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Rational integrals of quasi-homogeneous dynamical systems
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ARTICLE IN PRESS
JAMM-2276; No. of Pages 8
Journal of Applied Mathematics and Mechanics xxx (2015) xxx–xxx
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Rational integrals of quasi-homogeneous dynamical systems夽 V.V. Kozlov Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia
a r t i c l e
i n f o
Article history: Received 27 December 2014
a b s t r a c t Dynamical systems, described by quasi-homogeneous systems of differential equations with polynomial right-hand sides, are considered. The Euler–Poisson equations from solid-state dynamics, as well as the Euler–Poincaré equations in Lie algebras, which describe the dynamics of systems in Lie groups with a left-invariant kinetic energy, can be pointed out as examples. The conditions for the existence of rational first integrals of quasi-homogeneous systems are found. They include the conditions for the existence of invariant algebraic manifolds. Examples of systems with rational integrals which do not admit of first integrals that are polynomial with respect to the momenta are presented. Results of a general nature are also demonstrated in the example of a Hess–Appel’rot invariant manifold from the dynamics of an asymmetric heavy top. © 2015 Elsevier Ltd. All rights reserved.
1. Quasi-homogeneous systems of differential equations Differential equations in Rn = {x1 ,..., xn } (1.1) which are invariant under the similarity transformations (1.2) with several real g1 ,..., gn , are often encountered in applications. The criteria for the invariance of Eqs (1.1) are satisfied when the following relations hold: (1.3) Systems (1.1) with such a property are called quasi-homogeneous, and the numbers g1 ,..., gn are called quasi-homogeneity indices. Here are several examples. Example 1. If i are homogeneous polynomials of order m > 1, we can set g1 = ... = gn = g in formulae (1.2). Then, however, due to relations (1.3), g = (m – 1)−1 . An important special case is that of the Euler–Poincaré equations in an n-dimensional Lie algebra:
(1.4)
夽 Prikl. Mat. Mekh. Vol. 79, No. 3, pp. 307–316, 2015. E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001 0021-8928/© 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Kozlov VV. Rational integrals of quasi-homogeneous dynamical systems. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001
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The quasi-velocities 1 ,..., n (the coordinates in the Lie algebra) and the torques m1 ,..., mn are related by the linear relations
where ||Isp || is a positive-definite (n × n) matrix, i.e., the tensor of inertia of the system. The parameters ck ij = −ck ji are structural constants of the Lie algebra. They satisfy the well-known Jacobi identity. Relations (1.4) written in the variables are equations in the Lie algebra, and the relations written in the variables m are equations in a dual linear space. In both cases system (1.4) will be homogeneous with homogeneity index g = 1. Example 2. point:
Another example is provided by the Euler–Poisson equations, which describe the rotation of a heavy rigid body with a fixed
(1.5) Here p, q and r are components of the angular velocity of the body about the principal axes of inertia, A, B and C are the moments of inertia about these axes, ␥1 , ␥2 and ␥3 are the direction cosines of a vertical in the movable reference system, and , and are the products of the mass of the body and the components of the centre of mass of the body about its principal axes of inertia. The equations that are not written out in (1.5) are obtained by cyclic permutation of the independent variables and the constant parameters. System of differential equations (1.5) is quasi-homogeneous: the quasi-homogeneity indices for the variables p, q and r are equal to 1, and those for the variables ␥1 , ␥2 and ␥3 are equal to 2. This circumstance was had already been noted and utilized by Lyapunov when he studied the branching of the solutions of system (1.5) in the complex time plane.1 Example 3. The equations of motion of the problem of n gravitating bodies in natural Cartesian coordinates are also quasi-homogeneous. The quasi-homogeneity indices of the coordinates of the attracted bodies are equal to −2/3, and those of the momenta are equal to 1/3. Due to the quasi-homogeneity property, system (1.1) admits of the particular solutions
where the constants c1 ,..., cn satisfy the algebraic system of equations (1.6) These equations generally have non-zero complex roots. The points in the phase space c = (c1 ,..., cn ) are significant for the further analysis. We introduce the Kovalevskaya matrix K = ||Kij || with the elements
(1.7) where ␦ij is the Kronecker delta (in formula (1.7) there is no summation over j). Its eigenvalues 1 ,..., n are called Kovalevskaya exponents. A Kovalevskaya matrix was introduced in Yoshida’s paper,2 although it was used long before by Lyapunov (in a somewhat different form) in rigid body dynamics.1 The function f: Rn → R is called a quasi-homogeneous function of degree s, if
for all x ∈ Rn and ␣ > 0. Theorem 1.
