On the $\varvec{N}$-extended Euler system: generalized Jacobi elliptic functions

Abstract

We study the integrable system of first order differential equations $\omega _i(v)'=\alpha _i\,\prod _{j\ne i}\omega _j(v)$, $(1\le i, j\le N)$ as an initial value problem, with real coefficients $\alpha _i$ and initial conditions $\omega _i(0)$. The geometrical structure of the system allows to express it as a Poisson system. The analysis is based on its quadratic first integrals. For each dimension N, the system defines a family of functions, generically hyperelliptic functions. When $N=3$, this system generalizes the classic Euler system for the reduced flow of the free rigid body problem; thus, we call it N-extended Euler system (N-EES). In this paper, the cases $N=4$ and $N=5$ are studied, generalizing Jacobi elliptic functions which are defined as a 3-EES. The $N=4$ case was proposed in Hille (Lectures on Ordinary Differential Equations. Addison-Wesley, Reading, 1969), and the solution is presented in Abdel-Salam (Z Naturforsch A 64a:639–645, 2009; it is still expressed as elliptic functions. The hyperelliptic functions arise for the $N=5$ case, which also contain special solutions in elliptic form. Taking into account the nested structure of the N-EES, we propose reparametrizations of the type ${\mathrm{d}}v^*=g(\omega _i)\,{\mathrm{d}}v$ that separate geometry from dynamic. Some of those parametrizations turn out to be generalization of the Jacobi amplitude.

Appendices

Appendix 1: On the ratios of Jacobi $\theta _i$ functions as solutions of 3-EES

From Lawden [17] (Chp. 1) we borrow the following 3-EES differential systems satisfied by the ratios of the Jacobi $\theta _i$ functions

$$\begin{aligned} \frac{{\mathrm{d}}\phantom {-}}{{\mathrm{d}}v}\Big (\frac{\theta _1}{\theta _4}\Big )= & {} \theta _4^2(0)\frac{\theta _2}{\theta _4}\frac{\theta _3}{\theta _4}, \end{aligned}$$

(124)

$$\begin{aligned} \frac{{\mathrm{d}}\phantom {-}}{{\mathrm{d}}v}\Big (\frac{\theta _2}{\theta _4}\Big )= & {} -\theta _3^2(0)\frac{\theta _1}{\theta _4}\frac{\theta _3}{\theta _4}, \end{aligned}$$

(125)

$$\begin{aligned} \frac{{\mathrm{d}}\phantom {-}}{{\mathrm{d}}v}\Big (\frac{\theta _3}{\theta _4}\Big )= & {} -\theta _2^2(0)\frac{\theta _1}{\theta _4}\frac{\theta _2}{\theta _4}, \end{aligned}$$

(126)

etc. We find convenient to introduce the notation $x_{ij}=\theta _j/\theta _i$ and the reparametrization $v\rightarrow \tau$ given by ${\mathrm{d}}\tau =\sqrt{2\mathrm{K}/\pi }\,{\mathrm{d}}v$, with $x_{ij}'={\mathrm{d}}x_{ij}/{\mathrm{d}}\tau$. Thus, taking into account the values of $\theta _i(0)$, where $k^2=m$, $k^2+{k'}^2=1$ and $\mathrm{K}(m)$ are the the complete Legendre first elliptic integral, we write those IVP systems as follows. Note that, as was pointed out in Crespo and Ferrer [5], considering the sign of the coefficients, we may distinguish

• Two bounded systems:

  $$\begin{aligned}&x_{41}'=k'\,x_{42}\,x_{43},\\&x_{42}'=-\,x_{41}\,x_{43},\\&x_{43}'=-k\,x_{41}\,x_{42}, \quad (0,\sqrt{k/k'},1/\sqrt{k'}) \end{aligned}$$

  and

  $$\begin{aligned}&x_{31}'=x_{32}\,x_{34},\\&x_{32}'=-k'\,x_{31}\,x_{34},,\\&x_{34}'=k\,x_{31}\,x_{32}, \quad (0,\sqrt{k},\sqrt{k'}) \end{aligned}$$

