{i},h_{i})$; $i=1;2$ satisfying
(4.11) for $g_{i}$ and $h_{i}$ . Then, conditions (4.7) are satisfied. Now,
(4.6) defines a minimal Lorentz surface of general type $(\mathcal{M},p)$ in
$\mathbb{R}^{4}_{2}$ parametrized by canonical parameters, $p$ being the point
in $\mathcal{M}$ with vanishing coordinates. It is easy to see that (6.1)
transforms $(\mathcal{M},p)$ in
$\big{(}(\mathcal{M}_{g},p_{g}),(\mathcal{M}_{h},p_{h})\big{)}$ .
Consequently, the map (6.1) is surjective.

Now, we suppose that the surfaces $(\hat{\mathcal{M}}_{g},\hat{p}_{g})$ and
$(\mathcal{M}_{g},p_{g})$ in $\mathbb{R}^{3}_{1}$ are equivalent and
analogously, $(\hat{\mathcal{M}}_{h},\hat{p}_{h})$ and
$(\mathcal{M}_{h},p_{h})$ are equivalent. Note, that according to (2.2) and
(4.10), the change of $(\mathcal{M}_{g},p_{g})$ or $(\mathcal{M}_{h},p_{h})$
with its conjugate does not change the corresponding functions $g_{i}$ and
$h_{i}$, $i=1;2$.

Let $(\hat{\mathcal{M}}_{g},\hat{p}_{g})$ be obtained from
$(\mathcal{M}_{g},p_{g})$ by a motion (possibly non-proper) in
$\mathbb{R}^{3}_{1}$ and analogously for $(\hat{\mathcal{M}}_{h},\hat{p}_{h})$
and $(\mathcal{M}_{h},p_{h})$ . Then, (3.4), (3.6), (3.7), and (3.8) imply
that the functions $\hat{g}_{i}$ and $\hat{h}_{i}$ are related with $g_{i}$,
resp. $h_{i}$ for $i=1;2$ by linear-fractional maps with determinant $\pm 1$ .
According to Theorem 5.2 in [16] , if the determinants for $\hat{g}_{i}$ and
$\hat{h}_{i}$ have the same sign, then the corresponding surface
$(\hat{\mathcal{M}},\hat{p})$ in $\mathbb{R}^{4}_{2}$ can be obtained by a
proper motion from $(\mathcal{M},p)$ or from its conjugate. If the
determinants have different signs, then having in mind (6.2),
$(\hat{\mathcal{M}},\hat{p})$ can be obtained from $(\mathcal{M},p)$ or its
conjugate by an orientation-preserving anti-isometry in $\mathbb{R}^{4}_{2}$ .
Consequently, the corresponding surfaces $(\hat{\mathcal{M}},\hat{p})$ and
$(\mathcal{M},p)$ are equivalent, thus proving the injectivity. ∎

In what follows, we obtain a relation between the curvatures (Gauss curvature
$K$ and normal curvature $\varkappa$) of a minimal Lorentz surface of general
type $(\mathcal{M},p)$ in $\mathbb{R}^{4}_{2}$ and the Gauss curvatures
$K_{g}$ and $K_{h}$ of the corresponding minimal Lorentz surfaces in
$\mathbb{R}^{3}_{1}$.

######  Theorem 6.4.

Let $(\mathcal{M},p)$ be a minimal Lorentz surface of general type in
$\mathbb{R}^{4}_{2}$ with Gauss curvature $K$ and normal curvature
$\varkappa$. Let $(\mathcal{M}_{g},p_{g})$ and $(\mathcal{M}_{h},p_{h})$ be
the corresponding minimal surfaces in $\mathbb{R}^{3}_{1}$ with Gauss
curvatures $K_{g}$ and $K_{h}$, respectively.

