Quadratic Homogeneous ODE Systems of Jordan-Rigid Body Type

Dedicated to Acad. Radu Miron on the occasion of his 75 birthday

Mircea Crâşmăreanu

Abstract Trying to yield a Lax representation with the Lax pair of so(3)-type for some quadratic differential systems but with the Jordan bracket instead of usual Lie bracket, a class of homogeneous systems with remarkable properties is obtained.

Mathematics Subject Classification: 34A26, 17B80, 34A34 Key words: quadratic homogeneous system, Lax representation, Brockett perturbation, Jordan-rigid body systems.

The class of polynomial homogeneous ODE systems has special features, important from the point of view of applications ([3]). In this class, the subset of quadratic homogeneous systems presents interesting geometric aspects, cf. [4]. Such a quadratic system reads:

\frac{dx^{i}}{dt} = a_{jk}^{i}x^{j}x^{k}

and important examples are the Euler-Poincaré equations on Lie algebras ([10, p. 19])

and homogeneous sprays ([12]) if x^{i} = \frac{dy^{i}}{dt}.

For differential systems a useful method of study is provided by the so-called Lax representation which yields, in a systematic manner, a lot of first integrals, with mechanical significance, like energy or momentum ([10]).

Unfortunately, for some remarkable quadratic systems, one cannot find a Lax pair. Our paper gives two examples, namely the Nahm system and the system of elliptic Jacobi functions, treated in the second section. But we try to obtain a Lax pair for these systems, using another bracket instead of classical Lie bracket, more precisely the Jordan bracket. These types of deformation brackets appear in the first section where we add the so-called Brockett perturbation, a notion useful in rigid body control ([2], [5]). Let us note that we use the form of Lax pair given by the rigid body, which admits a classical Lax representation i.e. with respect to the Lie multiplication. For our aim, we must restrict to the upper-triangular part from a matriceal differential system and then a quadratic homogeneous ODE system is obtained called Jordan-rigid body type system. In the third section, using the Brockett perturbation for system of elliptic Jacobi functions, we obtain a so-called Brockett perturbation of trigonometric functions and a rational first integral of this last system is derived via Bernoulli type equations. In the following section, a Jordan counterpart of two systems, rigid body and Maxwell-Bloch, with usual Lax representation, is obtained. In the next section to a Jordan-rigid body type system we associate (i) two particular solutions, (ii) two quadratic first integrals and (iii) a numerical integrator preserving these first integrals. A search for so (4) version of our result and an appendix with the Maple computation of some matrices, useful for our models, and the Maple plot of three examples are inserted at the end of paper.

∗ Balkan Journal of Geometry and Its Applications, Vol.7, No.2, 2002, pp. 27-42.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2002.

An important remark is that in this new approach the above mentioned physical significance of Lax pairs is lost. Our initial aim was to point an algebraic character of Jordan-Lax type for some important systems but this searching for Lax yields a veritable round through algebra, geometry, differential equations and mechanics. For a survey on other applications of Jordan algebras in physics see [8].

1 Deformation algebras and Brockett perturbation

Let A be a real algebra in which the product is denoted by adjacency and let two numbers h, k ∈ IR. Let us consider the bracket [,]_{h,k} given by:

(1.1)  [a, b]_{h,k} = h · ab + k · ba

for every a, b ∈ A. Then the pair (A, [,]_{h,k}) is a new real algebra, appeared in [15] and [16]. Let us note that in the cited papers the bracket is denoted by ”∗” and (A, ∗) is called G-algebra. We prefer the bracket notations having in mind the following examples and as name the (h, k)-deformation algebra of A.

Examples ([16]):

(i) h ≡ 1, k ≡ −1 is the Lie algebra associated to A,

(ii) h = k ≡ 1 is the Jordan algebra associated to A.

Inspired by [5, p. 135] for a, b ∈ A we consider the Brockett perturbation with respect to [,]_{h,k} as the element [a, b]_{B,h,k} defined by

(1.2)  [a, b]_{B,h,k} = [a, b]_{h,k} + [a, [a, b]_{h,k}]_{h,k} .

A straightforward calculation gives

(1.3)  [a, b]_{B,h,k} = h · ab + k · ba + h^{2} · a^{2}b + 2hk · aba + k^{2} · ba^{2}

which for the considered examples becomes

(1.3i)  [a, b]_{B,1,−1} = ab − ba + a^{2}b − 2aba + ba^{2}

(1.3ii)  [a, b]_{B,1,1} = ab + ba + a^{2}b + 2aba + ba^{2}.

2 Systems of Jordan-rigid body type

It is well-known that some important ordinary differential systems admit a Lax formulation, that is, can be put in the equivalent form:

(2.1)  \frac{d}{dt}L = [L, B]

where L and B are matrices and [L, B] = LB − BL = [L, B]_{1,−1} is their commutator. The pair (L, B) is called the Lax pair of the given differential system and in Russian usage a Lax formulation is called sometimes Heisenberg representation ([10]). The utility of a Lax representation is given by the fact that the eigenvalues of L are first integrals of (2.1)([10, p. 98], [11]).

