# Integrable 2D Time-Irreversible Systems with a Cubic Second Integral

Authors: H. M. Yehia; A. A. Elmandouh
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

We construct a very rare integrable 2D mechanical system which admits a complementary integral of motion cubic in the velocities in the presence of conservative potential and velocity-dependent (gyroscopic) forces. Special cases are given interpretation as a motion of a particle on a sphere endowed with a Riemannian metric, a particle in the Euclidean plane, and new generalizations of two cases of motion of a rigid body with a cubic integral, known by names of Goriachev-Chaplygin and Goriachev.


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## Body

## 1. Introduction: History and Formulation of the Problem

The search for potentials of conservative motions of a particle in the plane, so that the motion admits an integral polynomial in the velocities, was initiated by Bertrand in the middle of the nineteenth century [1, 2]. His results were developed further by Darboux [3] for the case of a quadratic integral. A large number of works were devoted to construction of integrable potentials in the plane admitting a complementary integral of degree up to 6. Notable examples are [4–9]. For a detailed account of relevant results up to 1985, see [10].Birkhoff extended the method to accommodate general 2D mechanical systems acted upon by potential and gyroscopic forces. Those systems mostly live on Riemannian manifolds and the presence of gyroscopic forces makes their equations of motion time-irreversible. Birkhoff’s procedure was completed to the end only in two cases: for reversible systems with an integral quadratic in velocities and irreversible systems with an integral linear in velocities [11].Time-irreversible systems were considered in much fewer works. An almost complete list of those works is composed of [12–24]. Of those articles [12, 15–19, 22, 24] are exclusively devoted to irreversible systems with a quadratic complementary invariant.An essential modification of Birkhoff’s method in Yehia’s work [20] significantly reduced the number of PDEs determining the system and its integral and made it possible to tackle the time-reversible and irreversible cases with a polynomial integral. The culmination of the new method was the construction and classification of 41 irreversible systems admitting a quadratic integral [22] and the construction of a gigantic reversible system involving 21 parameters called “master system” with a complementary integral quartic in velocities [25].The new method also made it possible to construct for the first time irreversible integrable systems which admit a complementary integral cubic [21] and quartic [26] in velocities, based on the equations derived in [20].The present paper may be regarded as a continuation of [21]. Here we study mechanical systems described by or reduced to a two-dimensional system with Lagrangian(1)L=12a11q˙12+2a12q˙1q˙2+a22q˙22+a1q˙1+a2q˙2-V,where aij, aj, and V are functions of q1 and q2 and dots denote differentiation with respect to time t. As in [21] we use a point transformation to isometric coordinates and a change of the time variable(2)dt=Λdτ,and one can always reduce (1) to the form(3)L=12x′2+y′2+l1x′+l2y′+U,where Λ, l1, and l2 are certain functions of x and y and primes denote derivatives with respect to τ and(4)U=Λh-V.The equations of motion take the form(5)x′′+Ωy′=∂U∂x,y′′-Ωx′=∂U∂y,where(6)Ω=∂l1∂y-∂l2∂x.This system admits the zero-value Jacobi integral(7)I1=12x′2+y′2-U=0.The Jacobi constant h for the original system (1) enters as a parameter in the new force function (4) (see, e.g., [27]).From the results of [20, 21] the Lagrangian and the cubic integral can be written as(8)L=12x′2+y′2+13P2x′-Q2y′+U,(9)I=x′3+P2x′2+Q2x′y′+P1x′+Q1y′+R=const.,where Pj, Qj, and R are functions in x and y satisfying with U the nonlinear system of seven partial differential equations [21]: (10)∂P2∂x-∂Q2∂y=0,∂P2∂y+∂Q2∂x-3Ω=0,∂P1∂x-∂Q1∂y+2ΩQ2+3∂U∂x=0,∂P1∂y+∂Q1∂x-2ΩP2=0,P1∂U∂x+Q1∂U∂y+2U∂Q1∂y-2ΩUQ2=0,∂R∂x+ΩQ1+2P2∂U∂x+Q2∂U∂y+2U∂Q2∂y=0,∂R∂y-ΩP1+Q2∂U∂x=0.The irreversible case Ω≠0 was considered in [21], where several parameter systems admitting a cubic integral were found under the simplifying assumption that(11)Ω=Ω0y,(12)U=uy+vyΦx.In the present paper we will try, as in [21], to construct Lagrangian systems admitting a first integral polynomial of degree three in velocities, but instead of (11) we use the ansatz(13)Ω=Ω1ya1sin⁡x+a2cos⁡x+λΩ0y.As shown below, the problem is completely solved and an integrable system involving 15 parameters is constructed, adding two parameters to the system of [21]. Four new integrable problems are obtained as special cases of this system: a motion of a particle on a sphere endowed with a Riemannian metric, a particle in the plane, and two problems in the dynamics of a rigid body.


