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Two new exactly solvable cases of the Euler–Poisson equations
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 30 (2003) 203–205 www.elsevier.com/locate/mechrescom
Two new exactly solvable cases of the Euler–Poisson equations Puyun Gao
*
School of Astronautical and Materials Engineering, National University of Defence Technology, Changsha, Hunan 410073, PR China Received 21 October 2001; received in revised form 19 November 2001
Abstract In this paper, the direct method is used to find the first integrals and two new solvable cases of the Euler–Poisson equations are given. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Euler–Poisson equations; First integrals; Direct method; Solvable case; Exact solution
1. Introduction HamiltonÕs equations describing the motion of a top spinning about a fixed point are a system of six nonlinear first-order coupled ordinary differential equations (called the Euler–Poisson equations) (Arkhangelsky, 1977; Fomenko, 1988) 8 dp > > > A dt þ ðC BÞqr ¼ mgðz0 r2 y0 r3 Þ > > > dq > > B þ ðA CÞpr ¼ mgðx0 r3 z0 r1 Þ > > dt > > > > < C dr þ ðB AÞpq ¼ mgðy0 r1 x0 r2 Þ dt ð1Þ > dr > 1 ¼ rr2 qr3 > > dt > > > > dr 2 ¼ pr rr > > 3 1 > dt > > > : dr3 ¼ qr1 pr2 dt where, with respect to a moving Cartesian coordinate system based on the principal axes of inertia with origin at its fixed point, ðp; q; rÞ are the components of angular velocity, ðr1 ; r2 ; r3 Þ the direction cosines of the direction of gravity, ðA; B; CÞ the moments of inertia, and ðx0 ; y0 ; z0 Þ the position of the center of mass of the system; also m is the mass of the top and g the acceleration due to gravity. *
Fax: +86-0731-4512301. E-mail address: [email protected] (P. Gao).
0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0093-6413(02)00367-1
204
P. Gao / Mechanics Research Communications 30 (2003) 203–205
This system always possessed three independent first integrals (Arkhangelsky, 1977; Fomenko, 1988) H1 ¼ Ap2 þ Bq2 þ Cr2 þ 2mgðx0 r1 þ y0 r2 þ z0 r3 Þ H2 ¼ Apr1 þ Bqr2 þ Crr3 H3 ¼ r12 þ r22 þ r32 According to JacobiÕs theorem, to reduce these equations to quadratures, it suffices to know only four first integral, that is, one should add to the three quadratic integrals just stated an additional fourth integral. However, the Euler–Poisson equations are not exactly solvable in the general case (Kozlov, 1975, 1983). Since Eq. (1) depend on six parameters ðA; B; C; x0 ; y0 ; z0 Þ, the question arises, how to find special values of the six parameters for which Eq. (1) have the fourth first integral which is functionally independent of H1 , H2 and H3 . The quest to determine a fourth integral was a popular problem for 18th and 19th century mechanicians and mathematicians. However, it was only solved in four cases (Fomenko, 1988): 1. 2. 3. 4.
A ¼ B ¼ C, with integral px0 þ qy0 þ rz0 ¼ K1 ; x0 ¼ y0 ¼ z0 ¼ 0 (due to Euler), with integral A2 p2 þ B2 q2 þ C 2 r2 ¼ K2 ; A ¼ B and x0 ¼ y0 ¼ 0 (due to Lagrange), with integral Cr ¼ K3 ; A ¼ B ¼ 2C and y0 ¼ z0 ¼ 0 (due to Kovalevskaya), with integral (Kovalevskaya, 1889) mgx0 2 mgx0 2 r1 þ 2pq r2 ¼ K4 p 2 q2 C C
with K1 , K2 , K3 , K4 constants. The Kovalevskaya case has become (like the Euler case) of increasing interest because its multidimensional analogues have been discovered (Perelomov, 1982). The main purpose of this paper is to extend KovalevskayaÕs work (Kovalevskaya, 1889). Two new exactly solvable cases of the Euler–Poisson equations are found.