Examples of dynamical systems admitting a Lax pair: the 1-dimensional harmonic oscillator, the free rigid body, the Maxwell-Bloch equations from laser matter dynamics, the Toda lattice and so on (see the previous citations). In the following we recall in detail the rigid body, for which the equations of motion are ([5, p. 136]):

(2.2)  \{ \begin{matrix}\frac{dx_{1}}{dt} = a_{1}x_{2}x_{3} \\ \frac{dx_{2}}{dt} = a_{2}x_{3}x_{1} \\ \frac{dx_{3}}{dt} = a_{3}x_{1}x_{2} \end{matrix}

where

(2.3)  a_{1} = \frac{1}{I_{3}} − \frac{1}{I_{2}} , a_{2} = \frac{1}{I_{1}} − \frac{1}{I_{3}} , a_{3} = \frac{1}{I_{2}} − \frac{1}{I_{1}}

with I_{1}, I_{2}, I_{3} the principal moments of inertia and I_{1} > I_{2} > I_{3} > 0. Because the matrices of so (3)-type appear naturally in equations of motion for orthonormal basis in R^{3}, we search for a Lax pair of so (3)-type: (2.4)

L = ( \begin{matrix}0 & −x_{3} & x_{2} \\ x_{3} & 0 & −x_{1} \\ −x_{2} & x_{1} & 0 \end{matrix} ) , B = B (α_{1}, α_{2}, α_{3}) = ( \begin{matrix}0 & −α_{3}x_{3} & α_{2}x_{2} \\ α_{3}x_{3} & 0 & −α_{1}x_{1} \\ −α_{2}x_{2} & α_{1}x_{1} & 0 \end{matrix} )

with the unknowns α_{1}, α_{2}, α_{3}. The eq. (2.1) then becomes:

(2.5)  \{ \begin{matrix}\frac{dx_{1}}{dt} = (α_{3} − α_{2}) x_{2}x_{3} \\ \frac{dx_{2}}{dt} = (α_{1} − α_{3}) x_{3}x_{1} \\ \frac{dx_{3}}{dt} = (α_{2} − α_{1}) x_{1}x_{2} \end{matrix}

and comparing with (2.2) yields the equations:

\{ \begin{matrix}α_{3} − α_{2} = a_{1} = \frac{1}{I_{3}} − \frac{1}{I_{2}} \\ α_{1} − α_{3} = a_{2} = \frac{1}{I_{1}} − \frac{1}{I_{3}} \\ α_{2} − α_{1} = a_{3} = \frac{1}{I_{2}} − \frac{1}{I_{1}} \end{matrix}

with solution

(2.6)  α_{i} = \frac{1}{I_{i}} , 1 ≤ i ≤ 3.

It results that we obtain the Lax pair of the rigid body and then one have the associated Brockett perturbation ([5, p. 137]):

(2.7)  \{ \begin{matrix}\frac{dx_{1}}{dt} = a_{1}x_{2}x_{3} + x_{1} (a_{3}x_{2}^{2} − a_{2}x_{3}^{2}) \\ \frac{dx_{2}}{dt} = a_{2}x_{3}x_{1} + x_{2} (a_{1}x_{3}^{2} − a_{3}x_{1}^{2}) \\ \frac{dx_{3}}{dt} = a_{3}x_{1}x_{2} + x_{3} (a_{2}x_{1}^{2} − a_{1}x_{2}^{2}) \end{matrix} .

In the following we try the same program for other two remarkable quadratic systems ([6]):

A the Nahm system of static SU(2)-monopoles or Tzitzeica gradient flow

(2.8)  \{ \begin{matrix}\frac{dx_{1}}{dt} = x_{2}x_{3} \\ \frac{dx_{2}}{dt} = x_{3}x_{1} \\ \frac{dx_{3}}{dt} = x_{1}x_{2} \end{matrix}

B the differential system of elliptic Jacobi functions Tzitzeica-Lorentz gradient flow

(2.9)  \{ \begin{matrix}\frac{dx_{1}}{dt} = x_{2}x_{3} \\ \frac{dx_{2}}{dt} = −x_{3}x_{1} \\ \frac{dx_{3}}{dt} = −k^{2}x_{1}x_{2} \end{matrix}

with 0 < k^{2} < 1.

Comparing (2.5) with (2.8) and (2.9) we have:

A

\{ \begin{matrix}α_{3} − α_{2} = 1 \\ α_{1} − α_{3} = 1 \\ α_{2} − α_{1} = 1 \end{matrix}

and adding all equations it results 0 = 3, false B

\{ \begin{matrix}α_{3} − α_{2} = 1 \\ α_{1} − α_{3} = −1 \\ α_{2} − α_{1} = −k^{2} \end{matrix}

and adding it results 0 = −k^{2}, false.

Therefore the Lie bracket is not suitable for systems A and B with the pair (2.4) and then we try with the next (h, k) -deformation bracket, namely Jordan deformation. In this case the system:

(2.10)  \frac{d}{dt}L = [L, B]_{1,1} = LB + BL

for the pair (2.4) reads:

\frac{d}{dt} ( \begin{matrix}0 & −x_{3} & x_{2} \\ x_{3} & 0 & −x_{1} \\ −x_{2} & x_{1} & 0 \end{matrix} ) =

(2.11)  = ( \begin{matrix}−2 (α_{2}x_{2}^{2} + α_{3}x_{3}^{2}) & (α_{1} + α_{2}) x_{1}x_{2} & (α_{3} + α_{1}) x_{3}x_{1} \\ (α_{1} + α_{2}) x_{1}x_{2} & −2 (α_{3}x_{3}^{2} + α_{1}x_{1}^{2}) & (α_{2} + α_{3}) x_{2}x_{3} \\ (α_{3} + α_{1}) x_{3}x_{1} & (α_{2} + α_{3}) x_{2}x_{3} & −2 (α_{1}x_{1}^{2} + α_{2}x_{2}^{2}) \end{matrix} ) .