## 2. Solution of the Problem: The Conditional Integrable System

Regarding (10) and (14) and (13), a suitable ansatz for the reduced force function U has the structure(14)U=u0y+a12+a22u1y+λ2u2y+vyμ1sin⁡x+μ2cos⁡x+λwya2cos⁡x+a1sin⁡x-vy2a12-a22cos⁡2x-2a1a2sin⁡2x,where a1, a2, λ, μ1, and μ2 are arbitrary constants and u0, u1, u2, v, and w are functions to be determined of the single variable y. Then the coefficients of the integral (9) should take the following forms:(15)P2=f1ya1sin⁡x+a2cos⁡x+λf0y,Q2=f2ya1cos⁡x-a2sin⁡x,P1=f3ya12-a22cos⁡2x-2a1a2sin⁡2x+f4yμ1sin⁡x+μ2cos⁡x+λf5ya1sin⁡x+a2cos⁡x+a12+a22f6ay+λ2f6ay,Q1=f7y2a1a2cos⁡2x+a12-a22sin⁡2x-12f2yμ1cos⁡x-μ2sin⁡x+λf8ya1cos⁡x-a2sin⁡x,in which fi, i=1,…,8 are certain functions of y. Inserting those expressions in (10), we obtain the following system of ordinary differential equations: (16)λ3Ω0-dfdy=0,λ2df6bdy-2Ω0f0=0,μif2+2df4dy=0,μi6v+2f4+2df2dy=0,a12+a22df6ady-f1Ω1=0,aif1-df2dy=0,ai3Ω1-df1dy+f2=0,i=1,2,μ12+μ22f2dvdy+2vdf2dy-2vf4=0,λai2f2Ω0+f5+3w-df8dy=0,i=1,2,λμiddyΩ0f2-4vf0-2Ω0f4=0,aif2d2u0dy2+3df2dydu0dy+2d2f2dy2u0=0,i=1,2,μi2u0df2dy+f2du0dy=0,μi2u2df2dy+f2du2dy-2vf6b=0,i=1,2,λai-2u0df8dy+2u0Ω0f2-du0dyf8=0,λai2df5dy-2f8-4Ω1f0-4Ω0f1=0,i=1,2,a12+a22f2Ω1-2f3-df7dy+6v2=0,a12+a222f7+df3dy+f1Ω1=0,a23a12-a22-2v2f2+2df1dy+d2f2dy2+3v2f2-Ω1df7dy+3f3-f7dΩ1dy-2vdvdy3df2dy+4f1-2vd2vdy2f2=0,μia1a2-v2df2dy+2v2f4-vΩ1f2+f7dvdy-vf2dvdy+2vdf7dy-vf3=0,i=1,2,λai3-vf3-2vf6a+f7dvdy+f2du1dy+v2df2dy+2v2f4+2u1df2dy+vdvdyf2-vΩ1f2+vdf7dy=0,i=1,2,Rx,y=162df2dyv2+f7Ω1+4v2f1+2f2vdvdya23a12-a22cos⁡3x+a1a12-3a22sin⁡3x-18-4f1v-4vdf2dy+f2Ω1-2f2dvdyμ1a1-μ2a2cos⁡2x-μ1a2+μ2a1sin⁡2x-λ42wdf2dy+2Ω0f7+f8Ω1+2f1w+f2dwdy+8f0v2a12-a22cos⁡2x-2a1a2sin⁡2x-λ2-f2Ω0+4vf0μ1sin⁡x+μ2cos⁡x-a1sin⁡x+a2cos⁡xλ2Ω0f8+2f0w+2u2df2dy+f2du2dy+f2du0dy+2u0df2dy+12a12+a22Ω1f7+4v2f1+4u1df2dy-2v2df2dy-2vf2dvdy+2du1dyf2+ry,where r(y) is a new function determined from(17)drdy=12a1μ1+a2μ2Ω1f4-f2v+λ3Ω0f6b+λa12+a222Ω1f5-f2w+2f6aΩ0,λ2ai-2u2d2f2dy2-2f0dwdy-2wdf0dy-f8dΩ0dy-Ω0df8dy-f2d2u2dy2-3df2dydu2dy-Ω1f6b-Ω0f5=0,i=1,2.μ2a1-μ1a22f2dvdy+4vdf2dy-4f1v+f2Ω1=0,μ1a2+a1μ2-2dvdy2f1+3df2dy-4vf2+df1dy-d2f2dy2+Ω1df2dy-4f4+f2=0,λa1a2-wdf1dy+d2f2dy2+f2+dwdyf1+32df2dy-f22d2wdy2-Ω12df8dy+2f5+Ω02f3-df7dy-dΩ0dyf7-12dΩ1dyf8-4v2df0dy-8vf0dvdy=0,ai3-2d2fy2u1-f72dΩ1dy-3df2dydu1dy-f2d2u1dy2+v2f2+d2f2∘∘dy2-2df1dy+12Ω1f3-df7dy-2f6a+vdvdy3f2-4f1+dvdy2f2+vd2vdy2f2=0,i=1,2,a12+2a1a2-a22a12-2a1a2-a22v-2df7dy+Ω1f2+2f3-2f7dvdy=0,λa1a2v2-3f2Ω0+3f5-3w2Ω1f2+f3+12df7dy+3f72dwdy+3vf8dvdy=0,a12+a22f7du0dy+2u0df7dy-f2Ω1u0=0,a14+a24u12df7dy-Ω1f2+2v2f6a+f7du1dy=0,λμ1a1+μ2a22wf4-df2dy+2v2df8dy+f5-2f2Ω0+f8-f2dwdy=0,μia12+a22vΩ1f2-f3-2df7dy-v2df2dy-2f4-dvdyvf2-f7=0,λ3ai2u0Ω0f2-df8dy-f6bw-f8du2dy=0,i=1,2,λa1a22f8du1dy+4u1df8dy-wΩ1f2-2f6a-f3-2df7dy+2v2f5+f2Ω0-df8dy+4u1f2Ω0-2vdvdyf8+f7dwdy=0.Building