  1. (1)

If $\mathcal{M}$ is of first or second type, then

(6.3)
$K=\eta\sqrt[4]{|K_{g}K_{h}|}\>\frac{\sqrt{|K_{g}|}+\sqrt{|K_{h}|}}{2}\;;\quad\varkappa=\eta\sqrt[4]{|K_{g}K_{h}|}\>\frac{\sqrt{|K_{g}|}-\sqrt{|K_{h}|}}{2}\;,\;\;\eta=\mathop{\operatorname{sign}}\nolimits
K\,.$

  2. (2)

If $\mathcal{M}$ is of third type, then

(6.4)
$K=\eta\sqrt[4]{|K_{g}K_{h}|}\>\frac{\sqrt{|K_{g}|}-\sqrt{|K_{h}|}}{2}\;;\quad\varkappa=\eta\sqrt[4]{|K_{g}K_{h}|}\>\frac{\sqrt{|K_{g}|}+\sqrt{|K_{h}|}}{2}\;,\;\;\eta=\mathop{\operatorname{sign}}\nolimits\varkappa\,.$

###### Proof.

For simplicity we suppose that the first fundamental form of $\mathcal{M}$
satisfies $F<0$ . Let $\mathcal{M}$ be of first or second type, i.e.
$|K|>|\varkappa|$ . Then, (4.9) implies that $g^{\prime}_{1}g^{\prime}_{2}$
and $h^{\prime}_{1}h^{\prime}_{2}$ have the same sign, which coincides with
the sign of $K$. Denote this sign by $\eta$. Then, we have
$g^{\prime}_{1}g^{\prime}_{2}=\eta|g^{\prime}_{1}g^{\prime}_{2}|$,
$h^{\prime}_{1}h^{\prime}_{2}=\eta|h^{\prime}_{1}h^{\prime}_{2}|$. So,
equalities (4.9) take the following form:

(6.5)
$\begin{array}[]{llr}K&=&\displaystyle\frac{8\eta\sqrt{|g^{\prime}_{1}h^{\prime}_{1}g^{\prime}_{2}h^{\prime}_{2}|}}{|(g_{1}-g_{2})(h_{1}-h_{2})|}\left(\displaystyle\frac{|g^{\prime}_{1}g^{\prime}_{2}|}{(g_{1}-g_{2})^{2}}+\displaystyle\frac{|h^{\prime}_{1}h^{\prime}_{2}|}{(h_{1}-h_{2})^{2}}\right);\\\\[12.91663pt]
\varkappa&=&\displaystyle\frac{8\eta\sqrt{|g^{\prime}_{1}h^{\prime}_{1}g^{\prime}_{2}h^{\prime}_{2}|}}{|(g_{1}-g_{2})(h_{1}-h_{2})|}\left(\displaystyle\frac{|g^{\prime}_{1}g^{\prime}_{2}|}{(g_{1}-g_{2})^{2}}-\displaystyle\frac{|h^{\prime}_{1}h^{\prime}_{2}|}{(h_{1}-h_{2})^{2}}\right).\end{array}$

Let $(\mathcal{M}_{g},p_{g})$ and $(\mathcal{M}_{h},p_{h})$ be the minimal
Lorentz surfaces in $\mathbb{R}^{3}_{1}$ that correspond to $(\mathcal{M},p)$
according to (6.1) . Then, by use of (4.12) we obtain that the Gauss
curvatures $K_{g}$ and $K_{h}$ are expressed as follows:

(6.6)
$\sqrt{|K_{g}|}=\displaystyle\frac{4|g^{\prime}_{1}g^{\prime}_{2}|}{(g_{1}-g_{2})^{2}}\,;\qquad\sqrt{|K_{h}|}=\displaystyle\frac{4|h^{\prime}_{1}h^{\prime}_{2}|}{(h_{1}-h_{2})^{2}}\,.$

Hence, using (6.5), we obtain (6.3) .

Now, let $\mathcal{M}$ be of third type. According to Theorem 2.1, this is
equivalent to $|\varkappa|>|K|$ . Then, (4.9) implies that
$g^{\prime}_{1}g^{\prime}_{2}$ and $h^{\prime}_{1}h^{\prime}_{2}$ have
different signs. If $\varkappa>0$ , we obtain $g^{\prime}_{1}g^{\prime}_{2}>0$
and $h^{\prime}_{1}h^{\prime}_{2}<0$ . If $\varkappa<0$ , we get
$g^{\prime}_{1}g^{\prime}_{2}<0$ and $h^{\prime}_{1}h^{\prime}_{2}>0$ . Denote
$\eta=\mathop{\operatorname{sign}}\nolimits\varkappa$. Then
$g^{\prime}_{1}g^{\prime}_{2}=\eta|g^{\prime}_{1}g^{\prime}_{2}|$ and
$h^{\prime}_{1}h^{\prime}_{2}=-\eta|h^{\prime}_{1}h^{\prime}_{2}|$. Now, using
(4.9), we derive the following expressions of $K$ and $\varkappa$:

(6.7)
$\begin{array}[]{llr}K&=&\displaystyle\frac{8\eta\sqrt{|g^{\prime}_{1}h^{\prime}_{1}g^{\prime}_{2}h^{\prime}_{2}|}}{|(g_{1}-g_{2})(h_{1}-h_{2})|}\left(\displaystyle\frac{|g^{\prime}_{1}g^{\prime}_{2}|}{(g_{1}-g_{2})^{2}}-\displaystyle\frac{|h^{\prime}_{1}h^{\prime}_{2}|}{(h_{1}-h_{2})^{2}}\right);\\\\[12.91663pt]
\varkappa&=&\displaystyle\frac{8\eta\sqrt{|g^{\prime}_{1}h^{\prime}_{1}g^{\prime}_{2}h^{\prime}_{2}|}}{|(g_{1}-g_{2})(h_{1}-h_{2})|}\left(\displaystyle\frac{|g^{\prime}_{1}g^{\prime}_{2}|}{(g_{1}-g_{2})^{2}}+\displaystyle\frac{|h^{\prime}_{1}h^{\prime}_{2}|}{(h_{1}-h_{2})^{2}}\right).\end{array}$

From (6.6) and (6.7) we obtain (6.4) .

So, we proved formulas (6.3) and (6.4) in the case $F<0$. But, according to
Theorem 6.1, the map (6.1) do not depend on the canonical coordinates of
$\mathcal{M}$. The curvatures $K$, $\varkappa$, $K_{g}$ and $K_{h}$ are also
invariant. This proves the theorem. ∎

Now, we will obtain a relation between the first fundamental forms of
$(\mathcal{M},p)$ and its corresponding surfaces $(\mathcal{M}_{g},p_{g})$,
$(\mathcal{M}_{h},p_{h})$ . Using (4.8), we have

(6.8)
$|F|=\frac{|(g_{1}-g_{2})(h_{1}-h_{2})|}{8\sqrt{|g^{\prime}_{1}h^{\prime}_{1}g^{\prime}_{2}h^{\prime}_{2}|}}\>,$

$F$ being the non-vanishing coefficient of the first fundamental form of
$(\mathcal{M},p)$ . Analogously, for the coefficients $F_{g}$ and $F_{h}$ of
the surfaces $(\mathcal{M}_{g},p_{g})$ and $(\mathcal{M}_{h},p_{h})$ we have

(6.9)
$|F_{g}|=\frac{(g_{1}-g_{2})^{2}}{8|g^{\prime}_{1}g^{\prime}_{2}|}\>;\qquad|F_{h}|=\frac{(h_{1}-h_{2})^{2}}{8|h^{\prime}_{1}h^{\prime}_{2}|}\>.$

From (6.8) and (6.9) we obtain:

$|F|=\sqrt{|F_{g}F_{h}|}\,.$

As an application of the results presented above, at the end of this section,
we give an example showing how we can obtain a minimal Lorentz surface in
$\mathbb{R}^{4}_{2}$ from two given minimal Lorentz surfaces in
$\mathbb{R}^{3}_{1}$.

######  Example 6.1.

Let us consider the following Lorentz hyperbolic catenoids in
$\mathbb{R}^{3}_{1}$ (see [2, 6]):

  * •

$\mathcal{M}_{a}$ : $x_{2}^{2}-x_{1}^{2}=\cosh^{2}(x_{3})$ – _Lorentz
hypebolic catenoid of first kind_ ;

  * •

$\mathcal{M}_{b}$ : $x_{1}^{2}-x_{2}^{2}=\sinh^{2}(x_{3})$ – _Lorentz
hypebolic catenoid of second kind_.