The global equality (2.11) is impossible because the LHS is skew-symmetric and the RHS is symmetric. Then we restrict only to the upper-triangular system

(2.12)  \{ \begin{matrix}\frac{dx_{1}}{dt} = − (α_{2} + α_{3}) x_{2}x_{3} \\ \frac{dx_{2}}{dt} = (α_{3} + α_{1}) x_{3}x_{1} \\ \frac{dx_{3}}{dt} = − (α_{1} + α_{2}) x_{1}x_{2} \end{matrix}

because if (x_{1}, x_{2}, x_{3}) is a solution of this system then (y_{1}, y_{2}, y_{3}) = (−x_{1}, −x_{2}, −x_{3}) is solution for the lower-triangular system:

\{ \begin{matrix}\frac{dy_{1}}{dt} = (α_{2} + α_{3}) y_{2}y_{3} \\ \frac{dy_{2}}{dt} = − (α_{3} + α_{1}) y_{3}y_{1} \\ \frac{dy_{3}}{dt} = (α_{1} + α_{2}) y_{1}y_{2} \end{matrix} .

For our examples the system (2.12) becomes:

A

\{ \begin{matrix}α_{2} + α_{3} = −1 \\ α_{3} + α_{1} = 1 \\ α_{1} + α_{2} = −1 \end{matrix}

with solution

(2.13)  α_{1} = α_{3} = \frac{1}{2}, α_{2} = −\frac{3}{2}.

B

\{ \begin{matrix}α_{2} + α_{3} = −1 \\ α_{3} + α_{1} = −1 \\ α_{1} + α_{2} = k^{2} \end{matrix}

with solution

(2.14)  α_{1} = α_{2} = \frac{k^{2}}{2} , α_{3} = −\frac{k^{2}}{2} − 1

Using the boundary of parameter k^{2} it results for this example:

(2.15)  −\frac{3}{2} < α_{3} < −1 < 0 < α_{1} = α_{2} < \frac{1}{2}.

In conclusion, examples A and B admit a Lax pair of so (3)-type with respect to the Jordan bracket. Systems of (2.5)-type we call Lie-rigid body and systems of (2.12)-type we call Jordan-rigid body.

The systems (2.12) admit another form. Deriving the first eq. it results:

(2.16)  \frac{d^{2}x_{1}}{dt^{2}} = − (α_{2} + α_{3}) x_{1} [(α_{3} + α_{1}) x_{3}^{2} − (α_{1} + α_{2}) x_{2}^{3}]

and deriving again one obtains

\frac{d^{3}x_{1}}{dt^{3}} = \frac{dx_{1}}{dt} \frac{\frac{d^{2}x_{1}}{dt^{2}}}{x_{1}} + 4 (α_{1} + α_{2}) (α_{2} + α_{3}) (α_{3} + α_{1}) x_{1}^{2} \frac{dx_{1}}{dt}

or

\frac{x_{1} \frac{d^{3}x_{1}}{dt^{3}} − \frac{dx_{1}}{dt} \frac{d^{2}x_{1}}{dt^{2}}}{x_{1}^{2}} = 4 (α_{1} + α_{2}) (α_{2} + α_{3}) (α_{3} + α_{1}) x_{1} \frac{dx_{1}}{dt}

which can be integrated

(2.17)  \frac{\frac{d^{2}x_{1}}{dt^{2}}}{x_{1}} = 2 (α_{1} + α_{2}) (α_{2} + α_{3}) (α_{3} + α_{1}) x_{1}^{2} + C_{1}

with C_{1} a real constant. For C_{1} = 0 two particular solutions are obtained in section 5, see relations (5.3).

Using these Jordan-Lax representations and (1.3ii) we can write the Brockett perturbation

A

(2.18)  \{ \begin{matrix}\frac{dx_{1}}{dt} = x_{2}x_{3} − 2x_{1} (x_{1}^{2} − 2x_{2}^{2} + x_{3}^{2}) \\ \frac{dx_{2}}{dt} = x_{3}x_{1} + 6x_{2}^{3} \\ \frac{dx_{3}}{dt} = x_{1}x_{2} − 2x_{3} (x_{1}^{2} − 2x_{2}^{2} + x_{3}^{2}) \end{matrix}

B

(2.19)  \{ \begin{matrix}\frac{dx_{1}}{dt} = x_{2}x_{3} + x_{1} [−2k^{2}x_{1}^{2} − 2k^{2}x_{2}^{2} + (k^{2} + 3) x_{3}^{2}] \\ \frac{dx_{2}}{dt} = −x_{3}x_{1} + x_{2} [−2k^{2}x_{1}^{2} − 2k^{2}x_{2}^{2} + (k^{2} + 3) x_{3}^{2}] \\ \frac{dx_{3}}{dt} = −k^{2}x_{1}x_{2} + x_{3} [(1 − k^{2}) x_{1}^{2} + (1 − k^{2}) x_{2}^{2} + 2 (k^{2} + 2) x_{3}^{2}] \end{matrix}

Another remarkable quadratic homogeneous system is([9]):