The following conclusions are valid:
/ 0, = s is a Kovalevskaya exponent; 1) if f is a quasi-homogeneous integral of degree 2 of Eqs (1.1) and df(c) = 2) if system (1.1) admits of the integral invariant
and (c) = / 0, then 1 + ... + m = n; 3) if c = / 0, then = −1 is a Kovalevskaya exponent. Of course, it is assumed here that in the case of the complex values c1 ,..., cn the functions f and are extended to differentiable functions in the vicinity of the point x = c. Conclusion 1 is the well-known Yoshida theorem.2 Conclusions 2 and 3 (and their generalizations) were established3 when the problem of the tensor invariants of arbitrary structure of quasi-homogeneous system (1.1) was considered. As a simple example we will consider the Euler differential equations, which describe the free rotation of an asymmetric rigid body (the Euler–Poincaré equations in the SO(3) algebra). They are obtained from the first group of Eqs (1.5) if we set = = = 0. The constants c1 , c2 and c3 satisfy the algebraic system of equations
Please cite this article in press as: Kozlov VV. Rational integrals of quasi-homogeneous dynamical systems. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001
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where
Obviously, (1.8) It is easy to calculate the Kovalevskaya exponents. Regardless of the choice of signs in equality (1.8), the characteristic equation det||K – E|| = 0 has the trivial root = −1 and the two-fold root 2. The latter is attributed to the existence of two independent quadratic integrals: of the energy and of the square of the absolute value of the angular momentum. Finally, the sum of the Kovalevskaya exponents is equal to 3, which corresponds to preservation of the standard measure in R3 = {p, q, r}. 2. Rational integrals of the Euler–Poincaré equations in three-dimensional solvable Lie algebras First, we will make several general remarks regarding the properties of Euler–Poincaré equations. As it turns out, for n = 3 these equations can always be integrated in quadratures when one condition, which will be formulated below, holds. In order to demonstrate this, the Euler–Poincaré equations in the algebra must be expanded by adding the kinematic equations on the corresponding group (for example, for the SO(3) group, the well-known kinematic Euler equations should be added). As a result, we obtain an autonomous system in sixdimensional phase space, which has four first integrals. They are the energy integral and three more Noether integrals, which are linear in the velocities. The latter are generated by three linearly independent right-invariant vector poles in the Lie group. The fluxes of these fields are left-hand shifts, which presumably maintain the kinetic energy, i.e., a left-invariant Riemann metric on the group. We will assume that the energy integral is not expressed in terms of the three Noether integrals. For example, for the SO(3) group this is certainly so if the inertia tensor is not spherical. Since the phase flow of the expanded system maintains a natural measure in six-dimensional phase space (Liouville’s theorem), under the assumption indicated above, the integrability of the complete system (and, therefore, of a truncated system of Euler–Poincaré equations) follows from the classical Euler–Jacobi last-multiplier theorem. It should be stressed that, unlike the complete system, the truncated system of Euler–Poincaré equations does not always admit of an invariant measure with a smooth positive density. The condition for such a measure to exist does not depend on the tensor of inertia of the system and has the form
(2.1) As we know, such groups are called unimodular groups (for further details, see Ref. 4). When n = 3, conditions (2.1) may not be satisfied only for solvable algebras. The latter can be described using the following commutation relations: (2.2) where e1 , e2 , e3 is the basis of independent left-invariant fields on the group, and the matrix
(2.3) is non-degenerate. The structural constants ck ij = − ck ji , which appear in Eqs (1.4), are expressed in terms of elements of matrix (2.3) as follows:
The remaining constants are equal to zero. As a result, the Euler–Poincaré equations in the three-dimensional solvable Lie algebra take the form
(2.4) Unimodularity conditions (2.1) hold only in the case when the trace of matrix (2.3) is equal to zero: ␣ + ␦ = 0. Equations (2.4) admit of the energy integral
This is a positive-definite quadratic form in R3 = {m}. In addition, the first integrals of the linear system (2.5) will clearly be integrals of the original Euler–Poincaré system (2.4). Please cite this article in press as: Kozlov VV. Rational integrals of quasi-homogeneous dynamical systems. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001
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Consider the special case when matrix (2.3) is diagonal:  = ␥ = 0 (the general case when matrix (2.3) has unequal real eigenvalues reduces to this form). Then Eqs (2.4) admit of the first integral
(2.6) If the ratio ␣/␦ is irrational, function (2.6) is not algebraic. This simple remark contrasts with the elegant results of Bruns in the threebody problem and of Husson in the dynamics of a heavy top regarding the absence of new algebraic integrals, which have little bearing on the real dynamics. Note that non-algebraic first integrals are often encountered in systems with friction (see, for example, Ref. 5, Chapter XIV, and Ref. 6). However, in the case under consideration there is no dissipation of energy. We now set ␣ = ␦ = 1 ( = ␥ = 0). Then integral (2.6) becomes the rational function F = m1 /m2 . Remark.
Equation (2.4) will have an entire set of the rational integrals
where A, B, C and D are real numbers and AD – BC = / 0. However, all of them can be written as fractional rational expressions in terms of F. Since in the case under consideration the equilibrium position of system (2.5) is a singular node, Eqs (2.4) do not admit of a smooth first integral that is independent of the energy integral. In particular, there are no new integrals that are polynomial in the velocities. All these facts can be examined from the point of view of the Kovalevskaya–Lyapunov method. First, we must find non-trivial solutions of algebraic system (1.6), which takes the following form in the case under consideration: (2.7) The constants c and c’ are related by the linear relations
(2.8) Since non-trivial solutions are sought, at least one of the numbers c1 or c2 is non-zero (otherwise, from the last equation of system (2.7) we have c3 = 0). However, then c 3 = 1, and c1 and c2 have arbitrary values (but |c1 | + |c2 | = / 0). The last equation of (2.7) can be represented in the form
Clearly, it has a whole set of complex solutions. From formulae (2.8) we obtain the set of solutions c = (c1 , c2 , c3 ) of original algebraic system (2.7), in which c1 or c2 is non-zero. For each solution c = / 0 we can calculate the Kovalevskaya matrix K and compute its eigenvalues. They are the numbers −1, 2 and 0. / 0, = −1 is a Kovalevskaya exponent. Equations (2.4) Alternatively, this result can be obtained at once using Theorem 1. In fact, since c = admit of an energy integral in the form of a non-degenerate quadratic form. Therefore, according to Conclusion 1 of the theorem, = 2 will also be one of the Kovalevskaya exponents. Finally, Eqs (2.4) still have the rational integral F = m1 /m2 with a zero degree of homogeneity. It remained to be shown that dF(c) = / 0. In fact, if c2 = / 0, then
If c2 = 0 (and c1 = / 0), the rational integral F−1 should be taken instead of F. Note that the sum of the Kovalevskaya exponents is equal to 1 (rather than 3, as in the case when an invariant measure with a smooth positive density exists). In addition, the result that there is no invariant for Eqs (2.4) follows from more general results,4 since in the case under consideration, the solvable Lie group will not be unimodular (since ␣ + ␦ = 2 = / 0). 3. Conditions for the existence of rational integrals From now on we will assume that the right-hand sides of quasi-homogeneous differential equations (1.1) are polynomials in x1 ,..., xn . The system consisting of Examples 1 and 2 (Section 1) satisfies this condition, but the three-body problem (Example 3) does not. By virtue of the condition of polynomiality of the right-hand sides of system (1.1), it is natural to assume that the quasi-homogeneity exponents g1 ,..., gn are positive rational numbers. Consider the problem of the existence of integrals of system (1.1) in the form of the rational function
(3.1) where P and Q are quasi-homogeneous polynomials of x1 ,..., xn of degrees r and s, respectively, with constant real coefficients. Fraction (3.1) is assumed to be irreducible: the polynomials P and Q do not have a polynomial common denominator of degree ≥1. Theorem 2.