They are parametrized respectively by:

$\mathrm{x}_{a}=(\sinh(u)\cosh(v),\cosh(u)\cosh(v),v);\quad\mathrm{x}_{b}=(\sinh(u)\cosh(v),\sinh(u)\sinh(v),u).$

It is easy to see that in both cases $\mathcal{M}_{a}$ and $\mathcal{M}_{b}$
are minimal Lorentz surfaces in $\mathbb{R}^{3}_{1}$ parametrized by
isothermal coordinates $(u,v)$. Then, we can obtain the parametrization of
$\mathcal{M}_{a}$ and $\mathcal{M}_{b}$ in terms of isotropic coordinates
$(t_{1},t_{2})$ and in accordance with (2.1) we denote the corresponding pairs
of isotropic curves by $(\alpha_{1},\alpha_{2})$ and $(\beta_{1},\beta_{2})$,
respectively. With respect to the isotropic parameters the corresponding
tangent vectors are:

$\begin{array}[]{l}\mathcal{M}_{a}:\;\alpha^{\prime}_{1}=(\cosh(t_{1}),\sinh(t_{1}),1);\quad\alpha^{\prime}_{2}=(\cosh(t_{2}),\sinh(t_{2}),-1);\\\\[2.15277pt]
\mathcal{M}_{b}:\;\beta^{\prime}_{1}=(\cosh(t_{1}),\sinh(t_{1}),1);\quad\\!\beta^{\prime}_{2}=(\cosh(t_{2}),-\sinh(t_{2}),1).\end{array}$

Hence,
$(\alpha^{\prime\prime}_{1})^{2}=(\alpha^{\prime\prime}_{2})^{2}=(\beta^{\prime\prime}_{1})^{2}=(\beta^{\prime\prime}_{2})^{2}=1$
and according to Definition 2.2, $(t_{1},t_{2})$ are canonical isotropic
coordinates of $\mathcal{M}_{a}$ and $\mathcal{M}_{b}$.

We denote by $g_{1}$ and $g_{2}$ the functions that generate $\alpha_{1}$ and
$\alpha_{2}$, respectively, and analogously, we denote by $h_{1}$ and $h_{2}$
the functions corresponding to $\beta_{1}$ and $\beta_{2}$, respectively.
Applying (3.3) for each curve we obtain that:

$g_{1}(t_{1})=\mathrm{e}^{t_{1}}\,;\quad
g_{2}(t_{2})=-\mathrm{e}^{t_{2}}\,;\quad
h_{1}(t_{1})=\mathrm{e}^{t_{1}}\,;\quad h_{2}(t_{2})=\mathrm{e}^{-t_{2}}\,.$

Let us note that conditions (4.11) for $\mathcal{M}_{a}$ give no restriction
on $(t_{1},t_{2})$. On the other hand, conditions (4.11) for $\mathcal{M}_{b}$
imply $t_{1}+t_{2}\neq 0$ .

Let $\gamma_{1}$ be the isotropic curve in $\mathbb{R}^{4}_{2}$, defined by
the Weierstrass-type representation (3.10) with the pair $(g_{1},h_{1})$ and
$\omega_{1}=1$ . Analogously, we denote by $\gamma_{2}$ the isotropic curve
determined by (3.10) with $(g_{2},h_{2})$ and $\omega_{2}=1$ . Then, we have:

$\gamma^{\prime}_{1}=(\cosh(t_{1}),\sinh(t_{1}),0,1)\,;\qquad\gamma^{\prime}_{2}=(0,-1,\cosh(t_{2}),-\sinh(t_{2}))\,.$

Denote by $\mathcal{M}$ the surface in $\mathbb{R}^{4}_{2}$ defined by the
pair $(\gamma_{1},\gamma_{2})$ in accordance with (2.1), or equivalently,
$\mathcal{M}$ is the surface that corresponds to
$(\mathcal{M}_{a},\mathcal{M}_{b})$ according to (6.1). We have:

$\mathcal{M}:\;\mathrm{x}(t_{1},t_{2})=\frac{1}{2}\big{(}\sinh(t_{1}),\;\cosh(t_{1})-t_{2},\;\sinh(t_{2}),\;t_{1}-\cosh(t_{2})\big{)}\,;\quad
t_{1}+t_{2}\neq 0\,.$

By direct calculations we get
$(\gamma^{\prime\prime}_{1})^{2}=(\gamma^{\prime\prime}_{2})^{2}=1$ . Hence,
the surface $\mathcal{M}$ is of first type according to Definition 2.1 .

