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Discrete mechanics is presented as an alternative to the equations of fluid mechanics, in particular to the Navier-Stokes equation. The derivation of the discrete equation of motion is built from the intuitions of Galileo, the principles of…

This paper develops the approach to special relativity put forward by John S. Bell. The classical dynamics of an electron orbiting a nucleus in uniform motion is solved analytically and compared to numerical simulations for an accelerated…

We present the singular Euler--Maclaurin expansion, a new method for the efficient computation of large singular sums that appear in long-range interacting systems in condensed matter and quantum physics. In contrast to the traditional…

This is an annotated translation of Euler's 1777 manuscript "De Figura Quam Ventus Fluido Stagnanti Inducere Valet", where Euler models the shape which the wind gives to a still liquid surface.

We develop, via Arnold's geometric framework, a mechanism for constructing explicit, smooth, global-in-time, and typically non-stationary solutions of the incompressible Euler equations. The approach introduces a notion of generalized…

This paper proposes a new general methodology for finite-time singularity formation for moving interface problems involving the incompressible Euler equations in the plane. The first problem considered is the two-phase Euler vortex sheets…

We investigate the geometrically nonlinear deformation and buckling of a slender elastic beam subject to time-dependent `fictitious' (non-inertial) forces arising from unsteady rotation. Using a rotary apparatus that accurately imposes an…

In this work, we study the causality of near-wall inner and outer turbulent motions. The inner motions are defined as the self-sustained near-wall cycle, and the outer motions as those living in the logarithmic layer exhibiting footprints…

We show that the issue of the drag exerted by an incompressible fluid on a body in uniform motion has played a major role in the early development of fluid dynamics. In 1745 Euler came close, technically, to proving the vanishing of the…

We provide an exact infinite power series solution that describes the trajectory of a nonlinear simple pendulum undergoing librating and rotating motion for all time. Although the series coefficients were previously given in [V. Fair\'en,…

Soft robots, distinguished by their inherent compliance and continuum structures, present unique modeling challenges, especially when subjected to significant external loads such as gravity and payloads. In this study, we introduce an…

The classical irrotational capillary-gravity water wave problem described by the Euler equations with a nonlinear free surface boundary condition over a flat bed is considered. A modified flow force has been defined and a new formulation of…

Any nonspherical distribution of density inside planets and stars gives rise to a non-spherical external gravity and change of shape. If part or all of the observed zonal flows at the cloud deck of Jupiter and Saturn represent deep interior…

Einstein's special theory of relativity starts with assumptions about how observations conducted in relatively moving inertial frames must compare. From these assumptions, conclusions can be drawn regarding the laws of physics in any one…

We introduce a natural notion of incompressibility for fluids governed by the relativistic Euler equations on a fixed background spacetime, and show that the resulting equations reduce to the incompressible Euler equations in the classical…

Verlinde conjectured that gravitation is an emergent entropic force. This surprising conjecture was proved in [Physica A {\bf 505} (2018) 190] within a purely classical context. Here, we appeal to a quantum environment to deal with the…

In this paper, we study the existence and uniqueness of three dimensional steady Euler flows in rectangular nozzles when prescribing normal component of momentum at both the entrance and exit. If, in addition, the normal component of the…

A generalization of the Euler's elastic problem, i.e., finding a stationary configuration (planar elastica) of the Bernoulli's thin ideal elastic rod with boundary conditions defined through fixed endpoints and/or tangents at the endpoints,…

We present a semi-analytical correction to the seminal solution for the secular motion of a planet's orbit under gravitational influence of an external perturber derived by Heppenheimer (1978). A comparison between analytical predictions…

There is a remarkable and canonical problem in 3D geometry and topology: To understand existing models of 3D fluid motion or to create new ones that may be useful. We discuss from an algebraic viewpoint the PDE called Euler's equation for…