Rational function (3.1) is a first integral of system (1.1) if and only if there is a polynomial R(x), such that (3.2)
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For example, for Euler–Poincaré equations (2.4) with unit matrix (2.3) P = m1 , Q = m2 , and the role of polynomial R is played by the variable 3 , which is expressed linearly in terms of the momenta m1 , m2 and m3 . It is not difficult to prove (using equalities (1.3)) that the polynomial R will be a quasi-homogeneous function with a degree of quasi-homogeneity equal to 1 (which should not be confused with the ordinary degree of a polynomial). The sufficiency of condition (3.2) is obvious. We will prove its necessity. It is clear that F˙ = 0 if and only if (3.3) By virtue of the assumption that the right-hand sides of system (1.1) are polynomials, the total derivatives P˙ and Q˙ are polynomials of x1 ,..., xn . The left-hand side of equality (3.3) is divided evenly by P. On the other hand, the polynomials P and Q do not have non-trivial polynomial common divisors. Therefore (according to the theorem of the uniqueness of the expansion of a polynomial into irreducible factors), P˙ is divisible evenly by P: P˙ = PR1 . Similarly, Q˙ = QR2 . Substituting these expressions into equality (3.3), we obtain
Since PQ ≡ / 0 and there are no divisors of zero in a ring of polynomials, R1 = R2 , which it was required to prove. An important corollary follows from Theorem 2. Corollary.
The singularity sets {P(x) = 0} and {Q(x) = 0} of the rational integrals F and F−1 are invariant sets for system (1.1).
We will first investigate the question of what restrictions the existence of one invariant algebraic manifold imposes on a Kovalevskaya matrix. Theorem 3. Proof.
Suppose P˙ = PR, P(c) = 0, and dP(c) = / 0. Then r + R(c) is the Kovalevskaya exponent.
According to Euler’s formula for quasi-homogeneous functions,
(3.4) On the other hand,
(3.5) Differentiating relation (3.5) successively with respect to x1 ,..., xn , we obtain n equations, which can be represented in the following matrix form:
(3.6) where w is a column vector with components ∂P/∂x1 ,..., ∂P/∂xn , and the column vector u has the components ∂R/∂x1 ,..., ∂R/∂xn . Working similarly with Eq. (3.4), we obtain
(3.7) We now set x = c and take into account relations (1.6): (c) = −Gc. Summing Eqs (3.6) and (3.7) and recalling that P(c) = 0, we arrive at the equality
(3.8) According to the assumption
Therefore, r + R(c) is an eigenvalue of the matrix KT (that is, of the Kovalevskaya matrix K). The theorem is proved. If P is a first integral (R 0), Theorem 3 transforms into Yoshida’s theorem.2 Remark. The non-zero eigenvector w from equality (3.8) lies in the hyperplane (Gc, w) = 0. In order to prove this, we must set x = c in equality (3.4) and take into account that P(c) = 0 according to the assumption. In equalities (3.4) and (3.5) we set x = c, use relations (1.6) and sum these equalities. As a result, we obtain
If r + R(c) = / 0, the condition of Theorem 3 P(c) = 0 certainly holds. The following assertion regarding the spectrum of the Kovalevskaya matrix of a quasi-homogeneous system, which admits of a rational integral of form (3.1), is derived from Theorems 1–3. Please cite this article in press as: Kozlov VV. Rational integrals of quasi-homogeneous dynamical systems. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001
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Theorem 4. If Q(c) = / 0 (P(c) = / 0, respectively) and dF(c) = / 0 (dF−1 (c) = / 0), the Kovalevskaya matrix has the eigenvalue r – s (s – r, respectively). If P(c) = Q(c) = 0 and the covectors dP(c) and dQ(c) are linearly independent, the Kovalevskaya exponents include the numbers
Thus, if a quasi-homogeneous system of differential equations with polynomial right-hand sides has a rational first integral F, in a typical case either the Kovalevskaya exponents include the quasi-homogeneous degree of F, or two Kovalevskaya exponents, the difference between which is equal to the quasi-homogeneous degree of F, can be found. As an illustrative example we will consider a homogeneous system in a plane of the following form: (3.9) We will assume that a = / 0 and d = / 0. Equations (1.6) admit of two non-trivial solutions: (3.10) and (3.11) For solution (3.10) the Kovalevskaya matrix has the form
Its eigenvalues are (3.12) The presence of the eigenvalue = −1 follows from Conclusion 3 of Theorem 1. On the other hand, the other eigenvalue 2 can be obtained according to Theorem 3. In fact, the second equation of (3.9) has the form P˙ = PR, where P = x2 and R = cx1 + dx2 . According to solution (3.10), P(c) = 0, and R(c) = −c/a. Since deg P = 1 and dP(c) = / 0, 1 – c/a is one of the Kovalevskaya exponents. Similarly, solution (3.11) corresponds to a Kovalevskaya matrix with the eigenvalues (3.13) We will examine the values of the parameters a, b, c and d for which the second Kovalevskaya exponents in (3.12) and (3.13) are equal to the same integer k. Then