on the solution of the less general system of [21] and after some tedious manipulations, the solution of (16)-(17) was constructed. For convenience we introduce a new variable ν defined by the following relation [20]:(18)y=∫9α2+12βν-36αν2-12ν44ν3+6αν-βdν.We give here only the final form of the Lagrangian and the complementary cubic integral in the following form:(19)L0=12x′2-3FF12ν′2+13P2x′-Q¯2ν′+F1F-ν6+24αν4-16βν3-54α2ν2+12αβν-β24F+4ρ12ν+3ρ224ν3-β+6ρ1ρ22ν2-α6F+ρ3ν+ρ4+F13/2Fρ5+1Fρ22ν3-3αν+β+ρ12ν2+αcsin⁡x+dcos⁡x-F134F2c2-d2cos⁡2x-2cdsin⁡2x-F13/24Fasin⁡x+bcos⁡x,(20)I2=x′3+P2x′2+Q¯2x′ν′+P1x′+Q¯1ν′+R,where F=4ν4+12αν2-4βν-3α2, F1=4ν3+6αν-β, F2=2ν2+α, and ρi(i=1,2,…,5), a, b, c, and d are arbitrary parameters, introduced instead of the original parameters Ci and ai for convenience, and(21)P2=3Fρ1F2+ρ22ν3-3αν+β+38ν6+12αν4+8βν3+54α2ν2-12αβν+9α3+2β22FF1csin⁡x+dcos⁡x,Q¯2=9F2F13/2dsin⁡x-ccos⁡x,Q¯1=-942cdcos⁡2x+2c2-d2sin⁡2x+42F13ρ5Fd+bsin⁡x-a+ccos⁡x+ρ1F2+2ν3-3αν+βρ2dsin⁡x-ccos⁡x,P1=3F12F28ν6+36αν4-16βν3-18α2ν2-9α3-β2c2-d2cos⁡2x-2cdsin⁡2x+9F22F1asin⁡x+bcos⁡x-32F2F1ρ116ν8+32αν6-64βν5-120α2ν4-64αβν3-216α3ν2+8β2ν2+48βα2ν-9α4-4αβ2+ρ216ν9+288αν7-144βν6+216αν4αν-β+24ν3β2+3α3-108βα2ν2+81α4ν+36αβ2ν-18βα3-4β3+ρ54F2F1csin⁡x+dcos⁡x+3c2+d22F216ν9+96αν7-24βν6+216α2ν5-156αβν4+36β2-2α3ν3+54βα2ν2+3α2β2+27α3-9βα3-β3,R=9F1F2ρ13ν+ρ2ν2-α2asin⁡x+bcos⁡x+F158F340ν6+156αν4-56βν3-18α2ν2-12αβν-27α3-2β23c2d-d3cos⁡3x+c3-3cdsin⁡3x+-8F1ρ527F4νρ1+3ρ22ν2-α-8ρ3F1327+2F13c2+d22732ν9+288αν7-120βν6+576α2ν5-612αβν4-648α3ν3+144β2ν3+342α2βν2+270α4ν-27α3β-2β3-8F127F3ρ116αν3-12βν2-24α2ν+2αβ+ρ28ν6+60αν4-28βν3-18α2ν2-6αβν+9α3+2β2ρ1F2+ρ22ν3-3αν+βcsin⁡x+dcos⁡x+16F126ρ5F227F+ρ180ν8+256αν6-96βν5+120α2ν4-128αβν3-144α3ν2+12β2ν2+24α2βν-9α4-2αβ2+ρ280ν9+384αν7-144βν6+72α2ν5-120αβν4+12ν3β2-12α3+81α4ν+6αβ2ν-18βα3-2β3227F3c2-d2cos⁡2x-2cdsin⁡2x+64c2+d2927Fρ58αν3-6βν2-12α2ν+αβ-164αρ1+βρ2ν9-486α2ρ2+βρ1ν8+576αβρ2ν7-192ρ2β2-6α3ν6+288α3ρ1-504α2βρ2-48β2ρ1ν5-72α-6α3ρ2+7αβρ1+β2ρ2ν4+12β2ρ2+8αρ1-576α3αρ1+βρ2ν3-12ββ2ρ1-12α2αρ1+βρ2ν2-9αβρ23α2+β2+4αρ13α3+β2ν+2ρ227α6+18α3β2+2β4+αβρ12β2+9α3-ρ34FF∘∘4ρ1+6ν3-3αν+βρ2+1F3ρ1F2+ρ2ν3-3αν+β3-9ac+bd4F8αν3-6βν2-12α2ν+αβ.


