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On the motion of a heavy rigid body in an ideal fluid with circulation

Chaplygin's equations describing the planar motion of a rigid body in an unbounded volume of an ideal fluid involved in a circular flow around the body are considered. Hamiltonian structures, new integrable cases, and partial solutions are revealed, and their stability is examined. The problems of non-integrability of the equations of motion because of a chaotic behavior of the system are discussed.

1 Introduction and a review of the known results. S. A. Chaplygin [4] considered a general problem regarding the forces and momenta that impact an arbitrary rigid body involved in a parallel-plane motion in an unbounded volume of an ideal incompressible fluid. More specific formulations were studied earlier by N. E. Zhukovski [22,23], who considered the application of his formula for the lifting force to the description of heavy body falling in a fluid. However, these formulations yielded an unrealistic conclusion that the rotational and translational motions are independent, thus showing that additional analysis is required.
The necessary analysis was made by S. A. Chaplygin.
Chaplygin made general assumptions that the fluid motion is vortex-free with a zero velocity at the infinity, and the fluid circulation around the body is constant. He observed that this form of equations holds true if the propeller propulsion does not change its direction with respect to the body (aircraft) and the drag equilibrates this propulsion at any moment. Although this assumption resulted in the conclusion that the aircraft will have a directional stability, which is not true (the aircraft trajectory may be winding), some general conclusions that can be drawn from this study allow a remarkable mechanical interpretation. They are of both theoretical and practical interest even in the context of modern applied aerohydromechanics. S. A. Chapygin [3] considered the case of a circulation-free planar flow around a body and suggested a remarkable form of equations, which unfortunately he has not studied at all (equations with a similar form describe the falling of a body of rotation in a fluid under the same conditions). Originally written in 1890 as his student essay, this work was published only in 1933 as part of Chaplygin's Collected Works. Independently and at about the same time D. N. Goryachev [8] (1893) and V. A. Steklov [20] (1894) obtained the same equations and described their simplest qualitative properties. In particular, V. A. Steklov showed that, as the body falls, the amplitude of its oscillations with respect to the horizontal axis decreases while the frequency of these oscillations increases. He made this remark in an addendum to his book [20], in which the analysis of the asymptotic behavior was made with some mistakes. In essence, V. A. Steklov formulated the problem of asymptotic description of the behavior of a heavy rigid body during its fall. This problem was solved by V. V. Kozlov [9], who showed that, under almost all initial conditions, the body tends to fall at a uniform acceleration with its wider side up and oscillates around the horizontal axis with an increasing frequency of the order of t and a decreasing amplitude with an order of 1/ √ t. Asymptotic motions with different numbers of half-turns were analyzed numerically in [5].
The asymptotic behavior of a body falling without initial impact was studied in [18].
The effect of an abrupt ascent was described and studied in [6]. Under conditions of vortex-free flow around the body, it is assumed that, in the initial moment, the wider side of the body is horizontal and the body has a horizontal velocity. In the following moments, the body starts moving downward. However, if its apparent mass in the lateral direction is sufficiently large, the body will next abruptly move upward with its narrower side up and rise higher than its initial elevation.
More general equations describing a non-planar motion of a heavy rigid body in an unbounded volume of an ideal fluid involved in a vortex-free motion and resting at the infinity were also obtained in [3] (more compact form of these equations is presented in [10]). They generalize the well-known Kirchhoff equations, which are known to neglect the force of gravity. As was shown in [10], if a rigid body has three mutually orthogonal planes of symmetry, this body, when falling freely, asymptotically tends to take the position when its axis with the maximum added mass is vertical. Such body can also rotate around this axis.
A particular case of existence of the Hessian integral for the general equations [3] is described in [2]. In this case, the equations can be reduced to a simpler form analogous to the planar case. It is worth mentioning that a planar motion of a rigid body in a resisting medium is considered in [11]. For various models of hydrodynamic forces, some numerical and experimental results concerned with the falling motion of a heavy body in a fluid are presented in the papers [1,21,7,17,15]. An elementary analysis of the motion of a body in a resisting medium was first performed by Maxwell in [16]. Now let us return to the problem of a planar circular motion of a rigid body, which is the focus of this study. The equations of motion for this problem were suggested by S. A. Chaplygin in [4]. It was studied in [12] with some specific assumptions (as compared with [4]).