• Two unbounded systems:

  $$\begin{aligned}&x_{21}'=k\,x_{23}\,x_{24},\\&x_{23}'=k'\,x_{21}\,x_{24},,\\&x_{24}'=x_{21}\,x_{23}, \quad (0,1/\sqrt{k},\sqrt{k'/k}) \end{aligned}$$

  and

  $$\begin{aligned}&x_{12}'=-k\,x_{13}\,x_{14},\\&x_{13}'=-x_{12}\,x_{14},,\\&x_{14}'=-k'\,x_{12}\,x_{13},(1,\sqrt{(k'{+}1)/k},\sqrt{(k'+1)/k}). \end{aligned}$$

Then, we may express those ratios as functions the Jacobi elliptic functions and their Glashier ratios.

Appendix 2: Transformations and addition formulas for Jacobi elliptic functions

For the benefit of the reader, we bring here some well-known transformations involving the elliptic modulus. They may be found in any handbook of elliptic functions (remember that, depending on the authors, two notations are used: ‘modulus’ or ‘parameter’ related by $k^2\equiv m$, and their complementaries). Those formulas should be used for the reduction to the normal case of some of the particular cases mentioned along the paper.

• Negative parameter

Let m be a positive number and write

$$\begin{aligned} \mu =\frac{m}{1+m}, \quad \mu _1=\frac{1}{1+m}, \; v=\frac{u}{\sqrt{\mu _1}}. \end{aligned}$$

(127)

Then,

$$\begin{aligned}&\mathrm{sn}(u\,;-m)=\sqrt{\mu _1}\,\frac{\mathrm{sn}(v\,;\,\mu )}{\mathrm{dn}(v\,;\,\mu )},\\&\mathrm{cn}(u\,;-m)=\frac{\mathrm{cn}(v\,;\,\mu )}{\mathrm{dn}(v\,;\,\mu )},\\&\mathrm{dn}(u\,;-m)=\frac{1}{\mathrm{dn}(v\,;\,\mu )}. \end{aligned}$$

Thus, elliptic functions with negative parameter may be expressed by elliptic functions with a positive parameter. Note that $0<\mu <1$.

A final comment related to the complete elliptic integral of first kind is due here. Unlike Maple, the software Mathematica yields the following result

$$\begin{aligned} \int _0^{\pi /2}\frac{\mathrm {d}\phi }{\sqrt{1-m\sin ^2\phi }}=\frac{1}{\sqrt{1-m}}\,\mathrm {K}\left( \frac{m}{m-1}\right) \end{aligned}$$

(128)

for $\forall m\le 1$, instead of the expected result $\mathrm {K}(m)$. By applying the previous change (127), we have that, being m a positive number,

$$\begin{aligned} \mathrm {K}(-m)=\frac{1}{\sqrt{1+m}}\,\mathrm {K}\left( \frac{m}{1+m}\right) =\sqrt{\mu _1}\,\mathrm {K}(\mu ) \end{aligned}$$

(129)

which is exactly the same result given by Mathematica for $m<0$.

• Reciprocal parameter

Denoting now $v=\sqrt{m}u$, we have

$$\begin{aligned}&\mathrm{sn}(u\,;m)=\frac{1}{\sqrt{m}}\,\mathrm{sn}(v\,;m^{-1}),\\&\mathrm{cn}(u\,;m)=\mathrm{dn}(v\,;m^{-1}),\\&\mathrm{dn}(u\,;m)=\mathrm{cn}(v\,;m^{-1}). \end{aligned}$$

This is Jacobi’s real transformation. If $m>1$, then $m^{-1}<1$; thus, elliptic functions whose parameter is greater than 1 are related to the ones whose parameter is less than 1. In short, there is no loss of generality assuming $0\le m\le 1$.