## 3. The Generic Unconditional System

The Lagrangian (19) describes a system integrable on its zero-level of Jacobi’s integral I1. Following the method devised by Yehia [21, 25] (for a detailed account of this method, see [28]), we now proceed to construct the corresponding unconditional system by performing the inverse of time transformation (2). Our conditional system involves 4 energy-type parameters ρ3, ρ4, a, and b. We first express those parameters in terms of nine new parameters(22)ρ3=-12h1+ε1h,ρ4=-h2+ε2h,a=h3+ε3h,b=h4+ε4hand then we can perform the time transformation (2) with(23)Λ=4ν3+6αν-β3α2+4βν-12αν2-4ν4ε1ν+ε2+4ν3+6αν-βε3sin⁡x+ε4cos⁡xto the above system. Thus we obtain the Lagrangian(24)L=12ε1ν+ε2+4ν3+6αν-βε3sin⁡x+ε4cos⁡x4ν3+6αν-βx˙23α2+4βν-12αν2-4ν4+3ν˙24ν3+6αν-β+13P2x˙-Q¯2ν˙+1ε1ν+ε2+4ν3+6αν-βε3sin⁡x+ε4cos⁡xh12ν+h2+4ν3+6αν-βh3sin⁡x+h4cos⁡x-ρ12ν2+α+ρ22ν3-3αν+β4ν4+12αν2-4βν-3α2+ρ5csin⁡x+dcos⁡x+1124ν4+12αν2-4βν-3α234ν3+6αν-β2c2-d2cos⁡2x-2cdsin⁡2x-8ρ12ν-12ρ1ρ2α-2ν2-64ν3-βρ22+3c2+d28ν6+24αν4-16βν3-54α2ν2+12αβν-β2+h.The presence of the arbitrary parameter h in the last Lagrangian as an additive constant is insignificant and it can be ignored, as it does not contribute to the equations of motion. The same arbitrary constant h is now interpreted as the value of the Jacobi integral. Thus, we have the unconditional Jacobi integral(25)I1=12ε1ν+ε2+4ν3+6αν-βε3sin⁡x+ε4cos⁡x4ν3+6αν-βx˙23α2+4βν-12αν2-4ν4+3ν˙24ν3+6αν-β-1ε1ν+ε2+4ν3+6αν-βε3sin⁡x+ε4cos⁡xh12ν+h2+4ν3+6αν-βh3sin⁡x+h4cos⁡x-ρ12ν2+α+ρ22ν3-3αν+β4ν4+12αν2-4βν-3α2+ρ5csin⁡x+dcos⁡x+1124ν4+12αν2-4βν-3α234ν3+6αν-β2c2-d2cos⁡2x-2cdsin⁡2x-8ρ12ν-12ρ1ρ2α-2ν2-64ν3-βρ22+38ν6+24αν4-16βν3-54α2ν2+12αβν-β2c2+d2=h.The final form of the second integral can be obtained by replacing (x′,v′) in (20) by (Λx˙,Λν˙). The Lagrangian (24) characterizes a new integrable system. It contains fifteen arbitrary parameters α, β, ε1, ε2, ε3, ε4, ρ1, ρ2, ρ5, c, d, h1, h2, h3, and h4. Note that the angle variable x can be shifted by a phase angle in such a way to make one of the four parameters ε3, ε4, c, and d equal zero. The last system is an extension of the two systems with a cubic integral obtained in [21, 28] by adding the parameters c and d which invoke a part of the gyroscopic (irreversible) and potential terms.


## 4. Applications

In its full capacity, the fifteen-parameter system with the Lagrangian (24) has not yet found a mechanical interpretation for the full range of values of the parameters. In this section we provide four applications as special cases of that system: one integrable system on the sphere, one in the Euclidean plane, and two new integrable cases in rigid body dynamics. Those special cases indicate the richness of this system.


### 4.1. An Integrable System on the Sphere

The metric(26)ds2=ε1ν+ε2+4ν3+6αν-βε3sin⁡x+ε4cos⁡x4ν3+6αν-βdx23α2+4βν-12αν2-4ν4+3dν24ν3+6αν-β of the configuration space of the system described by (24) was considered in [28] and sufficient conditions for it to be Riemannian and well defined on S2 were found. Regarding this result we formulate the following.Theorem 1.
Suppose that8α3+β2<0 and let v1, v2, and v3 such that v1<v2<v3, v1+v2+v3=0 be three real roots of the cubic polynomial 4ν3+6αν-β. Let also ε1+ε2ν1>ε32+ε42[(-2α)3/2-β]. Then the Lagrangian (24) for v∈[v1,v2] describes an integrable time-irreversible system on S2.


### 4.2. A New Integrable System in the Plane

As in [21], the Lagrangian (24) acquires the simplest form when one sets α=β=ε2=ε3=ε4=0, ε1=1. Then, introducing the change of variables x→i3x, ν→e2y, we reduce the Lagrangian (24) to the form(27)L=12x˙2+y˙2+P1x˙-V+h,P1=α1e-4y+α2e-2y+α3ey+3x+α4ey-3x,V=-2α123e-8y-2α1α2e-6y-2α22e-4y-α6e-2y-α1e-3yα3e3x+α4e-3x-α2e-yα3e3x+α4e-3x+2ey27α4e3x-α5e-3x-e2y2α32e23x+α42e-23x-α3α4,where αi, i=1,2,…,6 are arbitrary parameters, introduced instead of the original parameters for convenience:(28)x¨+Ωy˙=-∂V∂x,y¨-Ωx˙=-∂V∂y,Ω=-4α1e-4y-2α2e-2y+α3ey+3x+α4ey-3x.Jacobi’s integral for this motion is(29)I1=12x˙2+y˙2+V=hand the cubic integral can be written as(30)I2=x˙3-3x˙y˙2+9α1e-4y+α2e-2yx˙2+33eyα3e3x-α4e-3xx˙y˙-3α1e-4y+α2e-2y+eyα3e3x+α4e-3xy˙2+3e2yα3α4-α32e23x-α42e-23x+2ey9α4e3x-α5e-3x+3α2e-yα3e3x+α4e-3x+9α1e-3yα3e3x+α4e-3x+6α6e-2y+24α22e-4y+16α12e-8y+36α1α2e-6yx˙+-33e2yα42e-23x-α32e23x-23ey9α4e3x+α5e-3x+33α2e-yα3e3x-α4e-3x+33α1e-3yα3e3x-α4e-3xy˙+10α12e-7yα3e3x+α4e-3x+18α1α2e-5yα3e3x+α4e-3x+49e-3ye3xα1α4+27α22α3+e-3x27α22α4-α1α5-2e2y9α4α5e-23x-α3e23x-2α3α5+2α42+2e-y9e3x4α2α4+27α3α6+e-3x27α4α6-4α2α5-e3y2α43e-33x+α33e33x-3α3α4α3e3x+α4e-3x+2e-6y8α23+3α1α6-6α2α42e-23x+α32e23x+8α13e-12y+28α2α12e-10y+6α2α6e-4y+36α1α22e-8y+9α1α3α4e-2y.This integrable system can be viewed as a generalization of a previously known one due to Yehia [21] by the introduction of two constants α3 and α4 to equations of motion. It also generalizes the reversible Toda-like system obtained by Hall [13] by the presence of the four parameters α1, α2, α3, and α4.