Circulation makes the motion of the rigid body more complicated. In [12], cases of stationary motion are found and their stability is examined. An integrable case and certain classes of particular solutions analogous to Zhukovski's solutions [22,23] are proposed.
In this paper, we will propose the Hamiltonian form of the general equations, and show new cases of integrability and classes of new particular solutions. We will also show that in the general case, the equations [4] are not integrable and their behavior is chaotic. Explicit evaluation of the forces and momenta associated with the circulation flow around a body was used in [4] to derive the following equations, which are similar to the Kirchhoff equations: In equations (1) a 1 , a 2 , b are added masses and momentum, λ = Iρ, ζ = Iρµ 1 , χ = Iρλ 1 . Parameters ζ, χ, proportional to the circulation I, are associated with the asymmetry of the body, and their evaluation is described in [4]. In the general case with circulation and asymmetry of the body ζ = 0, χ = 0, these parameters cannot be eliminated by any choice of a coordinate system fixed to the body.
Remark. Equations (1) were proposed by S.A.Chaplygin in 1926 and somewhat later (and presumably independently) were derived by Lamb and Glowert in 1929 [14]. Unlike Chaplygin [4], the behavior of solutions of these equations was not analyzed in these works.
It is worth mentioning that equations similar to (1) describe the planar motion of a rigid body in an ideal fluid with a uniform vorticity. The motion of a circular cylinder in a fluid with a uniformly distributed vorticity (for different boundary conditions) was analyzed by Prudeman and Taylor [14]. In this case, the equations of motion are reduced to a simplest linear system. The general (even planar) motion of an arbitrary rigid body is very complicated and has not been analyzed yet.
If all forces are potential, we have where the function U = U(x, y, ϕ) is the potential. In this case, equations (3) have the integral of energy where the kinetic energy T of the ("body + fluid") system is written in the diagonal form after translations and rotation of the moving coordinate system.
It can be shown that the forces caused by circulation are generalized potential forces and the equations of motion (in the case of potential external forces) can be written in the Lagrangian form where the Lagrangian function has the form Remark. The Lagrangian function is chosen in a symmetric calibration; recall that the summands that are linear with respect to velocities are determined to a total differential of an arbitrary function.
Equations (4)  in the momenta. It was found [11], that the equations of motion for generalized potential systems can be conveniently represented in the slightly modified variables: Using the Legendre transformation for new variables, we find the Hamiltonian: The obtained equations of motion have the form similar to that of the Poincaré-Chetaev equations: and the Poisson bracket of these variables contains "additional circulation terms" (hygroscopic terms): The rank of this Poisson structure is equal to 6; therefore, the system (7) can be reduced to a canonical system with three degrees of freedom.
3 Motion in the gravity field.
In this case, the potential energy of the system can be written in the form U = µ(y + ξ sin ϕ + η cos ϕ), and its Hamiltonian (7) has the form H = 1 2 The system (7), (9) admits an autonomous and a non-autonomous integral, which correspond to the projections of the momentum on the fixed axes Ox and Oy (see figure 1): which are not commutative {p x , p y } = λ.
To perform a reduction by one degree of freedom we express y from p x = c 1 and substitute it into the Hamiltonian (9): This reduced Hamiltonian depends only on the variables p 1 , p 2 , M and ϕ, the Poisson brackets of which, according to (8), form a closed subalgebra with a rank of 4. Thus, we have a reduced system with two degrees of freedom, which can be written in the canonical form (see below). However, we will not use the canonical form, but algebrise the reduced system to even greater extent with the help of dependent variables γ 1 = sin ϕ, γ 2 = cos ϕ : where p = (p 1 , p 2 ), γ = (γ 1 , γ 2 ), J = 0 1 −1 0 , s = (χ, ζ), r = (ξ, η).