Using (3.9) we find the functions $f_{1}$ and $f_{2}$ that correspond to
$(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$:

$f_{1}=\frac{1}{2}\mathrm{e}^{-t_{1}};\qquad f_{2}=\frac{1}{2}\ .$

Now, using (4.3) we obtain the Gauss curvature $K$ and the normal curvature
$\varkappa$ of $\mathcal{M}$:

$K=\frac{-4\cosh(t_{1})\cosh(t_{2})}{(\sinh(t_{1})+\sinh(t_{2}))^{3}}\,;\qquad\varkappa=\frac{4-4\cosh(t_{1})\cosh(t_{2})}{(\sinh(t_{1})+\sinh(t_{2}))^{3}}\,.$

It will be interesting to study the following question: _If we are given two
minimal Lorentz surfaces in $\mathbb{R}^{3}_{1}$, how do the geometric
properties of these two surfaces determine the geometric properties of the
corresponding surface in $\mathbb{R}^{4}_{2}$?_

Acknowledgments: The second and the third authors are partially supported by
the National Science Fund, Ministry of Education and Science of Bulgaria under
contract KP-06-N52/3.

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dimensional [25]. Thus, the homoclinic and heteroclinic orbits in the double
pendulum become two-dimensional invariant manifolds whose dynamics
asymptotically approach neighborhoods of the index-1 saddles of the triple
pendulum. Both situations require a delicate numerical treatment that will be
left to a follow-up investigation.

## Acknowledgments

The authors acknowledge funding support from the Army Research Office (ARO
W911NF-19-1-0045) and National Science Foundation AI Institute in Dynamic
Systems (grant number 2112085). The authors also would like to thank Shane
Ross for valuable discussions.

##  Appendix A Governing Equations of the PCR3BP

In Section 2 we illustrated the tube dynamics of index-1 saddles using
numerical simulations from the planar circular restricted three-body problem
(PCR3BP). We assume there are two massive bodies with masses $\mu_{1}$ and
$\mu_{2}$ which form their own two body problem, while a (relatively)
weightless body such as a satellite or shuttle moving in the gravitational
field generated by these objects. Here we present the relevant dynamical
system for the third weightless body restricted to the orbital plane of the
two massive bodies which constitute the Hamiltonian PCR3BP system. We define
the states of the system as $x,y,v_{x}$ and $v_{y}$, representing the position
of the third body in the plane and its velocity in each independent direction.

The potential energy of the system is given by

$V_{3BP}=-\frac{1}{2}\left(\mu_{1}r_{1}^{2}+\mu_{2}r_{2}^{2}\right)-\frac{\mu_{1}}{r_{1}}-\frac{\mu_{2}}{r_{2}},$
(A.1)

where

$\begin{split}r_{1}^{2}&=\left(x+\mu_{2}\right)^{2}+y^{2},\\\
r_{2}^{2}&=\left(x-\mu_{1}\right)^{2}+y^{2},\\\
\mu&=\frac{m_{2}}{m_{1}+m_{2}},\\\ \mu_{1}&=1-\mu,\\\
\mu_{2}&=\mu.\end{split}$ (A.2)

The kinetic energy is given by

$T_{3BP}=\frac{1}{2}(v_{x}^{2}+v_{y}^{2}),$ (A.3)

so that the total energy of the PCR3BP is

$\mathcal{H}_{3BP}=T_{3BP}+V_{3BP}.$ (A.4)

Using the above Hamiltonian, we can arrive at the corresponding first-order
ODE describing the equations of motion of the PCR3BP. The result is given by

$\begin{split}\dot{x}&=v_{x},\\\ \dot{y}&=v_{y},\\\
\dot{v}_{x}&=2v_{y}+x-\frac{(x-(1-\mu))\mu}{r_{1}^{3}}-\frac{(1-\mu)(x+\mu)}{r_{2}^{3}},\\\
\dot{v}_{y}&=-2v_{x}+y-\frac{\mu
y}{r_{1}^{3}}-\frac{(1-\mu)y}{r_{2}^{3}},\end{split}$ (A.5)

Here $\mu$ is the mass ratio of the two massive bodies. For the examples
provided in Section 2 we have followed [45] used taken $\mu=9.537\times
10^{-4}$, which is the mass ratio of Jupiter to the Sun in our solar system.