(2.20)  \{ \begin{matrix}\frac{dx_{1}}{dt} = x_{2}x_{3} − (x_{1})^{2} \\ \frac{dx_{2}}{dt} = x_{3}x_{1} − (x_{2})^{2} \\ \frac{dx_{3}}{dt} = x_{1}x_{2} − (x_{3})^{2} \end{matrix}

and because

B (x_{1}, −x_{2}, x_{3}) = ( \begin{matrix}0 & − (x_{3})^{2} & − (x_{2})^{2} \\ (x_{3})^{2} & 0 & − (x_{1})^{2} \\ (x_{2})^{2} & (x_{1})^{2} & 0 \end{matrix} )

it results that the Kasner system (2.18) is the upper-triangular part of relation:

\frac{dL}{dt} = [L, B (\frac{1}{2}, −\frac{3}{2}, \frac{1}{2})]_{1,1} + B (x_{1}, −x_{2}, x_{3}) .

3 Brockett perturbation of trigonometric functions

A well-known property for the system of elliptic Jacobi function is that when k^{2} → 0 the solution (x_{1}, x_{2}, x_{3}) of (2.9) satisfy:

(x_{1}, x_{2}, x_{3}) → (sin t, cos t, 1)

and this fact is used, for example, in order to obtain a generalization of Newtonian equations of motion in [19].

As consequence, the limit for k^{2} → 0 of system (2.17), namely:

(3.1)  \{ \begin{matrix}\frac{dx_{1}}{dt} = x_{2}x_{3} + 3x_{1}x_{3}^{2} \\ \frac{dx_{2}}{dt} = −x_{3}x_{1} + 3x_{2}x_{3}^{2} \\ \frac{dx_{3}}{dt} = x_{3} (x_{1}^{2} + x_{2}^{2} + 4x_{3}^{2}) \end{matrix}

we call the Brockett perturbation of trigonometric functions.

The aim of this section is to obtain a first integral of this differential system. On this respect we multiply the first eq. with x_{1} and the second eq. with x_{2}:

\{ \begin{matrix}x_{1} \frac{dx_{1}}{dt} = x_{1}x_{2}x_{3} + 3x_{1}^{2}x_{3}^{2} \\ x_{2} \frac{dx_{2}}{dt} = −x_{1}x_{2}x_{3} + 3x_{2}^{2}x_{3}^{2} \end{matrix}

and adding these relations it results

\frac{d}{dt} (x_{1}^{2} + x_{2}^{2}) = 6x_{3}^{2} (x_{1}^{2} + x_{2}^{2}) .

From the previous eq. and the last eq. (3.1) we have:

(3.2)  \frac{dx_{3}}{d (x_{1}^{2} + x_{2}^{2})} = \frac{x_{3} (x_{1}^{2} + x_{2}^{2} + 4x_{3}^{2})}{6x_{3}^{2} (x_{1}^{2} + x_{2}^{2})} = \frac{2}{3 (x_{1}^{2} + x_{2}^{2})}x_{3} + \frac{1}{6x_{3}} .

The eq. (3.2) considered in the unknown x_{3} as function of x_{1}^{2} + x_{2}^{2} is a Bernoulli type equation:

\frac{dx}{dt} + P (t) x = Q (t) x^{n}

with: x = x_{3}, t = x_{1}^{2}+x_{2}^{2}, P (t) = \frac{−2}{3t}, Q (t) = \frac{1}{6}, n = −1. The standard transformation

z = x^{1−n}, in our case z = x^{1−(−1)} = x^{2}, yields the equation:

\frac{dz}{dt} = 2x\frac{dx}{dt} = 2x (\frac{2}{3t}x + \frac{1}{6x}) = \frac{4}{3t}x^{2} + \frac{1}{3} = \frac{4}{3t}z + \frac{1}{3}

which is a linear one and then has the solution:

(3.3)  z = x_{3}^{2} = Ct^{\frac{4}{3}} − t = C (x_{1}^{2} + x_{2}^{2})^{\frac{4}{3}} − (x_{1}^{2} + x_{2}^{2})

with C a real constant. The final relation:

(3.4)  C^{3} = \begin{matrix}(x_{1}^{2} + x_{2}^{2} + x_{3}^{2}) & 3 \\ (x_{1}^{2} + x_{2}^{2}) & 4 \end{matrix}

is, in conclusion, a first integral for the system (3.1) of Brockett perturbation of trigonometric functions.

4 Jordan version of some classical Lax systems

At the beginning of section 2 there are given some examples of systems with usual Lax representation: the rigid body and Maxwell-Bloch equations. Is amazing that these systems appear also in the Jordan setting:

C From the rigid body system (2.2) and the system (2.12) it results:

(4.1)  \{ \begin{matrix}α_{2} + α_{3} = −a_{1} \\ α_{3} + α_{1} = a_{2} \\ α_{1} + α_{2} = −a_{3} \end{matrix}

with solution

(4.2)  α_{1} = −a_{3}, α_{2} = 0, α_{3} = −a_{1}

and the Brockett perturbation

(4.3)  \{ \begin{matrix}\frac{dx_{1}}{dt} = a_{1}x_{2}x_{3} + x_{1} [4a_{3}x_{1}^{2} + a_{3}x_{2}^{2} + (3a_{1} + a_{3}) x_{3}^{2}] \\ \frac{dx_{2}}{dt} = a_{2}x_{3}x_{1} + 3x_{2} (a_{3}x_{1}^{2} + a_{1}x_{3}^{2}) \\ \frac{dx_{3}}{dt} = a_{3}x_{1}x_{2} + x_{3} [(a_{1} + 3a_{3}) x_{1}^{2} + a_{1}x_{2}^{2} + 4a_{1}x_{3}] \end{matrix} .

