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Advances in Mathematical Physics
Volume 2016, Article ID 8958747, 10 pages
Research Article

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

1Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
2Faculty of Science, Mathematics Department, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

Received 20 September 2015; Accepted 14 February 2016

Academic Editor: Yao-Zhong Zhang

Copyright © 2016 H. M. Yehia and A. A. Elmandouh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.

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 [49]. 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 [1224]. Of those articles [12, 1519, 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 Lagrangianwhere , , and are functions of and and dots denote differentiation with respect to time . As in [21] we use a point transformation to isometric coordinates and a change of the time variableand one can always reduce (1) to the formwhere , , and are certain functions of and and primes denote derivatives with respect to andThe equations of motion take the formwhereThis system admits the zero-value Jacobi integralThe Jacobi constant 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 aswhere , , and are functions in and satisfying with the nonlinear system of seven partial differential equations [21]: The irreversible case was considered in [21], where several parameter systems admitting a cubic integral were found under the simplifying assumption that

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 ansatzAs 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 has the structurewhere , , , , and are arbitrary constants and , , , , and are functions to be determined of the single variable . Then the coefficients of the integral (9) should take the following forms:in which , are certain functions of . Inserting those expressions in (10), we obtain the following system of ordinary differential equations: where is a new function determined fromBuilding 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 defined by the following relation [20]:We give here only the final form of the Lagrangian and the complementary cubic integral in the following form:where , , , and , , , , and are arbitrary parameters, introduced instead of the original parameters and for convenience, and

3. The Generic Unconditional System

The Lagrangian (19) describes a system integrable on its zero-level of Jacobi’s integral . 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 , , , and . We first express those parameters in terms of nine new parametersand then we can perform the time transformation (2) withto the above system. Thus we obtain the LagrangianThe presence of the arbitrary parameter 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 is now interpreted as the value of the Jacobi integral. Thus, we have the unconditional Jacobi integralThe final form of the second integral can be obtained by replacing in (20) by . The Lagrangian (24) characterizes a new integrable system. It contains fifteen arbitrary parameters , , , , , , , , , , , , , , and . Note that the angle variable can be shifted by a phase angle in such a way to make one of the four parameters , , , and equal zero. The last system is an extension of the two systems with a cubic integral obtained in [21, 28] by adding the parameters and 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 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 were found. Regarding this result we formulate the following.

Theorem 1. Suppose that and let , , and such that , be three real roots of the cubic polynomial . Let also . Then the Lagrangian (24) for describes an integrable time-irreversible system on .

4.2. A New Integrable System in the Plane

As in [21], the Lagrangian (24) acquires the simplest form when one sets , . Then, introducing the change of variables , , we reduce the Lagrangian (24) to the formwhere , are arbitrary parameters, introduced instead of the original parameters for convenience:Jacobi’s integral for this motion isand the cubic integral can be written asThis integrable system can be viewed as a generalization of a previously known one due to Yehia [21] by the introduction of two constants and to equations of motion. It also generalizes the reversible Toda-like system obtained by Hall [13] by the presence of the four parameters , , , and .

4.3. Applications to Rigid Body Dynamics

The problem of motion of a rigid body whose principal moments of inertia are , , and , about a fixed point under forces with a scalar potential and vector potential , reduces after ignoring the cyclic angle of precession to the Routhianwhere , is the nutation angle, is the angle of proper rotation, and 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): , , , , , and

In this case, using the substitution , the Lagrangian (24) can be identified with the Routhian (31) if we assume that the moments of inertia satisfy , set the cyclic constant , and chooseThe cyclic integral can be written in the form The complementary cubic integral is