The $L_{1}$ and $L_{2}$ index-1 saddle Lagrangian points lie on the line
$y=0$. Such points are equilibria of (A.5) with $y=0$ and, per [44, Section
2.5], are given by $x=(1-\mu)\pm\gamma$, where $\gamma$ are solutions of the
quintic polynomials

$\gamma^{5}\mp(3-\mu)\gamma^{4}+(3-2\mu)\gamma^{3}-\mu\gamma^{2}\pm
2\mu\gamma-\mu=0.$ (A.6)

Precisely, the $L_{1}$ Lagrange point comes from a solution $\gamma$ with the
upper signs in the above polynomial, while the $L_{2}$ Lagrange point comes
from using lower signs. With the value $\mu=9.537\times 10^{-4}$, the
corresponding $\gamma$ values in (A.6) are $\gamma_{1}=0.06667655835869524$
and $\gamma_{2}=0.06978018274821962$, thus leading to the $x$ positions of the
$L_{1}$ and $L_{2}$ Lagrangian points

$\begin{split}x_{L_{1}}&=(1-\mu)-\gamma_{1}=0.9323697416413048,\\\
x_{L_{2}}&=(1-\mu)+\gamma_{2}=1.0688264827482197.\end{split}$ (A.7)

The coordinates $(x_{L_{i}},0,0,0)$, $i=1,2$, thus represent the location of
the index-1 saddles of interest in the PCR3BP (A.5).

##  Appendix B The Point-Mass Double Pendulum

Here we will provide the relevant equations of motion for the point-mass
double pendulum and describe the analogous linear analysis to that of §4.2.
The goal is to show that the Down-Up and Up-Down equilibria remain index-1
saddles for the simplified point-mass double pendulum. Furthermore, we comment
that although it is not reported here, there is a nearly identical tube
structure to what we have shown for the physical double pendulum, meaning that
our election to study the more physically realistic physical double pendulum
presents little difference to the more typically studied point-mass double
pendulum.

To begin, let $m_{1}$ denote the point mass of the first (base) arm and
$m_{2}$ be the point mass of the second arm. We take the length of the first
and second arms to be $\ell_{1}$ and $\ell_{2}$, respectively. The angle
$\theta_{1}$ represents the angular displacement of the first arm from the
downward vertical position, while $\theta_{2}$ is the angular displacement of
the second arm. We again refer the reader to the left panel of Figure 5 for
visual reference.

The kinetic energy of the double pendulum is given by

$T=\frac{1}{2}(m_{1}+m_{2})\ell_{1}^{2}\omega_{1}^{2}+\frac{1}{2}m_{2}\ell_{2}^{2}\omega_{2}^{2}+m_{2}\ell_{1}\ell_{2}\omega_{1}\omega_{2}\cos(\theta_{2}-\theta_{1})$
(B.1)

and the potential energy is given by

$V=-(m_{1}+m_{2})\ell_{1}g\cos(\theta_{1})-m_{2}\ell_{2}g\cos(\theta_{2}).$
(B.2)

Hence, the total energy of the system is given by the Hamiltonian function

$\mathcal{H}(\theta_{1},\theta_{2},\omega_{1},\omega_{2})=T+V$ (B.3)

composed of the sum of the kinetic (B.1) and potential (B.2) energies.
Following as in Section 3 we arrive at the corresponding first-order
conservative ODE describing the equations of motion for the point-mass double
pendulum:

$\begin{split}\dot{\theta}_{1}&=\omega_{1}\\\ \dot{\theta}_{2}&=\omega_{2}\\\
\dot{\omega}_{1}&=\frac{A_{1}+A_{2}+A_{3}}{\ell_{1}^{2}\ell_{2}(m_{1}+m_{2}-m_{2}\cos^{2}(\theta_{2}-\theta_{1}))}\\\
\dot{\omega}_{2}&=\frac{B_{1}+B_{2}+B_{3}}{\ell_{1}\ell_{2}^{2}m_{2}(m_{1}+m_{2}-m_{2}\cos^{2}(\theta_{2}-\theta_{1}))}\end{split}.$
(B.4)

Here we have

$\begin{split}A_{1}&=m_{2}\ell_{1}\ell_{2}^{2}\omega_{2}^{2}\sin(\theta_{2}-\theta_{1}),\\\
A_{2}&=\frac{1}{2}m_{2}\ell_{1}^{2}\ell_{2}\omega_{1}^{2}\sin(2\theta_{2}-2\theta_{1}),\\\
A_{3}&=\ell_{1}\ell_{2}g[m_{2}\sin(\theta_{2})\cos(\theta_{2}-\theta_{1})-(m_{1}+m_{2})\sin(\theta_{1})],\\\
B_{1}&=-\frac{1}{2}\ell_{1}\ell_{2}^{2}m_{2}^{2}\omega_{2}^{2}\sin(2\theta_{2}-2\theta_{1}),\\\
B_{2}&=-\ell_{1}^{2}\ell_{2}m_{2}(m_{1}+m_{2})\omega_{1}^{2}\sin(\theta_{2}-\theta_{1}),\\\
B_{3}&=-\ell_{1}\ell_{2}m_{2}g(m_{1}+m_{2})[\sin(\theta_{2})-\sin(\theta_{1})\cos(\theta_{2}-\theta_{1})],\end{split}$
(B.5)

where we have neglected any friction or control terms that could be added to
the system.

We now follow the linear analysis of §4.2 to show that again the Down-Up and
Up-Down steady-states are index-1 saddles of (B.4). Again, equilibria come in
the form $(\theta_{1},\theta_{2},\omega_{1},\omega_{2})=(\pi k_{1},\pi
k_{2},0,0)$ for every pair of integers $(k_{1},k_{2})\in\mathbb{Z}^{2}$ and
all of these equilibria are symmetric with respect to the action of the
reverser (4.1). As with the physical double pendulum, by periodicity of the
$\theta_{1},2$ components we have four equilibria that merit investigation:
Down-Down, Down-Up, Up-Down, and Up-Up, as described in (4.2).

Linearizing (B.4) about an equilibrium
$(\theta_{1},\theta_{2},\omega_{1},\omega_{2})=(\pi k_{1},\pi k_{2},0,0)$ with
$k_{1},k_{2}\in\mathbb{Z}$ results in the matrix

$\begin{bmatrix}0&0&1&0\\\ 0&0&0&1\\\ \frac{-g(m_{1}+m_{2})\cos(\pi
k_{1})}{\ell_{1}(m_{1}+m_{2}-m_{2}\cos^{2}(\pi k_{2}-\pi
k_{1}))}&\frac{gm_{2}\cos(\pi k_{2})\cos(\pi k_{2}-\pi
k_{1})}{\ell_{1}(m_{1}+m_{2}-m_{2}\cos^{2}(\pi k_{2}-\pi k_{1}))}&0&0\\\
\frac{g(m_{1}+m_{2})\cos(\pi k_{1})\cos(\pi k_{2}-\pi
k_{1})}{\ell_{2}(m_{1}+m_{2}-m_{2}\cos^{2}(\pi k_{2}-\pi
k_{1}))}&\frac{-g(m_{1}+m_{2})\cos(\pi
k_{2})}{\ell_{2}(m_{1}+m_{2}-m_{2}\cos^{2}(\pi k_{2}-\pi
k_{1}))}&0&0\end{bmatrix}.$ (B.6)

The block structure of (B.6) gives the square of its eigenvalues are equal to
those of the lower left $2\times 2$ matrix