D The system of Maxwell-Bloch equations is ([13], [14])

(4.4)  \{ \begin{matrix}\frac{dx_{1}}{dt} = x_{2} \\ \frac{dx_{2}}{dt} = x_{3}x_{1} \\ \frac{dx_{3}}{dt} = −x_{1}x_{2} \end{matrix}

and comparing with (2.12) it follows

(4.5)  \{ \begin{matrix}α_{2} + α_{3} = −\frac{1}{x_{3}} \\ α_{3} + α_{1} = 1 \\ α_{1} + α_{2} = 1 \end{matrix}

with solution

(4.6)  α_{1} = 1 + \frac{1}{2x_{3}} , α_{2} = α_{3} = −\frac{1}{2x_{3}}

and Brockett perturbation

(4.7)  \{ \begin{matrix}\frac{dx_{1}}{dt} = x_{2} − x_{1} [(4 + \frac{2}{x_{3}} ) x_{1}^{2} + (1 − \frac{1}{x_{3}} ) x_{2}^{2} + (1 − \frac{1}{x_{3}} ) x_{3}^{2}] \\ \frac{dx_{2}}{dt} = x_{3}x_{1} − x_{2} [(3 + \frac{1}{x_{3}} ) x_{1}^{2} − \frac{2}{x_{3}} x_{2}^{2} − 2x_{3}] \\ \frac{dx_{3}}{dt} = −x_{1}x_{2} − x_{3} [(3 + \frac{1}{x_{3}} ) x_{1}^{2} − \frac{2}{x_{3}} x_{2}^{2} − 2x_{3}] \end{matrix}

5 More about Jordan-rigid body systems

Recall the property of Lax representation with respect to the eigenvalues of L as first integrals. Unfortunately, this useful fact is not true in Jordan representation. So, the only non-zero eigenvalue of L is:

(5.1)  E = x_{1}^{2} + x_{2}^{2} + x_{3}^{2}

and then along the solutions of A and B we have:

\frac{dE}{dt} = \frac{∂E}{∂x_{1}} \frac{dx_{1}}{dt} + \frac{∂E}{∂x_{2}} \frac{dx_{2}}{dt} + \frac{∂E}{∂x_{3}} \frac{dx_{3}}{dt} =

= \{ \begin{matrix}=^{A} 2x_{1} (x_{2}x_{3}) + 2x_{2} (x_{3}x_{1}) + 2x_{3} (x_{1}x_{2}) = 6x_{1}x_{2}x_{3} ≠ 0 \\ =^{B} 2x_{1} (x_{2}x_{3}) + 2x_{2} (−x_{3}x_{1}) + 2x_{3} (−k^{2}x_{1}x_{2}) = −2k^{2}x_{1}x_{2}x_{3} ≠ 0. \end{matrix}

The Jordan-rigid body systems are particular cases of homogeneous systems treated in detail in [10, p. 311]:

(5.2)  \{ \begin{matrix}z^{.}_{1}= ε_{1}z_{2}z_{3} \\ z^{.}_{2}= ε_{2}z_{3}z_{1} \\ z^{.}_{3}= ε_{3}z_{1}z_{2} \end{matrix} .

Following the conclusion of [10, p. 311-312] and [17] about the above systems it results the existence of two particular solutions of the form z_{k} (t) = \frac{c_{k}}{t} with c_{k} real constants given by:

(5.3)  \{ \begin{matrix}c_{1} = ±\frac{1}{\sqrt{− (α_{3} + α_{1}) (α_{1} + α_{2})}} \\ c_{2} = ±\frac{1}{\sqrt{(α_{1} + α_{2}) (α_{2} + α_{3})}} \\ c_{3} = ±\frac{1}{\sqrt{− (α_{2} + α_{3}) (α_{3} + α_{1})}}. \end{matrix}

For our examples A

(5.4)  \{ \begin{matrix}(x_{1}, x_{2}, x_{3}) (t) = (\frac{1}{t}, −\frac{1}{t}, \frac{1}{t}) \\ (x_{1}, x_{2}, x_{3}) (t) = (\frac{1}{t}, \frac{1}{t}, −\frac{1}{t}) \end{matrix}

B

(5.5)  \{ \begin{matrix}(x_{1}, x_{2}, x_{3}) (t) = (\frac{1}{kt}, \frac{i}{kt}, \frac{i}{t}) \\ (x_{1}, x_{2}, x_{3}) (t) = (\frac{1}{kt}, −\frac{i}{kt}, −\frac{i}{t}) . \end{matrix}

Also, from the above citations, it results that systems (2.12) have two independent quadratic first integrals and then