2. Two new exactly solvable cases In this section, we shall given two new exactly solvable cases of the Euler–Poisson equations. Case 1. A ¼ B ¼ 2C and z0 ¼ 0. In this case, the inertia ellipsoid of the top is an ellipsoid of rotation and its center of gravity does not lie on any rotation axis if x0 y0 6¼ 0. The fourth first integral is given by 2
H4 ¼ C 2 ðp2 þ q2 Þ 2Cmg½ðp2 q2 Þðx0 r1 y0 r2 Þ þ 2pqðy0 r1 þ x0 r2 Þ þ m2 g2 ðx20 þ y02 Þðr12 þ r22 Þ It is clear that in this case, Eq. (1) can be rewritten as 8 dp > > > 2C dt Cqr ¼ mgy0 r3 > > > > dq > > 2C þ Cpr ¼ mgx0 r3 > dt > > > > < C dr ¼ mgðy0 r1 x0 r2 Þ dt dr > > 1 ¼ rr2 qr3 > > dt > > > dr > 2 ¼ pr rr > > 3 1 > dt > > > : dr3 ¼ qr pr 1 2 dt
ð2Þ
P. Gao / Mechanics Research Communications 30 (2003) 203–205
205
Write I1 ¼ C 2 ðp2 þ q2 Þ
2
I2 ¼ 2Cmg½ðp2 q2 Þðx0 r1 y0 r2 Þ þ 2pqðy0 r1 þ x0 r2 Þ I3 ¼ m2 g2 ðx20 þ y02 Þðr12 þ r22 Þ By Eq. (2), a straightforward computation shows that dI1 ¼ 2Cmgðp2 þ q2 Þðx0 q y0 pÞr3 dt dI2 ¼ 2Cmgðp2 þ q2 Þðx0 q y0 pÞr3 þ 2m2 g2 ðx20 þ y02 Þðpr2 qr1 Þr3 dt dI3 ¼ 2m2 g2 ðx20 þ y02 Þðpr2 qr1 Þr3 dt which imply dH4 d ¼ ðI1 I2 þ I3 Þ ¼ 0 dt dt It is not hard to verify that H4 is independent of H1 , H2 and H3 . Case 2. A ¼ B ¼ 2C and y0 ¼ 0. In this case, the fourth first integral is given by 2
H 4 ¼ C 2 ðp2 þ q2 Þ Cmg½2x0 ðp2 q2 Þr1 2z0 ðp2 þ q2 Þr3 þ z0 r2 r3 þ 4x0 pqr2 þ 2z0 prr1 þ 2z0 qrr2 þ m2 g2 ½ðx20 z20 Þðr12 þ r22 Þ þ 2x0 z0 r1 r3 Indeed, similarly to Case 1, a straightforward computation shows that H 4 is a first integral of Eq. (1) if the conditions A ¼ B ¼ 2C and y0 ¼ 0 hold. Cases 1 and 2 extend KovalevskayaÕs work.
3. Conclusions In the present work we find two new exactly solvable cases of the Euler–Poisson equations. It is well known that the general solution of Eq. (1) is expressed in the terms of hyperelliptic functions in the Kovalevskaya case. It is uncertain whether their general solution is expressed in the terms of hyperelliptic functions or not in Cases 1 and 2. This is an interesting open problem.
References Arkhangelsky, Y.A., 1977. Analytic Rigid-Body Dynamics. Nauka, Moscow. Fomenko, A.T., 1988. Integrability and Nonintegrability in Geometry and Mechanics. Kluwer Academic Publishers, Boston. Kozlov, V.V., 1975. Non-existence of an additional analytic integral in the problem of the motion of a non-symmetric heavy rigid body around a fixed point. Vestnik Mosk. Gos. Univers., Ser. Mat. Mekh. 1, 105–110. Kozlov, V.V., 1983. Integrability and nonintegrability in Hamiltonian mechanics. Uspekhi Mat. Nauk. 38, 3–67. Kovalevskaya, S., 1889. Sur le probleme de la rotation dÕun corps solid autour dÕun point fixe. Acta Math. 12, 177–232. Perelomov, A.M., 1982. Lax representation for systems of the type of S. Kovalevskaya. Funkts. Analiz i yego Prilozhen 16, 80–81.