and in this case the original system (3.9) admits of the homogeneous first integral (3.14) of degree k. When k ≤ 1, the function F will be a rational function, and when k ≥ 2, it will be a homogeneous polynomial. The existence of integral (3.14) is suggested by Theorem 4. Actually, it has only the necessary conditions for the existence of “typical” rational integrals. Remarks. 1. Equations (3.9) also admit of a rational first integral F−1 of degree –k. However, the Kovalevskaya exponents do not include the number –k, since the denominator of the fraction F−1 vanishes at points (3.10) and (3.11), and the numerator is non-zero. Therefore, Theorem 4 is not applicable here. 2. When k = 0, integral (3.14) reduces to the rational integral x1 /x2 . Here c = a, b = d, and the existence of such a rational integral follows at once from Theorem 2. 3. Function (3.14) will be a first integral for all real values of k. If k is rational, when it is raised to the appropriate degree, integral (3.14) is transformed into a rational function. For irrational values of k, the function F will be transcendental. 4. The Hess–Appel’rot invariant relation and Kovalevskaya exponents If the parameters of a rigid body are related by the conditions (4.1) Euler–Poisson equations (1.5) admit of the invariant relation P˙ = PR, where (4.2) Please cite this article in press as: Kozlov VV. Rational integrals of quasi-homogeneous dynamical systems. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001
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Here k is equal to
By virtue of the second condition of (4.1), these quantities are identical (for the details and history of this question, see Refs 7 and 8). The Hess–Appel’rot case played an important role in the problem of the conditions for uniqueness of the Euler–Poisson equations as functions of complex time. We will examine this case from the point of view of Kovalevskaya exponents (Theorem 3). A solution of algebraic system (1.6) must first be chosen under the assumption that the rigid body is dynamically asymmetrical. Following Lyapunov,1 we set (4.3) The constants cj are defined by formulae (1.8), in which the minus sign was chosen, and (4.4) Although the elements of the 6 × 6 Kovalevskaya matrix depend on the parameters of the rigid body, its characteristic polynomial does not depend on them; it is equal to (4.5) Remark. It should be borne in mind that the spectral parameter k in Lyapunov’s paper (Ref. 1) differs from the Kovalevskaya exponent ; they are related by the simple relation k = − 1. In fact, Lyapunov’s variations of the particular solutions of Eqs (1.1)
differ from the variations
which were used by Yoshida (and Kovalevskaya long before). All the roots of characteristic polynomial (4.5) are integers. The existence of the exponent = −1 corresponds to Conclusion 3 of Theorem 1. The value = 3 corresponds to the integral of the torque
which has the third degree of quasi-homogeneity. The root = 2 corresponds to the energy integral
which has a degree of quasi-homogeneity equal to 2. It should be noted that the roots of characteristic equation (4.5) do not include = 4, which is equal to the degree of quasi-homogeneity of the geometric integral ␥1 2 + ␥2 2 + ␥3 2 , since the point (4.3), (4.4) in phase space will be stationary for it. Characteristic equation (4.5) has the threefold root = 2. If conditions (4.1) hold, this root has simple elementary divisors: all the minors of the matrix K – 2E up to an order of ≤2 vanish.1 In particular, the matrix KT certainly has one more eigenvector with the eigenvalue = 2, which is linearly independent with the gradient of the energy integral at the point (4.3), (4.4) in phase space. Theorem 3 enables us to explain this puzzling property of the Hess–Appel’rot case. We turn to the relation P˙ = PR, in which P and R are given by formulae (4.2). At point (4.3) the linear form P vanishes according to conditions (4.1). Then,
when these conditions are taken into account. Since the polynomial P does not depend on the direction cosines, its degree of quasihomogeneity r is equal to 1. Therefore, according to Theorem 3, among the Kovalevskaya exponents there is one more number = 2, the / 0, the derivative of differentials of the energy integral and the function P being linearly independent at the point (4.3), (4.4) (since c2 = the energy with respect to q is non-zero, and P is totally independent of q). Acknowledgement The research was financed by the Russian Scientific Foundation (Project 14-50-00005). References 1. Lyapunov AM. A property of the differential equations of the problem of the motion of a heavy rigid body that has a fixed point. Soobshch Khar’k Mat Obsheh Ser 2 1894;4(3):123–40. 2. Yoshida H. Necessary condition for the existence of algebraic first integrals. Celest Mech 1983;31:363–99. 3. Kozlov VV. Tensor invariants of quasi-homogeneous systems of differential equations and the Kovalevskaya–Lyapunov asymptotic method. Mat Zametki 1992;51(2):46–52.
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Kozlov VV. Invariant measures of Euler–Poincaré equations in Lie algebras. Funkts Anal Prilozh 1988;22(1):69–70. Whittaker ET. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies with an Introduction to the Problem of Three Bodies. Cambridge: University Press; 1927. Trofimov VV, Shamolin MV. Geometrical and dynamical invariants of integral Hamiltonian and dissipative systems. Fund Prikl Mat 2010;16(4):3–229. Golubev VV. Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point. Jerusalem: Israel Program for Scientific Translations; 1960. Borisov AV, Mamayev IS. Rigid Body Dynamics. Hamiltonian Methods, Integrability, Chaos. Moscow; Izhevsk: Inst Komp’yut Issled; 2005.