H = \frac{1}{2} [σ_{1} (x_{1})^{2} + σ_{2} (x_{2})^{2} + σ_{3} (x_{3})^{2}]

as first integral if

(5.6)  σ_{1} (α_{2} + α_{3}) − σ_{2} (α_{3} + α_{1}) + σ_{3} (α_{1} + α_{2}) = 0

which for σ_{1} = 1 becomes

(α_{2} + α_{3}) − σ_{2} (α_{3} + α_{1}) + σ_{3} (α_{1} + α_{2}) = 0

with solutions

(5.7)  \{ \begin{matrix}σ_{2} = 0 ⇒ σ_{3} = −\frac{α_{2} + α_{3}}{α_{1} + α_{2}} \\ σ_{3} = 0 ⇒ σ_{2} = \frac{α_{2} + α_{3}}{α_{3} + α_{1}} . \end{matrix}

Then a Jordan-rigid body system has the first integrals

(5.8)  \{ \begin{matrix}H_{1} = \frac{1}{2} [(α_{3} + α_{1}) (x_{1})^{2} + (α_{2} + α_{3}) (x_{2})^{2}] \\ H_{2} = \frac{1}{2} [(α_{1} + α_{2}) (x_{1})^{2} − (α_{2} + α_{3}) (x_{3})^{2} ] \end{matrix}

which for our examples are ([6]):

A

(5.9)  H_{1} = \frac{1}{2} (x_{1}^{2} − x_{2}^{2}) , H_{2} = \frac{1}{2} (x_{1}^{2} − x_{3}^{2})

B

(5.10)  H_{1} = \frac{1}{2} (x_{1}^{2} + x_{2}^{2}) , H_{2} = \frac{1}{2} (k^{2}x_{1}^{2} + x_{3}^{2}) .

Unfortunately, these functions are not first integrals of associated Brockett perturbation. Indeed, from (2.16):

A

\begin{matrix}\frac{dH_{1}}{dt} = −2x_{1}^{2} (x_{1}^{2} − 2x_{2}^{2} + x_{3}^{2}) − 6x_{2}^{4} \\ \frac{dH_{2}}{dt} = 2 (x_{3}^{2} − x_{1}^{2}) (x_{1}^{2} − 2x_{2}^{2} + x_{3}^{2}) = −4H_{2} (x_{1}^{2} − 2x_{2}^{2} + x_{3}^{2}) \end{matrix}

and from (2.17)

\begin{aligned}\frac{dH_{1}}{dt} & = & (x_{1}^{2} + x_{2}^{2}) [−2k^{2}x_{1}^{2} − 2k^{2}x_{2}^{2} + (k^{2} + 3) x_{3}^{2}] = \\ & = & 2H_{1} [−4H_{1} + (k^{2} + 3) x_{3}^{2}] \end{aligned}

\begin{aligned}\frac{dH_{2}}{dt} & = & k^{2}x_{1}^{2} [−2k^{2}x_{1}^{2} − 2k^{2}x_{2}^{2} + (k^{2} + 3) x_{3}^{2}] + \\ & + & x_{3}^{2} [(1 − k^{2}) x_{1}^{2} + (1 − k^{2}) x_{2}^{2} + 2 (k^{2} + 2) x_{3}^{2}] . \end{aligned}

The first integrals (5.8) gives for x_{1} (t) the interpretation of inverse for an elliptic integral. So, from (5.8)

(5.11)  \{ \begin{matrix}x_{2}^{2} = \frac{1}{α_{2} + α_{3}} [2H_{1} − (α_{3} + α_{1}) x_{1}^{2}] \\ x_{3}^{2} = \frac{1}{α_{3} + α_{2}} [(α_{1} + α_{2}) x_{1}^{2} − 2H_{2}] \end{matrix}

and then, the first eq. (2.12) reads

\frac{dt}{dx_{1}} = \frac{α_{2} + α_{3}}{\sqrt{[2H_{1} − (α_{3} + α_{1}) x_{1}^{2}] [(α_{1} + α_{2}) x_{1}^{2} − 2H_{2}]}}

which yields

(5.12)  t = t (x_{1}) = ∫_{0}^{x_{1}} \frac{(α_{2} + α_{3}) du}{\sqrt{[2H_{1} − (α_{3} + α_{1}) u^{2}] [(α_{1} + α_{2}) u^{2} − 2H_{2}]}}

and also, the algebraic curve

(5.13)  (α_{2} + α_{3})^{2} y^{2} = [2H_{1} − (α_{3} + α_{1}) x^{2}] [(α_{1} + α_{2}) x^{2} − 2H_{2}] .

For the system B this computation appears in [7, p. 138].

Again associated with the first integrals (5.8) we obtain a numerical integrator which preserves these integrals. Recall that for an ODE system x^{.}_{i}= ϕ_{i} (x) the midpoint rule with step h is giving by ([1]):

\frac{x_{i}^{n+1} − x_{i}^{n}}{h} = ϕ_{i} (\frac{x^{n+1} + x^{n}}{2} ) .