### 4.3. Applications to Rigid Body Dynamics

The problem of motion of a rigid body whose principal moments of inertia areA, A, and C, about a fixed point under forces with a scalar potential V(γ) and vector potential l=(0,0,l3), reduces after ignoring the cyclic angle of precession ψ to the Routhian(31)R=12γ˙321-γ32+C1-γ32φ˙2A-A-Cγ32+fCγ3+Al31-γ32φ˙AA-A-Cγ32-1AV+f-l3γ322A-A-Cγ32,where γ3=cos⁡(θ), θ is the nutation angle, φ is the angle of proper rotation, and f is the value of cyclic integral. For more details see [21].As in [21], the Lagrangian (24) can be identified with the Routhian (31) in the following two cases.


#### 4.3.1. Case (a):A=4C, α=-1/2, β=1, ε1=-1/3, ε2=-1/6, and ε3=ε4=0

In this case, using the substitutionν=1/2(3γ32-1), the Lagrangian (24) can be identified with the Routhian (31) if we assume that the moments of inertia satisfy A=4C, set the cyclic constant f=0, and choose(32)l1=4Cnγ1,l2=4Cnγ2,l3=Ck+e02γ34-1γ32+e1γ12+γ22+e2γ1+e3γ2+nγ3,V=Ce5γ1+e6γ2+εγ32+e024γ36-2γ38-52γ34-e1k+e0-2e1γ12+γ22-e122γ12+γ222-e322γ22+γ32-e2e3γ1γ2+e222γ22+3n22γ32+e0γ32-2γ34-nγ3-e1γ12+γ22e2γ1+e3γ2+ne0γ32-2γ33-γ3k+e1γ12+γ22.The cyclic integral can be written in the form (33)I1=4pγ1+4qγ2+r+k+e2γ1+e3γ2+e1γ12+γ22+e02γ34-1γ32γ3+n4γ12+γ22+γ32=0.The complementary cubic integral is(34)I2=r-k+e2γ1+e3γ2+nγ3+e02-γ32γ34-e18γ12-1γ12+γ22p+nγ1+e22γ32+q+nγ2+e3γ322+ε2γ32+ke0γ34-e12-e12+e02-γ322γ34r+nγ3+e12+e02-γ322γ34e2γ1+e3γ2-e023γ34-6γ32+42γ38+e1γ12+γ22e11-8γ122+8γ12γ32-2-2γ34+9γ32-82γ34-γ32e1e2-e2k+e5p+nγ1+e2γ32+2e1e3-e3k+e6q+nγ2+e32γ3+ke0γ32+e11-2γ32γ12+γ22e2γ1+e3γ2+4e1γ12e0-2e1γ32γ32γ12+γ22+-4e1γ3γ12+γ22p+nγ1γ1+q+nγ2γ2-8e0e1e3γ2γ12γ32γ12+γ22+e0γ32-e1γ12+γ22e5γ1+e6γ2+4e1γ122e02γ36+εγ32γ12+γ22-8e1e2γ13γ12+γ22e1+e0γ32+4e0e12γ12γ34γ12+γ2228γ12γ32-2-γ12+γ22γ32-4+e1e32+e229γ12+γ2218γ12γ32-9γ34+13γ32-4+8e13γ131-4γ12γ12+γ222+8e1γ12-γ22γ12+γ22q+nγ22+2e1γ3γ12+γ22q+nγ2e35γ12-γ22-2e2γ1γ2+p+nγ1e23γ12+γ22-2e3γ1γ2-16e1γ1γ2γ12+γ22q+nγ2p+nγ1-e1e32γ1,where k, ε, n, and ei(i=0,1,…,6) are arbitrary parameters, introduced instead of the original parameters. This choice (32) characterizes a new integrable problem in the dynamics of a rigid body, which generalizes all previously known integrable cases with a cubic integral in this field, as in Table 1.Table 1
Conditions
Authors
Reference
e
0
=
e
1
=
ε
=
n
=
0
Sokolov and Tsiganov
[32] 2002
e
2
=
e
3
=
0
Yehia
[21] 2002
e
0
=
e
1
=
e
2
=
e
3
=
0
Yehia
[33] 1996
e
0
=
e
1
=
e
2
=
e
3
=
ε
=
n
=
0
Sretensky
[34] 1963
e
0
=
e
1
=
e
2
=
e
3
=
k
=
n
=
0
Goriachev
[29] 1915
e
0
=
e
1
=
e
2
=
e
3
=
k
=
ε
=
n
=
0
Goriachev
[35] 1900