Translated by P.S.
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Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Rational integrals of quasi-homogeneous dynamical systems夽 V.V. Kozlov Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia
a r t i c l e
i n f o
Article history: Received 27 December 2014
a b s t r a c t Dynamical systems, described by quasi-homogeneous systems of differential equations with polynomial right-hand sides, are considered. The Euler–Poisson equations from solid-state dynamics, as well as the Euler–Poincaré equations in Lie algebras, which describe the dynamics of systems in Lie groups with a left-invariant kinetic energy, can be pointed out as examples. The conditions for the existence of rational first integrals of quasi-homogeneous systems are found. They include the conditions for the existence of invariant algebraic manifolds. Examples of systems with rational integrals which do not admit of first integrals that are polynomial with respect to the momenta are presented. Results of a general nature are also demonstrated in the example of a Hess–Appel’rot invariant manifold from the dynamics of an asymmetric heavy top. © 2015 Elsevier Ltd. All rights reserved.
1. Quasi-homogeneous systems of differential equations Differential equations in Rn = {x1 ,..., xn } (1.1) which are invariant under the similarity transformations (1.2) with several real g1 ,..., gn , are often encountered in applications. The criteria for the invariance of Eqs (1.1) are satisfied when the following relations hold: (1.3) Systems (1.1) with such a property are called quasi-homogeneous, and the numbers g1 ,..., gn are called quasi-homogeneity indices. Here are several examples. Example 1. If i are homogeneous polynomials of order m > 1, we can set g1 = ... = gn = g in formulae (1.2). Then, however, due to relations (1.3), g = (m – 1)−1 . An important special case is that of the Euler–Poincaré equations in an n-dimensional Lie algebra:
(1.4)
夽 Prikl. Mat. Mekh. Vol. 79, No. 3, pp. 307–316, 2015. E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001 0021-8928/© 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Kozlov VV. Rational integrals of quasi-homogeneous dynamical systems. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001
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The quasi-velocities 1 ,..., n (the coordinates in the Lie algebra) and the torques m1 ,..., mn are related by the linear relations
where ||Isp || is a positive-definite (n × n) matrix, i.e., the tensor of inertia of the system. The parameters ck ij = −ck ji are structural constants of the Lie algebra. They satisfy the well-known Jacobi identity. Relations (1.4) written in the variables are equations in the Lie algebra, and the relations written in the variables m are equations in a dual linear space. In both cases system (1.4) will be homogeneous with homogeneity index g = 1. Example 2. point:
Another example is provided by the Euler–Poisson equations, which describe the rotation of a heavy rigid body with a fixed
(1.5) Here p, q and r are components of the angular velocity of the body about the principal axes of inertia, A, B and C are the moments of inertia about these axes, ␥1 , ␥2 and ␥3 are the direction cosines of a vertical in the movable reference system, and , and are the products of the mass of the body and the components of the centre of mass of the body about its principal axes of inertia. The equations that are not written out in (1.5) are obtained by cyclic permutation of the independent variables and the constant parameters. System of differential equations (1.5) is quasi-homogeneous: the quasi-homogeneity indices for the variables p, q and r are equal to 1, and those for the variables ␥1 , ␥2 and ␥3 are equal to 2. This circumstance was had already been noted and utilized by Lyapunov when he studied the branching of the solutions of system (1.5) in the complex time plane.1 Example 3. The equations of motion of the problem of n gravitating bodies in natural Cartesian coordinates are also quasi-homogeneous. The quasi-homogeneity indices of the coordinates of the attracted bodies are equal to −2/3, and those of the momenta are equal to 1/3. Due to the quasi-homogeneity property, system (1.1) admits of the particular solutions
where the constants c1 ,..., cn satisfy the algebraic system of equations (1.6) These equations generally have non-zero complex roots. The points in the phase space c = (c1 ,..., cn ) are significant for the further analysis. We introduce the Kovalevskaya matrix K = ||Kij || with the elements
(1.7) where ␦ij is the Kronecker delta (in formula (1.7) there is no summation over j). Its eigenvalues 1 ,..., n are called Kovalevskaya exponents. A Kovalevskaya matrix was introduced in Yoshida’s paper,2 although it was used long before by Lyapunov (in a somewhat different form) in rigid body dynamics.1 The function f: Rn → R is called a quasi-homogeneous function of degree s, if
for all x ∈ Rn and ␣ > 0. Theorem 1.