For a Jordan-rigid body system this rule is

(5.14)  \{ \begin{matrix}x_{1}^{n+1} − x_{1}^{n} = −\frac{h}{4} (α_{2} + α_{3}) (x_{2}^{n+1} + x_{2}^{n}) (x_{3}^{n+1} + x_{3}^{n}) \\ x_{2}^{n+1} − x_{2}^{2} = \frac{h}{4} (α_{3} + α_{1}) (x_{3}^{n+1} + x_{3}^{n}) (x_{1}^{n+1} + x_{1}^{n}) \\ x_{3}^{n+1} − x_{3}^{n} = −\frac{h}{4} (α_{1} + α_{2}) (x_{1}^{n+1} + x_{1}^{n}) (x_{2}^{n+1} + x_{2}^{n}) \end{matrix}

and let us prove that preserves the first integrals (5.8). Indeed

H_{1}^{n+1} − H_{1}^{n} = \frac{1}{2} (α_{3} + α_{1}) [(x_{1}^{n+1})^{2} − (x_{1}^{n})^{2}] + \frac{1}{2} [(x_{2}^{n+1})^{2} − (x_{2}^{n})^{2}] =
= −\frac{h}{8} (α_{3} + α_{1}) (α_{2} + α_{3}) (x_{1}^{n+1} + x_{1}^{n}) (x_{2}^{n+1} + x_{2}^{n}) (x_{3}^{n+1} + x_{3}^{n}) +
+ \frac{h}{8} (α_{2} + α_{3}) (α_{3} + α_{1}) (x_{1}^{n+1} + x_{1}^{n}) (x_{2}^{n+1} + x_{2}^{n}) (x_{3}^{n+1} + x_{3}^{n}) = 0

H_{2}^{n+1} − H_{2}^{n} = \frac{1}{2} (α_{1} + α_{2}) [(x_{1}^{n+1})^{2} − (x_{1}^{n})^{2}] − \frac{1}{2} (α_{2} + α_{3}) [(x_{2}^{n+1})^{2} − (x_{2}^{n})^{2}] =
= −\frac{h}{8} (α_{1} + α_{2}) (α_{2} + α_{3}) (x_{1}^{n+1} + x_{1}^{n}) (x_{2}^{n+1} + x_{2}^{n}) (x_{3}^{n+1} + x_{3}^{n}) +
+ \frac{h}{8} (α_{2} + α_{3}) (α_{1} + α_{2}) (x_{1}^{n+1} + x_{1}^{n}) (x_{2}^{n+1} + x_{2}^{n}) (x_{3}^{n+1} + x_{3}^{n}) = 0.

Quadratic Homogeneous ODE Systems 39

6 The so (4)-version

In this section we sketch the so (4) version of Jordan-rigid body type systems. For

L = ( \begin{matrix}0 & x_{1} & x_{2} & x_{3} \\ −x_{1} & 0 & x_{4} & x_{5} \\ −x_{2} & −x_{4} & 0 & x_{6} \\ −x_{3} & −x_{5} & −x_{6} & 0 \end{matrix} )

and

B = B (α_{1}, …, α_{6}) = ( \begin{matrix}0 & α_{1}x_{1} & α_{2}x_{2} & α_{3}x_{3} \\ −α_{1}x_{1} & 0 & α_{4}x_{4} & α_{5}x_{5} \\ −α_{2}x_{2} & −α_{4}x_{4} & 0 & α_{6}x_{6} \\ −α_{3}x_{3} & −α_{5}x_{5} & −α_{6}x_{6} & 0 \end{matrix} )

one have, using Maple:

LB =
( \begin{matrix}−x_{1}^{2}α_{1} − x_{2}^{2}α_{2} − x_{3}^{2}α_{3} & −x_{2}α_{4}x_{4} − x_{3}α_{5}x_{5} & x_{1}α_{4}x_{4} − x_{3}α_{6}x_{6} & x_{1}α_{5}x_{5} + x_{2}α_{6}x_{6} \\ −x_{4}α_{2}x_{2} − x_{5}α_{3}x_{3} & −x_{1}^{2}α_{1} − x_{4}^{2}α_{4} − x_{5}^{2}α_{5} & −x_{1}α_{2}x_{2} − x_{5}α_{6}x_{6} & −x_{1}α_{3}x_{3} + x_{4}α_{6}x_{6} \\ x_{4}α_{1}x_{1} − x_{6}α_{3}x_{3} & −x_{2}α_{1}x_{1} − x_{6}α_{5}x_{5} & −x_{2}^{2}α_{2} − x_{4}^{2}α_{4} − x_{6}^{2}α_{6} & −x_{2}α_{3}x_{3} − x_{4}α_{5}x_{5} \\ x_{5}α_{1}x_{1} + x_{6}α_{2}x_{2} & −x_{3}α_{1}x_{1} + x_{6}α_{4}x_{4} & −x_{3}α_{2}x_{2} − x_{5}α_{4}x_{4} & −x_{3}^{2}α_{3} − x_{5}^{2}α_{5} − x_{6}^{2}α_{6} \end{matrix} )
BL =
( \begin{matrix}−x_{1}^{2}α_{1} − x_{2}^{2}α_{2} − x_{3}^{2}α_{3} & −x_{4}α_{2}x_{2} − x_{5}α_{3}x_{3} & x_{4}α_{1}x_{1} − x_{6}α_{3}x_{3} & x_{5}α_{1}x_{1} + x_{6}α_{2}x_{2} \\ −x_{2}α_{4}x_{4} − x_{3}α_{5}x_{5} & −x_{1}^{2}α_{1} − x_{4}^{2}α_{4} − x_{5}^{2}α_{5} & −x_{2}α_{1}x_{1} − x_{6}α_{5}x_{5} & −x_{3}α_{1}x_{1} + x_{6}α_{4}x_{4} \\ x_{1}α_{4}x_{4} − x_{3}α_{6}x_{6} & −x_{1}α_{2}x_{2} − x_{5}α_{6}x_{6} & −x_{2}^{2}α_{2} − x_{4}^{2}α_{4} − x_{6}^{2}α_{6} & −x_{3}α_{2}x_{2} − x_{5}α_{4}x_{4} \\ x_{1}α_{5}x_{5} + x_{2}α_{6}x_{6} & −x_{1}α_{3}x_{3} + x_{4}α_{6}x_{6} & −x_{2}α_{3}x_{3} − x_{4}α_{5}x_{5} & −x_{3}^{2}α_{3} − x_{5}^{2}α_{5} − x_{6}^{2}α_{6} \end{matrix} )

which means that a so (4) Jordan-rigid body system reads:

(6.1)  \{ \begin{matrix}\frac{dx_{1}}{dt} = − (α_{2} + α_{4}) x_{2}x_{4} − (α_{3} + α_{5}) x_{3}x_{5} \\ \frac{dx_{2}}{dt} = (α_{1} + α_{4}) x_{1}x_{4} − (α_{3} + α_{5}) x_{3}x_{6} \\ \frac{dx_{3}}{dt} = (α_{1} + α_{5}) x_{1}x_{5} + (α_{2} + α_{6}) x_{2}x_{6} \\ \frac{dx_{4}}{dt} = − (α_{1} + α_{2}) x_{1}x_{2} − (α_{5} + α_{6}) x_{5}x_{6} \\ \frac{dx_{5}}{dt} = − (α_{1} + α_{3}) x_{1}x_{3} + (α_{4} + α_{6}) x_{4}x_{6} \\ \frac{dx_{6}}{dt} = − (α_{2} + α_{3}) x_{2}x_{3} − (α_{4} + α_{5}) x_{4}x_{5}. \end{matrix}

We treat only the main subject of equations, namely first integrals. Searching for

H = \frac{1}{2} (ρ_{1}x_{1}^{2} + … + ρ_{6}x_{6}^{2})

it results the system

(6.2)  \{ \begin{matrix}ρ_{1} (α_{2} + α_{4}) − ρ_{2} (α_{1} + α_{4}) + ρ_{4} (α_{1} + α_{2}) = 0 \\ ρ_{1} (α_{3} + α_{5}) − ρ_{3} (α_{1} + α_{5}) + ρ_{5} (α_{1} + α_{3}) = 0 \\ ρ_{2} (α_{3} + α_{6}) − ρ_{3} (α_{2} + α_{6}) + ρ_{6} (α_{2} + α_{3}) = 0 \\ ρ_{4} (α_{4} + α_{6}) − ρ_{5} (α_{4} + α_{6}) + ρ_{6} (α_{4} + α_{5}) = 0. \end{matrix}

40 M. Crâşmăreanu

Appendix

Using Maple, for matrices (2.4) the following formulae hold:

L^{2} = ( \begin{matrix}−x_{2}^{2} − x_{3}^{2} & x_{1}x_{2} & x_{1}x_{3} \\ x_{1}x_{2} & −x_{3}^{2} − x_{1}^{2} & x_{2}x_{3} \\ x_{1}x_{3} & x_{2}x_{3} & −x_{1}^{2} − x_{2}^{2}, \end{matrix} )

L^{2}B + BL^{2} = ( \begin{matrix}0 & ω_{3}x_{3} & −ω_{2}x_{2} \\ −ω_{3}x_{3} & 0 & ω_{1}x_{1} \\ ω_{2}x_{2} & −ω_{1}x_{1} & 0 \end{matrix} )

where

\{ \begin{matrix}ω_{1} = 2α_{1}x_{1}^{2} + (α_{1} + α_{2}) x_{2}^{2} + (α_{1} + α_{3}) x_{3}^{2} \\ ω_{2} = (α_{1} + α_{2}) x_{1}^{2} + 2α_{2}x_{2}^{2} + (α_{2} + α_{3}) x_{3}^{2} \\ ω_{3} = (α_{1} + α_{3}) x_{1}^{2} + (α_{2} + α_{3}) x_{2}^{2} + 2α_{3}x_{3}^{2}, \end{matrix}

LBL = ( \begin{matrix}0 & ρx_{3} & −ρx_{2} \\ −ρx_{3} & 0 & ρx_{1} \\ ρx_{2} & −ρx_{1} & 0, \end{matrix} )

where

ρ = α_{1}x_{1}^{2} + α_{2}x_{2}^{2} + α_{3}x_{3}^{2}.

B^{2} = ( \begin{matrix}− (α_{2}^{2}x_{2}^{2} + α_{3}^{2}x_{3}^{2}) & α_{1}α_{2}x_{1}x_{2} & α_{1}α_{3}x_{1}x_{3} \\ α_{1}α_{2}x_{1}x_{2} & − (α_{3}^{2}x_{3}^{2} + α_{1}^{2}x_{1}^{2}) & α_{2}α_{3}x_{2}x_{3} \\ α_{1}α_{3}x_{1}x_{3} & α_{2}α_{3}x_{2}x_{3} & − (α_{1}^{2}x_{1}^{2} + α_{2}^{2}x_{2}^{2}) \end{matrix} ) .

The Maple plot of our systems:

A

B

for k^{2} = \frac{1}{2} :

The Kasner system

Acknowledgement. The author is indebted to Prof. Mihai Anastasiei for useful remarks.

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University ”Al.I. Cuza” of Iassy Faculty of Mathematics Iaşi 6600, România e-mail: mcrasm@uaic.ro