#### 4.3.2. Case (b):A=4/3C, α=0, β=1, and ε2=ε3=ε4=0

For this case we use the substitutionν=γ32/3. We construct the integrable case of a rigid body dynamics in which A=B=4/3, C=1, and(35)l3=n+1γ12+γ223n+e0γ32/3+e12+γ32γ32/3+e2γ1+e3γ2γ32/3,V=e4γ1+e5γ2+e6γ32/3+e32-e22γ12-γ22-4e2e3γ1γ24γ34/3+1γ12+γ222e024-7γ326γ32/3-e0e15γ32-2-3e0nγ38/3-e1213γ34-8γ32+42γ34/3-3ne1γ34/3γ32+2+3n2γ322γ32-4-e22+e324γ34/35γ32+1-e2γ1+e3γ23γ12+γ223e0+3e1γ32/3+e7+9nγ34/3+6e1γ34/3-e7γ32/3,where n and ei, i=0,1,…,7, are arbitrary constant. This choice (35) gives a new integrable case in a rigid body dynamics. This case adds two arbitrary parameters e2 and e3 to the case found by Yehia [21] and has five arbitrary parameters n, e0, e1, e2, and e3, more than the original case found by Goriachev [29] in 1915.Although having no obvious physical meaning, Goriachev’s case has received a growing interest in the last years [30, 31]. It turns out to be the first example of a mechanical system whose complex invariant varieties are strata of Jacobians of a nonhyperelliptic curve, here a trigonal curve of genus 3 [31].


## 4.1. An Integrable System on the Sphere

The metric(26)ds2=ε1ν+ε2+4ν3+6αν-βε3sin⁡x+ε4cos⁡x4ν3+6αν-βdx23α2+4βν-12αν2-4ν4+3dν24ν3+6αν-β of the configuration space of the system described by (24) was considered in [28] and sufficient conditions for it to be Riemannian and well defined on S2 were found. Regarding this result we formulate the following.Theorem 1.
Suppose that8α3+β2<0 and let v1, v2, and v3 such that v1<v2<v3, v1+v2+v3=0 be three real roots of the cubic polynomial 4ν3+6αν-β. Let also ε1+ε2ν1>ε32+ε42[(-2α)3/2-β]. Then the Lagrangian (24) for v∈[v1,v2] describes an integrable time-irreversible system on S2.


## 4.2. A New Integrable System in the Plane

As in [21], the Lagrangian (24) acquires the simplest form when one sets α=β=ε2=ε3=ε4=0, ε1=1. Then, introducing the change of variables x→i3x, ν→e2y, we reduce the Lagrangian (24) to the form(27)L=12x˙2+y˙2+P1x˙-V+h,P1=α1e-4y+α2e-2y+α3ey+3x+α4ey-3x,V=-2α123e-8y-2α1α2e-6y-2α22e-4y-α6e-2y-α1e-3yα3e3x+α4e-3x-α2e-yα3e3x+α4e-3x+2ey27α4e3x-α5e-3x-e2y2α32e23x+α42e-23x-α3α4,where αi, i=1,2,…,6 are arbitrary parameters, introduced instead of the original parameters for convenience:(28)x¨+Ωy˙=-∂V∂x,y¨-Ωx˙=-∂V∂y,Ω=-4α1e-4y-2α2e-2y+α3ey+3x+α4ey-3x.Jacobi’s integral for this motion is(29)I1=12x˙2+y˙2+V=hand the cubic integral can be written as(30)I2=x˙3-3x˙y˙2+9α1e-4y+α2e-2yx˙2+33eyα3e3x-α4e-3xx˙y˙-3α1e-4y+α2e-2y+eyα3e3x+α4e-3xy˙2+3e2yα3α4-α32e23x-α42e-23x+2ey9α4e3x-α5e-3x+3α2e-yα3e3x+α4e-3x+9α1e-3yα3e3x+α4e-3x+6α6e-2y+24α22e-4y+16α12e-8y+36α1α2e-6yx˙+-33e2yα42e-23x-α32e23x-23ey9α4e3x+α5e-3x+33α2e-yα3e3x-α4e-3x+33α1e-3yα3e3x-α4e-3xy˙+10α12e-7yα3e3x+α4e-3x+18α1α2e-5yα3e3x+α4e-3x+49e-3ye3xα1α4+27α22α3+e-3x27α22α4-α1α5-2e2y9α4α5e-23x-α3e23x-2α3α5+2α42+2e-y9e3x4α2α4+27α3α6+e-3x27α4α6-4α2α5-e3y2α43e-33x+α33e33x-3α3α4α3e3x+α4e-3x+2e-6y8α23+3α1α6-6α2α42e-23x+α32e23x+8α13e-12y+28α2α12e-10y+6α2α6e-4y+36α1α22e-8y+9α1α3α4e-2y.This integrable system can be viewed as a generalization of a previously known one due to Yehia [21] by the introduction of two constants α3 and α4 to equations of motion. It also generalizes the reversible Toda-like system obtained by Hall [13] by the presence of the four parameters α1, α2, α3, and α4.


## 4.3. Applications to Rigid Body Dynamics

The problem of motion of a rigid body whose principal moments of inertia areA, A, and C, about a fixed point under forces with a scalar potential V(γ) and vector potential l=(0,0,l3), reduces after ignoring the cyclic angle of precession ψ to the Routhian(31)R=12γ˙321-γ32+C1-γ32φ˙2A-A-Cγ32+fCγ3+Al31-γ32φ˙AA-A-Cγ32-1AV+f-l3γ322A-A-Cγ32,where γ3=cos⁡(θ), θ is the nutation angle, φ is the angle of proper rotation, and f is the value of cyclic integral. For more details see [21].As in [21], the Lagrangian (24) can be identified with the Routhian (31) in the following two cases.