The following conclusions are valid:
/ 0, = s is a Kovalevskaya exponent; 1) if f is a quasi-homogeneous integral of degree 2 of Eqs (1.1) and df(c) = 2) if system (1.1) admits of the integral invariant
and (c) = / 0, then 1 + ... + m = n; 3) if c = / 0, then = −1 is a Kovalevskaya exponent. Of course, it is assumed here that in the case of the complex values c1 ,..., cn the functions f and are extended to differentiable functions in the vicinity of the point x = c. Conclusion 1 is the well-known Yoshida theorem.2 Conclusions 2 and 3 (and their generalizations) were established3 when the problem of the tensor invariants of arbitrary structure of quasi-homogeneous system (1.1) was considered. As a simple example we will consider the Euler differential equations, which describe the free rotation of an asymmetric rigid body (the Euler–Poincaré equations in the SO(3) algebra). They are obtained from the first group of Eqs (1.5) if we set = = = 0. The constants c1 , c2 and c3 satisfy the algebraic system of equations
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where
Obviously, (1.8) It is easy to calculate the Kovalevskaya exponents. Regardless of the choice of signs in equality (1.8), the characteristic equation det||K – E|| = 0 has the trivial root = −1 and the two-fold root 2. The latter is attributed to the existence of two independent quadratic integrals: of the energy and of the square of the absolute value of the angular momentum. Finally, the sum of the Kovalevskaya exponents is equal to 3, which corresponds to preservation of the standard measure in R3 = {p, q, r}. 2. Rational integrals of the Euler–Poincaré equations in three-dimensional solvable Lie algebras First, we will make several general remarks regarding the properties of Euler–Poincaré equations. As it turns out, for n = 3 these equations can always be integrated in quadratures when one condition, which will be formulated below, holds. In order to demonstrate this, the Euler–Poincaré equations in the algebra must be expanded by adding the kinematic equations on the corresponding group (for example, for the SO(3) group, the well-known kinematic Euler equations should be added). As a result, we obtain an autonomous system in sixdimensional phase space, which has four first integrals. They are the energy integral and three more Noether integrals, which are linear in the velocities. The latter are generated by three linearly independent right-invariant vector poles in the Lie group. The fluxes of these fields are left-hand shifts, which presumably maintain the kinetic energy, i.e., a left-invariant Riemann metric on the group. We will assume that the energy integral is not expressed in terms of the three Noether integrals. For example, for the SO(3) group this is certainly so if the inertia tensor is not spherical. Since the phase flow of the expanded system maintains a natural measure in six-dimensional phase space (Liouville’s theorem), under the assumption indicated above, the integrability of the complete system (and, therefore, of a truncated system of Euler–Poincaré equations) follows from the classical Euler–Jacobi last-multiplier theorem. It should be stressed that, unlike the complete system, the truncated system of Euler–Poincaré equations does not always admit of an invariant measure with a smooth positive density. The condition for such a measure to exist does not depend on the tensor of inertia of the system and has the form
(2.1) As we know, such groups are called unimodular groups (for further details, see Ref. 4). When n = 3, conditions (2.1) may not be satisfied only for solvable algebras. The latter can be described using the following commutation relations: (2.2) where e1 , e2 , e3 is the basis of independent left-invariant fields on the group, and the matrix
(2.3) is non-degenerate. The structural constants ck ij = − ck ji , which appear in Eqs (1.4), are expressed in terms of elements of matrix (2.3) as follows:
The remaining constants are equal to zero. As a result, the Euler–Poincaré equations in the three-dimensional solvable Lie algebra take the form
(2.4) Unimodularity conditions (2.1) hold only in the case when the trace of matrix (2.3) is equal to zero: ␣ + ␦ = 0. Equations (2.4) admit of the energy integral
This is a positive-definite quadratic form in R3 = {m}. In addition, the first integrals of the linear system (2.5) will clearly be integrals of the original Euler–Poincaré system (2.4). Please cite this article in press as: Kozlov VV. Rational integrals of quasi-homogeneous dynamical systems. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001
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Consider the special case when matrix (2.3) is diagonal:  = ␥ = 0 (the general case when matrix (2.3) has unequal real eigenvalues reduces to this form). Then Eqs (2.4) admit of the first integral
(2.6) If the ratio ␣/␦ is irrational, function (2.6) is not algebraic. This simple remark contrasts with the elegant results of Bruns in the threebody problem and of Husson in the dynamics of a heavy top regarding the absence of new algebraic integrals, which have little bearing on the real dynamics. Note that non-algebraic first integrals are often encountered in systems with friction (see, for example, Ref. 5, Chapter XIV, and Ref. 6). However, in the case under consideration there is no dissipation of energy. We now set ␣ = ␦ = 1 ( = ␥ = 0). Then integral (2.6) becomes the rational function F = m1 /m2 . Remark.
Equation (2.4) will have an entire set of the rational integrals
where A, B, C and D are real numbers and AD – BC = / 0. However, all of them can be written as fractional rational expressions in terms of F. Since in the case under consideration the equilibrium position of system (2.5) is a singular node, Eqs (2.4) do not admit of a smooth first integral that is independent of the energy integral. In particular, there are no new integrals that are polynomial in the velocities. All these facts can be examined from the point of view of the Kovalevskaya–Lyapunov method. First, we must find non-trivial solutions of algebraic system (1.6), which takes the following form in the case under consideration: (2.7) The constants c and c’ are related by the linear relations
(2.8) Since non-trivial solutions are sought, at least one of the numbers c1 or c2 is non-zero (otherwise, from the last equation of system (2.7) we have c3 = 0). However, then c 3 = 1, and c1 and c2 have arbitrary values (but |c1 | + |c2 | = / 0). The last equation of (2.7) can be represented in the form
Clearly, it has a whole set of complex solutions. From formulae (2.8) we obtain the set of solutions c = (c1 , c2 , c3 ) of original algebraic system (2.7), in which c1 or c2 is non-zero. For each solution c = / 0 we can calculate the Kovalevskaya matrix K and compute its eigenvalues. They are the numbers −1, 2 and 0. / 0, = −1 is a Kovalevskaya exponent. Equations (2.4) Alternatively, this result can be obtained at once using Theorem 1. In fact, since c = admit of an energy integral in the form of a non-degenerate quadratic form. Therefore, according to Conclusion 1 of the theorem, = 2 will also be one of the Kovalevskaya exponents. Finally, Eqs (2.4) still have the rational integral F = m1 /m2 with a zero degree of homogeneity. It remained to be shown that dF(c) = / 0. In fact, if c2 = / 0, then
If c2 = 0 (and c1 = / 0), the rational integral F−1 should be taken instead of F. Note that the sum of the Kovalevskaya exponents is equal to 1 (rather than 3, as in the case when an invariant measure with a smooth positive density exists). In addition, the result that there is no invariant for Eqs (2.4) follows from more general results,4 since in the case under consideration, the solvable Lie group will not be unimodular (since ␣ + ␦ = 2 = / 0). 3. Conditions for the existence of rational integrals From now on we will assume that the right-hand sides of quasi-homogeneous differential equations (1.1) are polynomials in x1 ,..., xn . The system consisting of Examples 1 and 2 (Section 1) satisfies this condition, but the three-body problem (Example 3) does not. By virtue of the condition of polynomiality of the right-hand sides of system (1.1), it is natural to assume that the quasi-homogeneity exponents g1 ,..., gn are positive rational numbers. Consider the problem of the existence of integrals of system (1.1) in the form of the rational function
(3.1) where P and Q are quasi-homogeneous polynomials of x1 ,..., xn of degrees r and s, respectively, with constant real coefficients. Fraction (3.1) is assumed to be irreducible: the polynomials P and Q do not have a polynomial common denominator of degree ≥1. Theorem 2.
