[Paper Review] Floating Bodies of Equilibrium in Three Dimensions. The central symmetric case

Franz Wegner|ArXiv.org|Mar 7, 2008

Mathematics and Applications9 references3 citations

TL;DR

This paper investigates three-dimensional, centrally symmetric bodies that float in equilibrium in any orientation, extending Ulam's problem beyond spheres. Using spherical harmonic expansions and nonlinear shape equations, it proves that for relative density $\rho = \frac{1}{2}$, non-spherical solutions exist if holes are allowed, and for $\rho \ne \frac{1}{2}$, deformed spheres with a large number of solutions emerge via a formal perturbative expansion, assuming no degenerate zeros of associated Legendre functions.

ABSTRACT

Three-dimensional central symmetric bodies different from spheres that can float in all orientations are considered. For relative density rho=1/2 there are solutions, if holes in the body are allowed. For rho different from 1/2 the body is deformed from a sphere. A set of nonlinear shape-equations determines the shape in lowest order in the deformation. It is shown that a large number of solutions exists. An expansion scheme is given, which allows a formal expansion in the deformation to arbitrary order under the assumption that apart from x=0,+1,-1 there is no x, which obeys P_{p,2}(x)=0 for two different integer ps, where P are Legendre functions.

Motivation & Objective

• To determine whether non-spherical, centrally symmetric three-dimensional bodies can float in indifferent equilibrium in all orientations, extending Ulam's problem beyond the sphere.
• To analyze the existence and structure of floating bodies of equilibrium for arbitrary relative densities $\rho \ne \frac{1}{2}$, beyond the known 2D case.
• To derive and solve nonlinear shape equations governing the deformation of such bodies from a sphere using spherical harmonic expansions.
• To establish a formal perturbative expansion scheme for the shape of floating bodies, valid under the condition that no $\theta_0$ satisfies $P_{p,-2}^{(\text{odd})}(\cos\theta_0) = 0$ for two different $p$.
• To explore the role of symmetry, particularly mirror and inversion symmetry, in shaping the solutions of the floating body problem.

Proposed method

• The shape of the floating body is expanded in spherical harmonics around a sphere, with deformation measured from the center of mass.
• The potential energy is minimized under the constraint of constant submerged and buoyant volumes, leading to conditions on the centers of mass lying on spheres.
• Nonlinear shape equations are derived in first and second order of deformation, involving associated Legendre functions and coefficients from Legendre polynomial expansions.
• The equations are analyzed under symmetry assumptions (e.g., rotational and mirror symmetry), restricting solutions to those invariant under subgroups of O(3).
• A formal expansion scheme is constructed for higher-order corrections, assuming no degeneracy in zeros of $P_{p,-2}(\cos\theta_0)$ across different $p$, ensuring solvability.
• Rodriguez’s formula and recursion relations for associated Legendre functions are used to compute coefficients in the expansion, particularly for $G^{(d/2-1)}_{2n}(x)$.

Experimental results

Research questions

• RQ1 Are there non-spherical, centrally symmetric three-dimensional bodies that float in indifferent equilibrium in all orientations for relative density $\rho = \frac{1}{2}$?
• RQ2 Can such floating bodies exist for $\rho \ne \frac{1}{2}$, and if so, what constraints govern their shape?
• RQ3 What is the structure of the nonlinear shape equations that determine the deformation of a sphere into a floating body of equilibrium?
• RQ4 How does the symmetry of the body—particularly mirror and inversion symmetry—affect the existence and form of solutions?
• RQ5 Can a formal perturbative expansion be constructed to determine the shape of such bodies to arbitrary order in deformation?

Key findings

• For $\rho = \frac{1}{2}$, non-spherical solutions exist if holes are allowed in the body, demonstrating that the sphere is not the only floating body of equilibrium in this case.
• For $\rho \ne \frac{1}{2}$, the body is deformed from a sphere, and the shape is governed by a set of nonlinear shape equations derived from spherical harmonic expansions.
• A large number of solutions to the shape equations are found under symmetry assumptions, particularly when the body is invariant under subgroups of O(3), leading to mirror-symmetric shapes.
• The second-order shape equations allow only deformations whose squared projection onto the same harmonic mode is proportional to the original deformation, restricting solutions to even $p$ for central symmetry.
• The formal expansion scheme is valid under the assumption that no $\theta_0 \in (-1,1)$ satisfies $P_{p,-2}(\cos\theta_0) = 0$ for two different $p$, ensuring non-degenerate solutions.
• Numerical checks suggest that the function $K_p(x)$, related to the denominator of the coefficient $\gamma$, is strictly positive for real $x$ up to $p=36$, supporting the regularity of the expansion.