• Decrease of parameter

$$\begin{aligned} \mu =\Big (\frac{1-\sqrt{m_1}}{1+\sqrt{m_1}}\Big )^2, \quad v=\frac{u}{1+\sqrt{\mu }}. \end{aligned}$$

(130)

$$\begin{aligned}&\mathrm{sn}(u\,;m)=\frac{(1+\sqrt{\mu })\mathrm{sn}(v\,;\,\mu )}{1+\sqrt{\mu }\,\mathrm{sn}^2(v\,;\,\mu )},\\&\mathrm{cn}(u\,;m)=\frac{\mathrm{cn}(v\,;\,\mu )\,\mathrm{dn}(v\,;\,\mu )}{1+\sqrt{\mu }\,\mathrm{sn}^2(v\,;\,\mu )},\\&\mathrm{dn}(u\,;m)=\frac{1-\sqrt{\mu }\,\mathrm{sn}^2(v\,;\,\mu )}{1+\sqrt{\mu }\,\mathrm{sn}^2(v\,;\,\mu )}. \end{aligned}$$

This is Gauss transformation or the descending Landen transformation, which makes elliptic functions to depend on functions with a smaller parameter.

Note that, making use of the double angle, we may also write

$$\begin{aligned} \mathrm{dn}(u\,;m)=\frac{\sqrt{\mu }\,\mathrm{cn}(2v\,;\,\mu ) + \mathrm{dn}(2v\,;\,\mu )}{1+\sqrt{\mu }}. \end{aligned}$$

(131)

There are analogous expressions for the increase of parameter. For a recent study where generalized formules are given, see [16].

• Addition formulae

Complementing previous transformations, we collect also here the addition formulae

$$\begin{aligned} \mathrm{sn}(\alpha +\beta )= & {} \frac{\mathrm{sn}\,\alpha \, \mathrm{cn}\,\beta \,\mathrm{dn}\,\beta + \mathrm{sn}\,\beta \, \mathrm{cn}\,\alpha \, \mathrm{dn}\,\alpha }{1-m\,\mathrm{sn}^2\alpha \, \mathrm{sn}^2\beta },\\ \mathrm{cn}(\alpha +\beta )= & {} \frac{\mathrm{cn}\,\alpha \, \mathrm{cn}\,\beta - \mathrm{sn}\,\alpha \, \mathrm{sn}\,\beta \,\mathrm{dn}\alpha \, \mathrm{dn}\,\beta }{1-m\,\mathrm{sn}^2\alpha \, \mathrm{sn}^2\beta },\\ \mathrm{dn}(\alpha +\beta )= & {} \frac{\mathrm{dn}\,\alpha \, \mathrm{dn}\beta - m\, \mathrm{sn}\,\alpha \, \mathrm{sn}\,\beta \,\mathrm{cn}\,\alpha \,\mathrm{cn}\,\beta }{1-m\,\mathrm{sn}^2\alpha \, \mathrm{sn}^2\beta }, \end{aligned}$$

which we have generalized for the new functions; more precisely, this has been done for the 4-EES Mahler system.

A précis on Generalized Nambu dynamics

With the aim of making this work as self-contained as possible, we review here, without giving a proof, some of the basics concepts and facts relative to Nambu dynamics. All of them and further details may be found in [23, 24] and the references therein.

In 1973, Yoichiro Nambu introduced a generalization of the Hamiltonian dynamics by introducing a new bracket called the Nambu bracket [23]. This new structure generalizes the Poisson bracket and enables to define the Nambu-Poisson manifolds (see [24]). Here we recall some of the basic ideas about this generalization.