Rational function (3.1) is a first integral of system (1.1) if and only if there is a polynomial R(x), such that (3.2)
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For example, for Euler–Poincaré equations (2.4) with unit matrix (2.3) P = m1 , Q = m2 , and the role of polynomial R is played by the variable 3 , which is expressed linearly in terms of the momenta m1 , m2 and m3 . It is not difficult to prove (using equalities (1.3)) that the polynomial R will be a quasi-homogeneous function with a degree of quasi-homogeneity equal to 1 (which should not be confused with the ordinary degree of a polynomial). The sufficiency of condition (3.2) is obvious. We will prove its necessity. It is clear that F˙ = 0 if and only if (3.3) By virtue of the assumption that the right-hand sides of system (1.1) are polynomials, the total derivatives P˙ and Q˙ are polynomials of x1 ,..., xn . The left-hand side of equality (3.3) is divided evenly by P. On the other hand, the polynomials P and Q do not have non-trivial polynomial common divisors. Therefore (according to the theorem of the uniqueness of the expansion of a polynomial into irreducible factors), P˙ is divisible evenly by P: P˙ = PR1 . Similarly, Q˙ = QR2 . Substituting these expressions into equality (3.3), we obtain
Since PQ ≡ / 0 and there are no divisors of zero in a ring of polynomials, R1 = R2 , which it was required to prove. An important corollary follows from Theorem 2. Corollary.
The singularity sets {P(x) = 0} and {Q(x) = 0} of the rational integrals F and F−1 are invariant sets for system (1.1).
We will first investigate the question of what restrictions the existence of one invariant algebraic manifold imposes on a Kovalevskaya matrix. Theorem 3. Proof.
Suppose P˙ = PR, P(c) = 0, and dP(c) = / 0. Then r + R(c) is the Kovalevskaya exponent.
According to Euler’s formula for quasi-homogeneous functions,
(3.4) On the other hand,
(3.5) Differentiating relation (3.5) successively with respect to x1 ,..., xn , we obtain n equations, which can be represented in the following matrix form:
(3.6) where w is a column vector with components ∂P/∂x1 ,..., ∂P/∂xn , and the column vector u has the components ∂R/∂x1 ,..., ∂R/∂xn . Working similarly with Eq. (3.4), we obtain
(3.7) We now set x = c and take into account relations (1.6): (c) = −Gc. Summing Eqs (3.6) and (3.7) and recalling that P(c) = 0, we arrive at the equality
(3.8) According to the assumption
Therefore, r + R(c) is an eigenvalue of the matrix KT (that is, of the Kovalevskaya matrix K). The theorem is proved. If P is a first integral (R 0), Theorem 3 transforms into Yoshida’s theorem.2 Remark. The non-zero eigenvector w from equality (3.8) lies in the hyperplane (Gc, w) = 0. In order to prove this, we must set x = c in equality (3.4) and take into account that P(c) = 0 according to the assumption. In equalities (3.4) and (3.5) we set x = c, use relations (1.6) and sum these equalities. As a result, we obtain
If r + R(c) = / 0, the condition of Theorem 3 P(c) = 0 certainly holds. The following assertion regarding the spectrum of the Kovalevskaya matrix of a quasi-homogeneous system, which admits of a rational integral of form (3.1), is derived from Theorems 1–3. Please cite this article in press as: Kozlov VV. Rational integrals of quasi-homogeneous dynamical systems. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001
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Theorem 4. If Q(c) = / 0 (P(c) = / 0, respectively) and dF(c) = / 0 (dF−1 (c) = / 0), the Kovalevskaya matrix has the eigenvalue r – s (s – r, respectively). If P(c) = Q(c) = 0 and the covectors dP(c) and dQ(c) are linearly independent, the Kovalevskaya exponents include the numbers
Thus, if a quasi-homogeneous system of differential equations with polynomial right-hand sides has a rational first integral F, in a typical case either the Kovalevskaya exponents include the quasi-homogeneous degree of F, or two Kovalevskaya exponents, the difference between which is equal to the quasi-homogeneous degree of F, can be found. As an illustrative example we will consider a homogeneous system in a plane of the following form: (3.9) We will assume that a = / 0 and d = / 0. Equations (1.6) admit of two non-trivial solutions: (3.10) and (3.11) For solution (3.10) the Kovalevskaya matrix has the form
Its eigenvalues are (3.12) The presence of the eigenvalue = −1 follows from Conclusion 3 of Theorem 1. On the other hand, the other eigenvalue 2 can be obtained according to Theorem 3. In fact, the second equation of (3.9) has the form P˙ = PR, where P = x2 and R = cx1 + dx2 . According to solution (3.10), P(c) = 0, and R(c) = −c/a. Since deg P = 1 and dP(c) = / 0, 1 – c/a is one of the Kovalevskaya exponents. Similarly, solution (3.11) corresponds to a Kovalevskaya matrix with the eigenvalues (3.13) We will examine the values of the parameters a, b, c and d for which the second Kovalevskaya exponents in (3.12) and (3.13) are equal to the same integer k. Then