### 4.3.1. Case (a):A=4C, α=-1/2, β=1, ε1=-1/3, ε2=-1/6, and ε3=ε4=0

In this case, using the substitutionν=1/2(3γ32-1), the Lagrangian (24) can be identified with the Routhian (31) if we assume that the moments of inertia satisfy A=4C, set the cyclic constant f=0, and choose(32)l1=4Cnγ1,l2=4Cnγ2,l3=Ck+e02γ34-1γ32+e1γ12+γ22+e2γ1+e3γ2+nγ3,V=Ce5γ1+e6γ2+εγ32+e024γ36-2γ38-52γ34-e1k+e0-2e1γ12+γ22-e122γ12+γ222-e322γ22+γ32-e2e3γ1γ2+e222γ22+3n22γ32+e0γ32-2γ34-nγ3-e1γ12+γ22e2γ1+e3γ2+ne0γ32-2γ33-γ3k+e1γ12+γ22.The cyclic integral can be written in the form (33)I1=4pγ1+4qγ2+r+k+e2γ1+e3γ2+e1γ12+γ22+e02γ34-1γ32γ3+n4γ12+γ22+γ32=0.The complementary cubic integral is(34)I2=r-k+e2γ1+e3γ2+nγ3+e02-γ32γ34-e18γ12-1γ12+γ22p+nγ1+e22γ32+q+nγ2+e3γ322+ε2γ32+ke0γ34-e12-e12+e02-γ322γ34r+nγ3+e12+e02-γ322γ34e2γ1+e3γ2-e023γ34-6γ32+42γ38+e1γ12+γ22e11-8γ122+8γ12γ32-2-2γ34+9γ32-82γ34-γ32e1e2-e2k+e5p+nγ1+e2γ32+2e1e3-e3k+e6q+nγ2+e32γ3+ke0γ32+e11-2γ32γ12+γ22e2γ1+e3γ2+4e1γ12e0-2e1γ32γ32γ12+γ22+-4e1γ3γ12+γ22p+nγ1γ1+q+nγ2γ2-8e0e1e3γ2γ12γ32γ12+γ22+e0γ32-e1γ12+γ22e5γ1+e6γ2+4e1γ122e02γ36+εγ32γ12+γ22-8e1e2γ13γ12+γ22e1+e0γ32+4e0e12γ12γ34γ12+γ2228γ12γ32-2-γ12+γ22γ32-4+e1e32+e229γ12+γ2218γ12γ32-9γ34+13γ32-4+8e13γ131-4γ12γ12+γ222+8e1γ12-γ22γ12+γ22q+nγ22+2e1γ3γ12+γ22q+nγ2e35γ12-γ22-2e2γ1γ2+p+nγ1e23γ12+γ22-2e3γ1γ2-16e1γ1γ2γ12+γ22q+nγ2p+nγ1-e1e32γ1,where k, ε, n, and ei(i=0,1,…,6) are arbitrary parameters, introduced instead of the original parameters. This choice (32) characterizes a new integrable problem in the dynamics of a rigid body, which generalizes all previously known integrable cases with a cubic integral in this field, as in Table 1.Table 1
Conditions
Authors
Reference
e
0
=
e
1
=
ε
=
n
=
0
Sokolov and Tsiganov
[32] 2002
e
2
=
e
3
=
0
Yehia
[21] 2002
e
0
=
e
1
=
e
2
=
e
3
=
0
Yehia
[33] 1996
e
0
=
e
1
=
e
2
=
e
3
=
ε
=
n
=
0
Sretensky
[34] 1963
e
0
=
e
1
=
e
2
=
e
3
=
k
=
n
=
0
Goriachev
[29] 1915
e
0
=
e
1
=
e
2
=
e
3
=
k
=
ε
=
n
=
0
Goriachev
[35] 1900


### 4.3.2. Case (b):A=4/3C, α=0, β=1, and ε2=ε3=ε4=0

For this case we use the substitutionν=γ32/3. We construct the integrable case of a rigid body dynamics in which A=B=4/3, C=1, and(35)l3=n+1γ12+γ223n+e0γ32/3+e12+γ32γ32/3+e2γ1+e3γ2γ32/3,V=e4γ1+e5γ2+e6γ32/3+e32-e22γ12-γ22-4e2e3γ1γ24γ34/3+1γ12+γ222e024-7γ326γ32/3-e0e15γ32-2-3e0nγ38/3-e1213γ34-8γ32+42γ34/3-3ne1γ34/3γ32+2+3n2γ322γ32-4-e22+e324γ34/35γ32+1-e2γ1+e3γ23γ12+γ223e0+3e1γ32/3+e7+9nγ34/3+6e1γ34/3-e7γ32/3,where n and ei, i=0,1,…,7, are arbitrary constant. This choice (35) gives a new integrable case in a rigid body dynamics. This case adds two arbitrary parameters e2 and e3 to the case found by Yehia [21] and has five arbitrary parameters n, e0, e1, e2, and e3, more than the original case found by Goriachev [29] in 1915.Although having no obvious physical meaning, Goriachev’s case has received a growing interest in the last years [30, 31]. It turns out to be the first example of a mechanical system whose complex invariant varieties are strata of Jacobians of a nonhyperelliptic curve, here a trigonal curve of genus 3 [31].