Hamiltonian dynamics takes place on a Poisson manifold, i. e., a pair made of a smooth manifold M endowed with a bilinear operation $\{\,,\,\}$ on $\mathcal {F}(M)=C^{\infty }(M)$ satisfying skew-symmetry, the Jacobi identity and the Leibniz rule. Nambu’s generalization hinges on the introduction of the Nambu-Poisson manifold of order N. It is a smooth manifold M endowed with a N-multilinear skew-symmetric operation on $\mathcal {F}(M)$ called the Nambu bracket of order N, satisfying the fundamental identity, which generalizes the Jacobi identity and one more property extending the Leibniz rule

• (i) The Nambu bracket

  $$\begin{aligned} \{\,,\,\}:\mathcal {F}(M)\otimes ^N\longrightarrow \mathcal {F}(M) \end{aligned}$$

  (132)

  is a N-multilinear operation.

• (ii) Skew-symmetry,

  $$\begin{aligned} \{f_1,\ldots ,f_N\}={\epsilon (i_1,\ldots ,i_N)}\{f_{i_1},\ldots ,f_{i_N}\}. \end{aligned}$$

  (133)

  Where $\epsilon$ is the N-dimensional Levi-Civita symbol.

• (iii) The fundamental identity holds

  $$\begin{aligned}&\{\{f_1,\ldots \,f_N\},f_{N+1},\ldots ,f_{2N-1}\}\nonumber \\&\quad +\{f_{N},\{f_1,\ldots \,,f_{N-1},f_{N+1}\},f_{N+2},\nonumber \\&\quad \ldots ,f_{2N-1}\} +\cdots \cdots +\{f_N,\ldots ,f_{2N-2},\nonumber \\&\quad \{f_1,\ldots ,\,f_{N-1},f_{2N-1}\}\}\nonumber \\&\quad =\{f_1,\ldots ,f_{N-1},\{f_N,\ldots ,\,f_{2N-1}\}\}. \end{aligned}$$

  (134)

• (iv) $\{\;,\,\}$ Leibniz rule is satisfied in each factor

  $$\begin{aligned}&\{f_1f_2,\ldots ,f_{N+1}\}=f_1\{f_2,\ldots ,f_{N+1}\}\nonumber \\&\quad +f_2\{f_1,\ldots ,f_{N+1}\}, \forall \; f_1,\ldots ,f_{N+1}\in \mathcal {F}(M).\nonumber \\ \end{aligned}$$

  (135)

According to the above properties, Nambu bracket may be rendered in a geometrical sense as a smooth section of the vector space of exterior N-forms on the tangent bundle of M, i.e., $\bigwedge ^NTM$. In other words, the Nambu bracket is realized as the N-contravariant tensor

$$\begin{aligned} \{f_1,\ldots ,f_{N}\}\mathop {=}\limits ^{\mathrm{def}}\beta (df_1,\ldots ,df_N), \end{aligned}$$

(136)

where $\beta \in \bigwedge ^NTM$ is called the Nambu tensor. It is given, in local coordinates $x=(\omega _1,\ldots ,\omega _N)$, by the following expression

$$\begin{aligned} \beta =\sum _{i_1,\ldots ,i_N=1}^N \beta _{i_1,\ldots ,i_N}(x)\frac{\partial }{\partial \omega _{i_1}}\wedge \ldots \wedge \frac{\partial }{\partial \omega _{i_N}}, \end{aligned}$$

(137)

Notice that $\beta$ may be expressed, for suitable local coordinates, by $\beta _{i_1,\ldots ,i_N}(x)=\epsilon (i_1,\ldots ,i_N)$, where $\epsilon$ is the Levi-Civita tensor. In what follows, we name the N-contravariant tensor given by Levi-Civita tensor as the standard Nambu bracket on $\mathcal {F}(M)$, which will be denoted by the usual bracket $\{\,,\,\}$. That is to say, the standard Nambu bracket is given by the determinant of the gradients of the functions involved.

Even though Nambu is a generalization of Hamiltonian dynamics, there are also fundamental differences between them. For example, in [15], it is proven that every Nambu-Poisson bracket with $N\ge 3$ is essentially a determinant. This is not true for Poisson ones.