Mechanics Research Communications 30 (2003) 203–205 www.elsevier.com/locate/mechrescom
Two new exactly solvable cases of the Euler–Poisson equations Puyun Gao
*
School of Astronautical and Materials Engineering, National University of Defence Technology, Changsha, Hunan 410073, PR China Received 21 October 2001; received in revised form 19 November 2001
Abstract In this paper, the direct method is used to find the first integrals and two new solvable cases of the Euler–Poisson equations are given. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Euler–Poisson equations; First integrals; Direct method; Solvable case; Exact solution
1. Introduction HamiltonÕs equations describing the motion of a top spinning about a fixed point are a system of six nonlinear first-order coupled ordinary differential equations (called the Euler–Poisson equations) (Arkhangelsky, 1977; Fomenko, 1988) 8 dp > > > A dt þ ðC BÞqr ¼ mgðz0 r2 y0 r3 Þ > > > dq > > B þ ðA CÞpr ¼ mgðx0 r3 z0 r1 Þ > > dt > > > > < C dr þ ðB AÞpq ¼ mgðy0 r1 x0 r2 Þ dt ð1Þ > dr > 1 ¼ rr2 qr3 > > dt > > > > dr 2 ¼ pr rr > > 3 1 > dt > > > : dr3 ¼ qr1 pr2 dt where, with respect to a moving Cartesian coordinate system based on the principal axes of inertia with origin at its fixed point, ðp; q; rÞ are the components of angular velocity, ðr1 ; r2 ; r3 Þ the direction cosines of the direction of gravity, ðA; B; CÞ the moments of inertia, and ðx0 ; y0 ; z0 Þ the position of the center of mass of the system; also m is the mass of the top and g the acceleration due to gravity. *
Fax: +86-0731-4512301. E-mail address: [email protected] (P. Gao).
0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0093-6413(02)00367-1
204
P. Gao / Mechanics Research Communications 30 (2003) 203–205
This system always possessed three independent first integrals (Arkhangelsky, 1977; Fomenko, 1988) H1 ¼ Ap2 þ Bq2 þ Cr2 þ 2mgðx0 r1 þ y0 r2 þ z0 r3 Þ H2 ¼ Apr1 þ Bqr2 þ Crr3 H3 ¼ r12 þ r22 þ r32 According to JacobiÕs theorem, to reduce these equations to quadratures, it suffices to know only four first integral, that is, one should add to the three quadratic integrals just stated an additional fourth integral. However, the Euler–Poisson equations are not exactly solvable in the general case (Kozlov, 1975, 1983). Since Eq. (1) depend on six parameters ðA; B; C; x0 ; y0 ; z0 Þ, the question arises, how to find special values of the six parameters for which Eq. (1) have the fourth first integral which is functionally independent of H1 , H2 and H3 . The quest to determine a fourth integral was a popular problem for 18th and 19th century mechanicians and mathematicians. However, it was only solved in four cases (Fomenko, 1988): 1. 2. 3. 4.
A ¼ B ¼ C, with integral px0 þ qy0 þ rz0 ¼ K1 ; x0 ¼ y0 ¼ z0 ¼ 0 (due to Euler), with integral A2 p2 þ B2 q2 þ C 2 r2 ¼ K2 ; A ¼ B and x0 ¼ y0 ¼ 0 (due to Lagrange), with integral Cr ¼ K3 ; A ¼ B ¼ 2C and y0 ¼ z0 ¼ 0 (due to Kovalevskaya), with integral (Kovalevskaya, 1889) mgx0 2 mgx0 2 r1 þ 2pq r2 ¼ K4 p 2 q2 C C
with K1 , K2 , K3 , K4 constants. The Kovalevskaya case has become (like the Euler case) of increasing interest because its multidimensional analogues have been discovered (Perelomov, 1982). The main purpose of this paper is to extend KovalevskayaÕs work (Kovalevskaya, 1889). Two new exactly solvable cases of the Euler–Poisson equations are found.