$\begin{bmatrix}\frac{-g(m_{1}+m_{2})\cos(\pi
k_{1})}{\ell_{1}(m_{1}+m_{2}-m_{2}\cos^{2}(\pi k_{2}-\pi
k_{1}))}&\frac{gm_{2}\cos(\pi k_{2})\cos(\pi k_{2}-\pi
k_{1})}{\ell_{1}(m_{1}+m_{2}-m_{2}\cos^{2}(\pi k_{2}-\pi k_{1}))}\\\
\frac{g(m_{1}+m_{2})\cos(\pi k_{1})\cos(\pi k_{2}-\pi
k_{1})}{\ell_{2}(m_{1}+m_{2}-m_{2}\cos^{2}(\pi k_{2}-\pi
k_{1}))}&\frac{-g(m_{1}+m_{2})\cos(\pi
k_{2})}{\ell_{2}(m_{1}+m_{2}-m_{2}\cos^{2}(\pi k_{2}-\pi
k_{1}))}\end{bmatrix}.$ (B.7)

We therefore arrive at the following results.

Down-Down. The Down-Down state is obtained by setting $k_{1}=k_{2}=0$. In this
case (B.7) becomes

$\begin{bmatrix}\frac{-g(m_{1}+m_{2})}{\ell_{1}m_{1}}&\frac{gm_{2}}{\ell_{1}m_{1}}\\\
\frac{g(m_{1}+m_{2})}{\ell_{2}m_{1}}&\frac{-g(m_{1}+m_{2})}{\ell_{2}m_{1}}\end{bmatrix}.$
(B.8)

Since the trace of the above matrix is negative and the determinant is
positive for all $m_{1},m_{2},\ell_{1},\ell_{2}>0$, it follows that both
eigenvalues are distinct and strictly negative. Therefore, the linearization
(B.6) about the Down-Down state has two pairs of purely complex eigenvalues,
making it a linear center.

Down-Up. The Down-Up equilibrium is obtained by setting $k_{1}=0$ and
$k_{2}=1$. The matrix (B.7) becomes

$\begin{bmatrix}\frac{-g(m_{1}+m_{2})}{\ell_{1}m_{1}}&\frac{gm_{2}}{\ell_{1}m_{1}}\\\
\frac{-g(m_{1}+m_{2})}{\ell_{2}m_{1}}&\frac{g(m_{1}+m_{2})}{\ell_{2}m_{1}}\end{bmatrix}$
(B.9)

which has a negative determinant for all relevant parameter values. Therefore,
the above matrix has one positive and one negative real eigenvalue, thus
making the Down-Up equilibrium an index-1 saddle since (B.6) has one positive
eigenvalue, one negative eigenvalue, a pair purely imaginary eigenvalues when
$k_{1}=0$ and $k_{2}=1$.

Up-Down. We obtain the Up-Down equilibrium by setting $k_{1}=1$ and $k_{2}=0$.
The matrix (B.7) is then given by

$\begin{bmatrix}\frac{g(m_{1}+m_{2})}{\ell_{1}m_{1}}&\frac{-gm_{2}}{\ell_{1}m_{1}}\\\
\frac{g(m_{1}+m_{2})}{\ell_{2}m_{1}}&\frac{-g(m_{1}+m_{2})}{\ell_{2}m_{1}}\end{bmatrix}$
(B.10)

and following as in the Down-Up equilibrium, we find that the Up-Down
equilibrium is also an index-1 saddle.

Up-Up. The Up-Up state has $k_{1}=k_{2}=1$, resulting in (B.7) taking the form

$\begin{bmatrix}\frac{g(m_{1}+m_{2})}{\ell_{1}m_{1}}&\frac{-gm_{2}}{\ell_{1}m_{1}}\\\
\frac{-g(m_{1}+m_{2})}{\ell_{2}m_{1}}&\frac{g(m_{1}+m_{2})}{\ell_{2}m_{1}}\end{bmatrix}.$
(B.11)

The above matrix is the result of negating all of the entries of the Down-Down
analysis by $-1$. It follows that the linearization (B.6) about the Up-Up
state has two positive eigenvalues and two negative eigenvalues. From the
nomenclature above, the Up-Up state is an index-2 saddle.

From above we can see that the Down-Up and Up-Down equilibria are index-1
saddles, similar to the physical pendulum.

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