This bracket allows to study the variation in $f\in \mathcal {F}(M)$ on Nambu-Poisson manifolds when it is restricted to be in the intersection of $N-1$ hypermanifolds $H_i$

$$\begin{aligned} \dfrac{df}{dt}=\{f,H_1,\ldots ,H_{N-1}\}. \end{aligned}$$

(138)

In this vein, the Nambu formulation has been applied to Hamiltonian systems with constraints (see [14] and the references therein). It is straightforward to extend the above formalism to the dynamics of points $\omega =(\omega _1,\ldots ,\omega _N)$ in the phase space M by means of the Nambu–Hamilton equations of motion as they were first given in [23]

$$\begin{aligned} \dfrac{d\omega _i}{dt}= & {} \{\omega _i,H_1,\ldots ,H_{N-1}\} \phantom {\quad \quad \quad \frac{\partial H_1}{\partial \omega _{i_1}}\ldots \frac{\partial H_{N-1}}{\partial \omega _{i_{N-1}}}}\nonumber \\= & {} \sum _{i_1..i_{N-1}\ne i} \epsilon (i,i_1,\ldots ,i_{N-1})\frac{\partial H_1}{\partial \omega _{i_1}}\ldots \frac{\partial H_{N-1}}{\partial \omega _{i_{N-1}}},\nonumber \\ \end{aligned}$$

(139)

where $H_i$ are called the Hamiltonian functions.

Next we gather below some basic features of the Nambu structures, which will be of high relevance in the subsequent development. All of them, except the Remark 8, can be found in [24].

Theorem 3

(Nambu nested structure) The set of all possible Nambu structures on M is isomorphic to the Grassmann algebra $\bigwedge TM$. Since it is a graded associative algebra, every Nambu structure is considered as an N-degree element of $\bigwedge TM$ and by fixing $f_1,\ldots ,f_k$ in (136), with $k\le N-2$; we are left with a Nambu structure of order $N-k$. More precisely for the case $k=N-2$, the Nambu structure obtained is a Poisson structure and the fixed integrals $f_1,\ldots ,f_{N-2}$ are the Casimirs.

Theorem 4

($SL(N,\mathbb {R})$ Nambu bracket invariance) The Nambu bracket is invariant under the action (left or right) of the special linear group $SL(N,\mathbb {R})$. That is to say, let $\phi$ be the action of $SL(N,\mathbb {R})$ on $\mathcal {F}(M)\otimes ^N$ given by

$$\begin{aligned}&\phi :SL(N,\mathbb {R})\times \mathcal {F}(M)\otimes ^N\longrightarrow \mathcal {F}(M)\otimes ^N,\nonumber \\&\quad (A,F)\rightarrow F', \end{aligned}$$

(140)

where $A\in SL(N,\mathbb {R})$ and $F,F'\in \mathcal {F}(M)\otimes ^N$ are the N-tuples given by $F=(f_1,\ldots f_N)$ and $F'=A\,F=(f_1',\ldots f_N')$. Thus, the following identity holds

$$\begin{aligned} \{f_1,\ldots ,f_{N}\}=\{f_1',\ldots ,f_{N}'\} \end{aligned}$$

(141)

Theorem 5

(Liouville Condition) The corresponding phase flow on the phase space of the Nambu–Hamilton equations of motion is divergence-free and preserves the standard volume form $d\omega _1\wedge \dots d\omega _N$.

However, the reciprocal of Theorem 5 is not true, i.e., divergence-free systems can be written into the Nambu formalism. Such a statement was made in [11], but the error is shown in [4], see [22] for further details.

Remark 8

(Geometric interpretation) Let us consider $\mathbb {R}^N$ together with the standard Nambu bracket and $N-1$ hypermanifolds $M_{i}^{h_i}$ given by the level sets $H_i=h_i$ of the functions $H_i\in \mathcal {F}(M)$ for $i=1,\ldots ,N-1$. Then, the Nambu–Hamilton equations of motion given in (138) may be interpreted as a parametrization of the intersection curves resulting from $\bigcap _{i=1}^{N-1} M_{i}^{h_i}$.