## 4.3.1. Case (a):A=4C, α=-1/2, β=1, ε1=-1/3, ε2=-1/6, and ε3=ε4=0

In this case, using the substitutionν=1/2(3γ32-1), the Lagrangian (24) can be identified with the Routhian (31) if we assume that the moments of inertia satisfy A=4C, set the cyclic constant f=0, and choose(32)l1=4Cnγ1,l2=4Cnγ2,l3=Ck+e02γ34-1γ32+e1γ12+γ22+e2γ1+e3γ2+nγ3,V=Ce5γ1+e6γ2+εγ32+e024γ36-2γ38-52γ34-e1k+e0-2e1γ12+γ22-e122γ12+γ222-e322γ22+γ32-e2e3γ1γ2+e222γ22+3n22γ32+e0γ32-2γ34-nγ3-e1γ12+γ22e2γ1+e3γ2+ne0γ32-2γ33-γ3k+e1γ12+γ22.The cyclic integral can be written in the form (33)I1=4pγ1+4qγ2+r+k+e2γ1+e3γ2+e1γ12+γ22+e02γ34-1γ32γ3+n4γ12+γ22+γ32=0.The complementary cubic integral is(34)I2=r-k+e2γ1+e3γ2+nγ3+e02-γ32γ34-e18γ12-1γ12+γ22p+nγ1+e22γ32+q+nγ2+e3γ322+ε2γ32+ke0γ34-e12-e12+e02-γ322γ34r+nγ3+e12+e02-γ322γ34e2γ1+e3γ2-e023γ34-6γ32+42γ38+e1γ12+γ22e11-8γ122+8γ12γ32-2-2γ34+9γ32-82γ34-γ32e1e2-e2k+e5p+nγ1+e2γ32+2e1e3-e3k+e6q+nγ2+e32γ3+ke0γ32+e11-2γ32γ12+γ22e2γ1+e3γ2+4e1γ12e0-2e1γ32γ32γ12+γ22+-4e1γ3γ12+γ22p+nγ1γ1+q+nγ2γ2-8e0e1e3γ2γ12γ32γ12+γ22+e0γ32-e1γ12+γ22e5γ1+e6γ2+4e1γ122e02γ36+εγ32γ12+γ22-8e1e2γ13γ12+γ22e1+e0γ32+4e0e12γ12γ34γ12+γ2228γ12γ32-2-γ12+γ22γ32-4+e1e32+e229γ12+γ2218γ12γ32-9γ34+13γ32-4+8e13γ131-4γ12γ12+γ222+8e1γ12-γ22γ12+γ22q+nγ22+2e1γ3γ12+γ22q+nγ2e35γ12-γ22-2e2γ1γ2+p+nγ1e23γ12+γ22-2e3γ1γ2-16e1γ1γ2γ12+γ22q+nγ2p+nγ1-e1e32γ1,where k, ε, n, and ei(i=0,1,…,6) are arbitrary parameters, introduced instead of the original parameters. This choice (32) characterizes a new integrable problem in the dynamics of a rigid body, which generalizes all previously known integrable cases with a cubic integral in this field, as in Table 1.Table 1
Conditions
Authors
Reference
e
0
=
e
1
=
ε
=
n
=
0
Sokolov and Tsiganov
[32] 2002
e
2
=
e
3
=
0
Yehia
[21] 2002
e
0
=
e
1
=
e
2
=
e
3
=
0
Yehia
[33] 1996
e
0
=
e
1
=
e
2
=
e
3
=
ε
=
n
=
0
Sretensky
[34] 1963
e
0
=
e
1
=
e
2
=
e
3
=
k
=
n
=
0
Goriachev
[29] 1915
e
0
=
e
1
=
e
2
=
e
3
=
k
=
ε
=
n
=
0
Goriachev
[35] 1900


## 4.3.2. Case (b):A=4/3C, α=0, β=1, and ε2=ε3=ε4=0

For this case we use the substitutionν=γ32/3. We construct the integrable case of a rigid body dynamics in which A=B=4/3, C=1, and(35)l3=n+1γ12+γ223n+e0γ32/3+e12+γ32γ32/3+e2γ1+e3γ2γ32/3,V=e4γ1+e5γ2+e6γ32/3+e32-e22γ12-γ22-4e2e3γ1γ24γ34/3+1γ12+γ222e024-7γ326γ32/3-e0e15γ32-2-3e0nγ38/3-e1213γ34-8γ32+42γ34/3-3ne1γ34/3γ32+2+3n2γ322γ32-4-e22+e324γ34/35γ32+1-e2γ1+e3γ23γ12+γ223e0+3e1γ32/3+e7+9nγ34/3+6e1γ34/3-e7γ32/3,where n and ei, i=0,1,…,7, are arbitrary constant. This choice (35) gives a new integrable case in a rigid body dynamics. This case adds two arbitrary parameters e2 and e3 to the case found by Yehia [21] and has five arbitrary parameters n, e0, e1, e2, and e3, more than the original case found by Goriachev [29] in 1915.Although having no obvious physical meaning, Goriachev’s case has received a growing interest in the last years [30, 31]. It turns out to be the first example of a mechanical system whose complex invariant varieties are strata of Jacobians of a nonhyperelliptic curve, here a trigonal curve of genus 3 [31].



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*Source: 8958747-2016-05-04.xml*