The system has two obvious integrals -energy H and geometric integral γ 2 1 + γ 2 2 = 1. Since the system is Hamiltonian, for being integrable it requires one more additional integral (although this system can be integrated by the Euler-Jacobi method because the equations (12) conserve the standard invariant measure). Note that the system (12) has an important mechanical meaning, its form is as simple as that of the Euler-Poisson equations, and analogous formulations of problems are quite meaningful for this system. The first aspect relates to the integrability of the system (12).
Canonical variables. Using the algorithm described in [2], it is easy to construct canonical variables analogous to the Anduaye variables in the rigid body dynamics. In 4 Integrable cases.
The following integrable cases are known.
Additional quadratic integral has the form and the system (12) can be reduced, similar to the classical Euler-Poinsot case, to a system of three equations for variables M, p 1 , and p 2 . Chaplygin in [4] noted that explicit integration involves some very complicated quadrature, which is expressed in elliptic functions under the condition χ = ζ = 0.
2. ζ = χ = 0, a 1 = a 2 = 1 (V. V. Kozlov, 1993 [12]) -a case of dynamical symmetry. This case is analogous to the Lagrange case, although the additional integral is quadratic with respect to momenta Let us consider a new case of integrability.
3. Let us assume that χ = −λη, ζ = λξ (i. e. s = −λJr). Additional quadratic integral has the form Except the cases mentioned above . In this case, and at the level F = 0 we haveḞ = 0.

Chaplygin's case. Bifurcation analysis.
Let us consider the system (9) at µ = 0, ρ = χ = 0. As was noted above, equations for M, p 1 , p 2 can be separated and have the forṁ The common level of the first integrals is represented by closed curves formed at the intersection of an ellipsoid and an elliptic paraboloid; these curves are analogous to centroid lines in the classical Euler-Poinsot problem (figure 2). It is easy to find and describe the particular solutions of (18) which are analogous to permanent rotations in the latter problem.
For the sake of definiteness, let us assume that λ > 0 and a 1 > a 2 .
In this case, the body rotates uniformly (φ = c b = const) around the origin of the moving coordinate system O 1 (see figure 1), though this origin is fixed (and in the general case does not coincide with the center of mass). This solution is unstable when λb a 1 < c < λb a 2 and stable otherwise (the stability on the boundaries requires special analysis).
II, III. There exists a pair of analogous solutions of the form where in one case, i = 1, j = 2, while in the other case, i = 2, j = 1. Each of these solutions exists under condition c > bλ a j , and the constants of the integrals for them are related by In the case of such motion, the solid body uniformly rotates around the origin of a fixed coordinate system, and the principal axes of the body always pass through where d 2 calculated along asymptotic solutions of the unperturbed problem. Here F 0 is an integral of the unperturbed system.
If the integral (as a function of the parameter on the separatrix) has a simple zero, then separatrices split and transversally intersect. In the case of two degrees of freedom this makes the perturbed problem non-integrable [13].
We take the integral (19) as F 0 .
On homoclinic asymptotic solutions (22), the integral (23) can be explicitly eval-uated using residues: The integral (24) has a simple zero, except for the special case of B = 0. This allows us to conclude that the perturbed system (9) is non-integrable, except for the case of a 1 = a 2 . Note that it is under this condition that the integrals (14), (15) were obtained. This condition is necessary but not sufficient for the integrability.
The chaotic behavior of the system (9), associated with the splitting of separatrices and non-integrability, is illustrated in figure 4, which represents a Poincaré section of the phase flow of the system (12)  Let us discuss the issue of the possible fall of a body in the presence of circulation.
Recall that the body moving in the absence of circulation asymptotically tends to fall with its wider side down [9].
If λ = 0, the moving body will stay in a finite-width band parallel to the x axis.
Indeed, according to (10), we have We assume that p 1 , p 2 are limited and rewrite (11). The right-hand part of this equality is clearly a limited function at λ = 0, therefore p 1 , p 2 are also limited.
According to (25), the body moves in the horizontal direction with a mean velocity of − µ λ . This result was obtained in [15] for χ = ζ = 0.