and in this case the original system (3.9) admits of the homogeneous first integral (3.14) of degree k. When k ≤ 1, the function F will be a rational function, and when k ≥ 2, it will be a homogeneous polynomial. The existence of integral (3.14) is suggested by Theorem 4. Actually, it has only the necessary conditions for the existence of “typical” rational integrals. Remarks. 1. Equations (3.9) also admit of a rational first integral F−1 of degree –k. However, the Kovalevskaya exponents do not include the number –k, since the denominator of the fraction F−1 vanishes at points (3.10) and (3.11), and the numerator is non-zero. Therefore, Theorem 4 is not applicable here. 2. When k = 0, integral (3.14) reduces to the rational integral x1 /x2 . Here c = a, b = d, and the existence of such a rational integral follows at once from Theorem 2. 3. Function (3.14) will be a first integral for all real values of k. If k is rational, when it is raised to the appropriate degree, integral (3.14) is transformed into a rational function. For irrational values of k, the function F will be transcendental. 4. The Hess–Appel’rot invariant relation and Kovalevskaya exponents If the parameters of a rigid body are related by the conditions (4.1) Euler–Poisson equations (1.5) admit of the invariant relation P˙ = PR, where (4.2) Please cite this article in press as: Kozlov VV. Rational integrals of quasi-homogeneous dynamical systems. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001
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Here k is equal to
By virtue of the second condition of (4.1), these quantities are identical (for the details and history of this question, see Refs 7 and 8). The Hess–Appel’rot case played an important role in the problem of the conditions for uniqueness of the Euler–Poisson equations as functions of complex time. We will examine this case from the point of view of Kovalevskaya exponents (Theorem 3). A solution of algebraic system (1.6) must first be chosen under the assumption that the rigid body is dynamically asymmetrical. Following Lyapunov,1 we set (4.3) The constants cj are defined by formulae (1.8), in which the minus sign was chosen, and (4.4) Although the elements of the 6 × 6 Kovalevskaya matrix depend on the parameters of the rigid body, its characteristic polynomial does not depend on them; it is equal to (4.5) Remark. It should be borne in mind that the spectral parameter k in Lyapunov’s paper (Ref. 1) differs from the Kovalevskaya exponent ; they are related by the simple relation k = − 1. In fact, Lyapunov’s variations of the particular solutions of Eqs (1.1)
differ from the variations
which were used by Yoshida (and Kovalevskaya long before). All the roots of characteristic polynomial (4.5) are integers. The existence of the exponent = −1 corresponds to Conclusion 3 of Theorem 1. The value = 3 corresponds to the integral of the torque
which has the third degree of quasi-homogeneity. The root = 2 corresponds to the energy integral
which has a degree of quasi-homogeneity equal to 2. It should be noted that the roots of characteristic equation (4.5) do not include = 4, which is equal to the degree of quasi-homogeneity of the geometric integral ␥1 2 + ␥2 2 + ␥3 2 , since the point (4.3), (4.4) in phase space will be stationary for it. Characteristic equation (4.5) has the threefold root = 2. If conditions (4.1) hold, this root has simple elementary divisors: all the minors of the matrix K – 2E up to an order of ≤2 vanish.1 In particular, the matrix KT certainly has one more eigenvector with the eigenvalue = 2, which is linearly independent with the gradient of the energy integral at the point (4.3), (4.4) in phase space. Theorem 3 enables us to explain this puzzling property of the Hess–Appel’rot case. We turn to the relation P˙ = PR, in which P and R are given by formulae (4.2). At point (4.3) the linear form P vanishes according to conditions (4.1). Then,
when these conditions are taken into account. Since the polynomial P does not depend on the direction cosines, its degree of quasihomogeneity r is equal to 1. Therefore, according to Theorem 3, among the Kovalevskaya exponents there is one more number = 2, the / 0, the derivative of differentials of the energy integral and the function P being linearly independent at the point (4.3), (4.4) (since c2 = the energy with respect to q is non-zero, and P is totally independent of q). Acknowledgement The research was financed by the Russian Scientific Foundation (Project 14-50-00005). References 1. Lyapunov AM. A property of the differential equations of the problem of the motion of a heavy rigid body that has a fixed point. Soobshch Khar’k Mat Obsheh Ser 2 1894;4(3):123–40. 2. Yoshida H. Necessary condition for the existence of algebraic first integrals. Celest Mech 1983;31:363–99. 3. Kozlov VV. Tensor invariants of quasi-homogeneous systems of differential equations and the Kovalevskaya–Lyapunov asymptotic method. Mat Zametki 1992;51(2):46–52.
Please cite this article in press as: Kozlov VV. Rational integrals of quasi-homogeneous dynamical systems. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001
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Kozlov VV. Invariant measures of Euler–Poincaré equations in Lie algebras. Funkts Anal Prilozh 1988;22(1):69–70. Whittaker ET. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies with an Introduction to the Problem of Three Bodies. Cambridge: University Press; 1927. Trofimov VV, Shamolin MV. Geometrical and dynamical invariants of integral Hamiltonian and dissipative systems. Fund Prikl Mat 2010;16(4):3–229. Golubev VV. Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point. Jerusalem: Israel Program for Scientific Translations; 1960. Borisov AV, Mamayev IS. Rigid Body Dynamics. Hamiltonian Methods, Integrability, Chaos. Moscow; Izhevsk: Inst Komp’yut Issled; 2005.
Translated by P.S.
Please cite this article in press as: Kozlov VV. Rational integrals of quasi-homogeneous dynamical systems. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.001