Abstract
The present chapter is devoted to the investigation of the problem of motion of a rigid body by inertia in an ideal incompressible fluid, infinitely extending in all directions and at rest at infinity. Strictly speaking, this problem belongs to the field of fluid dynamics. The problem evolved namely in this way. The ordinary differential equations of motion of the solid are simultaneously solved with partial differential equation governing the motion of the liquid under boundary conditions satisfied on the surface of the moving solid. In this process, the pressure of the liquid had to be explicitly calculated at every point of the surface of the body. Nevertheless, after the study of some simple cases, and mainly in the works of Thomson and Tait [352] and of Kirchhoff [219], it became clear that the body and the liquid can be treated as forming together one dynamical system of six degrees of freedom, so that the detailed picture of the pressure of the fluid on the surface of the body is completely avoided. This system, composed of the body and liquid, was reduced to the motion of a rigid body with modified characteristics to compensate the motion of the liquid. When referred to a coordinate frame fixed in the body, the kinetic energy of this system is expressed as a quadratic form of the components of the angular and linear velocities of the body with constant coefficients. This step was decisive in the evolution of the subject along the next few decades.
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Notes
- 1.
A weaker type of equivalence will be treated below involves isomorphism on the level of Routh-reduced equations of motion. The full Lagrangian systems are not isomorphic to each other, but any integrable case of one of them leads to an integrable case of the other.
- 2.
In Euler’s case we have solved only the dynamical equations of motion and adopted a very special solution of Poisson’s equations in which the vectors \(\boldsymbol{\gamma }\) and \(\mathbf {G}\) are parallel. Hoever, the transformation applies equally well to the general solution of thewhole Euler-Poisson system.
- 3.
Here, periodicity relates only to the Euler-Poisson variables \(\boldsymbol{\omega ,\gamma .}\) The motion is periodic relative to the body system of axes. The motion can be periodic in space only under commensurability condition between the periods of the relative and the precessional motions.
- 4.
Here we mean the alternative problem of motion about a fixed point. In the Chaplygin problem, it corresponds to a steady translational motion of the body in the liquid.
- 5.
In fact, the condition \(\bar{\mathbf {b}}=\mathbf {0},\) is over-restrictive. The result holds when \(\bar{\mathbf {b}}\) is propertional to \(\bar{\mathbf {I}}= \mathbf {\frac{1}{2}\mathop {\mathrm {tr}}\nolimits }(\mathbf {I})\boldsymbol{\delta }-\mathbf {I=\frac{1}{2}\mathop {\mathrm {tr}}\nolimits }( \bar{\mathbf {a}}^{\mathbf {-1}} {)\boldsymbol{\delta } -}\bar{\mathbf {a}}^{-\mathbf {1}}{.}\) Compare with Case 2 of Table 10.1. The full form, consistent with that in Table 10.1, was given in [36].
- 6.
In fact, articles were rejected from publication in the Russian journal PMM J. Appl. Math. Mech. (See [200]). Namely, this rejection evoked the publication of the whole series of papers in “Mekh. Tverd. Tela”.
- 7.
The “main form” means the second representation, i.e. the one used by Kharlamov (See the last paragraph).
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Yehia, H.M. (2022). The Problem of Motion of a Body in a Liquid. In: Rigid Body Dynamics. Advances in Mechanics and Mathematics, vol 45. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-96336-1_10
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