Related papers: Two-dimensional Newton's Problem of Minimal Resist…
In the cosmos, any two bodies share a gravitational attraction. When in proximity to one another in empty space, their motions can be modeled by Newtonian gravity. Newton found their orbits when the two bodies are infinitely small, the…
We consider the motion of several rigid bodies immersed in a two-dimensional incompress-ible perfect fluid, the whole system being bounded by an external impermeable fixed boundary. The fluid motion is described by the incompressible Euler…
General properties of the three-body problem in a model of modified dynamics are investigated. It is shown that the three-body problem in this model shares some characters with the similar problem in Newtonian dynamics. Moreover, the planar…
The global existence of weak solutions to the three space dimensional Prandtl equations is studied under some constraint on its structure. This is a continuation of our recent study on the local existence of classical solutions with the…
Equations are derived for the shape of a hypersurface in $\mathbb{R}^N$ for which a rigid motion yields a minimal surface in $\mathbb{R}^{N+1}$. Some elementary, but unconventional, aspects of the classical case $N=2$ (solved by H.F. Scherk…
We outline two subjects of relativistic mechanics: (i) the set of allowable world lines, and (ii) the origin of the relativistic law of dynamics governing point particles. We show that: (i) allowable world lines in the classical theory of…
Several N-body problems in ordinary (3-dimensional) space are introduced which are characterized by Newtonian equations of motion (``acceleration equal force;'' in most cases, the forces are velocity-dependent) and are amenable to exact…
An equilibrium of a linear elastic body subject to loading and satisfying the friction and contact conditions can be described by a variational inequality of the second kind and the respective discrete model attains the form of a…
Small-scale plasticity problems are often characterised by different patterning behaviours ranging from macroscopic down to the atomistic scale. In successful models of such complex behaviour, its origin lies within non-convexity of the…
In 1934, Reinhardt conjectured that the shape of the centrally symmetric convex body in the plane whose densest lattice packing has the smallest density is a smoothed octagon. This conjecture is still open. We formulate the Reinhardt…
This work considers the problem of approximating initial condition and time-dependent optimal control and trajectory surfaces using multivariable Fourier series. A modified Augmented Lagrangian algorithm for translating the optimal control…
Infinite-dimensional linear conic formulations are described for nonlinear optimal control problems. The primal linear problem consists of finding occupation measures supported on optimal relaxed controlled trajectories, whereas the dual…
This article offers a broad-brush account of the Newtonian three-body problem, from its origins with Newton to its vibrant present, emphasizing its enduring influence on theoretical physics. It unfolds through a series of self-contained…
When a physical system is modeled by a nonlinear function, the unknown parameters can be estimated by fitting experimental observations by a least-squares approach. Newton's method and its variants are often used to solve problems of this…
A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such…
In his fondamental "Essay on the 3-body problem", Lagrange, well before Jacobi's "reduction of the node", carries out the first complete reduction of symetries. Discovering the so-called homographic motions, he shows that they necessarily…
We study global optimization of non-convex functions through optimal control theory. Our main result establishes that (quasi-)optimal trajectories of a discounted control problem converge globally and practically asymptotically to the set…
In 1775, Fagnano introduced the following geometric optimization problem: inscribe a triangle of minimal perimeter in a given acute-angled triangle. A widely accessible solution is provided by the Hungarian mathematician L. Fejer in 1900.…
In order to describe the velocity and the anaerobic energy of two runners competing against each other for middle-distance races, we present a mathematical model relying on an optimal control problem for a system of ordinary differential…
The 3-body problem in $\mathbb{R}^4$ has 24 dimensions and is invariant under translations and rotations. We do the full symplectic symmetry reduction and obtain a reduced Hamiltonian in local symplectic coordinates on a reduced phase space…