2. Two new exactly solvable cases In this section, we shall given two new exactly solvable cases of the Euler–Poisson equations. Case 1. A ¼ B ¼ 2C and z0 ¼ 0. In this case, the inertia ellipsoid of the top is an ellipsoid of rotation and its center of gravity does not lie on any rotation axis if x0 y0 6¼ 0. The fourth first integral is given by 2
H4 ¼ C 2 ðp2 þ q2 Þ 2Cmg½ðp2 q2 Þðx0 r1 y0 r2 Þ þ 2pqðy0 r1 þ x0 r2 Þ þ m2 g2 ðx20 þ y02 Þðr12 þ r22 Þ It is clear that in this case, Eq. (1) can be rewritten as 8 dp > > > 2C dt Cqr ¼ mgy0 r3 > > > > dq > > 2C þ Cpr ¼ mgx0 r3 > dt > > > > < C dr ¼ mgðy0 r1 x0 r2 Þ dt dr > > 1 ¼ rr2 qr3 > > dt > > > dr > 2 ¼ pr rr > > 3 1 > dt > > > : dr3 ¼ qr pr 1 2 dt
ð2Þ
P. Gao / Mechanics Research Communications 30 (2003) 203–205
205
Write I1 ¼ C 2 ðp2 þ q2 Þ
2
I2 ¼ 2Cmg½ðp2 q2 Þðx0 r1 y0 r2 Þ þ 2pqðy0 r1 þ x0 r2 Þ I3 ¼ m2 g2 ðx20 þ y02 Þðr12 þ r22 Þ By Eq. (2), a straightforward computation shows that dI1 ¼ 2Cmgðp2 þ q2 Þðx0 q y0 pÞr3 dt dI2 ¼ 2Cmgðp2 þ q2 Þðx0 q y0 pÞr3 þ 2m2 g2 ðx20 þ y02 Þðpr2 qr1 Þr3 dt dI3 ¼ 2m2 g2 ðx20 þ y02 Þðpr2 qr1 Þr3 dt which imply dH4 d ¼ ðI1 I2 þ I3 Þ ¼ 0 dt dt It is not hard to verify that H4 is independent of H1 , H2 and H3 . Case 2. A ¼ B ¼ 2C and y0 ¼ 0. In this case, the fourth first integral is given by 2
H 4 ¼ C 2 ðp2 þ q2 Þ Cmg½2x0 ðp2 q2 Þr1 2z0 ðp2 þ q2 Þr3 þ z0 r2 r3 þ 4x0 pqr2 þ 2z0 prr1 þ 2z0 qrr2 þ m2 g2 ½ðx20 z20 Þðr12 þ r22 Þ þ 2x0 z0 r1 r3 Indeed, similarly to Case 1, a straightforward computation shows that H 4 is a first integral of Eq. (1) if the conditions A ¼ B ¼ 2C and y0 ¼ 0 hold. Cases 1 and 2 extend KovalevskayaÕs work.
3. Conclusions In the present work we find two new exactly solvable cases of the Euler–Poisson equations. It is well known that the general solution of Eq. (1) is expressed in the terms of hyperelliptic functions in the Kovalevskaya case. It is uncertain whether their general solution is expressed in the terms of hyperelliptic functions or not in Cases 1 and 2. This is an interesting open problem.
References Arkhangelsky, Y.A., 1977. Analytic Rigid-Body Dynamics. Nauka, Moscow. Fomenko, A.T., 1988. Integrability and Nonintegrability in Geometry and Mechanics. Kluwer Academic Publishers, Boston. Kozlov, V.V., 1975. Non-existence of an additional analytic integral in the problem of the motion of a non-symmetric heavy rigid body around a fixed point. Vestnik Mosk. Gos. Univers., Ser. Mat. Mekh. 1, 105–110. Kozlov, V.V., 1983. Integrability and nonintegrability in Hamiltonian mechanics. Uspekhi Mat. Nauk. 38, 3–67. Kovalevskaya, S., 1889. Sur le probleme de la rotation dÕun corps solid autour dÕun point fixe. Acta Math. 12, 177–232. Perelomov, A.M., 1982. Lax representation for systems of the type of S. Kovalevskaya. Funkts. Analiz i yego Prilozhen 16, 80–81.