相关论文: On the invariant formulation of fluid mechanics

We show that various functionals related to the supremum of a real function defined on an arbitrary set or a measure space are Hadamard directionally differentiable. We specifically consider the supremum norm, the supremum, the infimum, and…

A systematic exposition is given of the theory of invariant differential operators on a not necessarily reductive homogeneous space. This exposition is modelled on Helgason's treatment of the general reductive case and the special…

A methodology for deriving dual variational principles for the classical Newtonian mechanics of mass points in the presence of applied forces, interaction forces, and constraints, all with a general dependence on particle velocities and…

Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the…

We present a construction of a large class of Laplace invariants for linear hyperbolic partial differential operators of fairly general form and arbitrary order.

Steady incompressible potential flows of an inviscid or viscous fluid are considered in infinite N-dimensional cylinders with tangential boundary conditions. We show that such flows, if away from stagnation, are constant and parallel to the…

We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…

We present a variational formulation of electrodynamics using de Rham even and odd differential forms. Our formulation relies on a variational principle more complete than the Hamilton principle and thus leads to field equations with…

Fluidity, the ability of liquids to flow, is the key property distinguishing liquids from solids. This fluidity is set by the mobile transit atoms moving from one quasi-equilibrium point to the next. The nature of this transit motion is…

We give a formulation of classical mechanics in the language of operators acting on a Hilbert space. The formulation given comes from a unitary irreducible representation of the Galilei group that is compatible with the basic postulates of…

It has recently been proven that the generalised Cauchy fractional derivative (also known as the Riemann-Liouville fractional derivative) is equal to the Grunwald-Letnikov derivative. However, we observe that there are "Grunwald…

Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp. 499-505, Podvigina et al., J. Comput. Phys.,…

Minimal flow spaces of dimension 1 are among the most fundamental limit sets in dynamical systems. These invariant sets occur as the typical minimal sets in surface flows, the minimal sets of suspensions of subshifts (for example, in Lorenz…

In this paper the generalisation of previous author's formulation of fractional continuum mechanics to the case of anisotropic non-locality is presented. The considerations include the review of competitive formulations available in…

We consider invariant covariant derivatives on reductive homogeneous spaces corresponding to the well-known invariant affine connections. These invariant covariant derivatives are expressed in terms of horizontally lifted vector fields on…

In this article, I present a volume average regularization for the second functional derivative operator that appears in the metric-basis Wheeler-DeWitt equation. Naively, the second functional derivative operator in the Wheeler-DeWitt…

We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in three-dimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The…

Fractional variation is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. Fractional velocity can be suitable for characterizing singular behavior of derivatives…

We prove that a general class of nonlinear, non-autonomous ODEs in Fr\'echet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that…

We study operators that are generalizations of the classical Riemann-Liouville fractional integral, and of the Riemann-Liouville and Caputo fractional derivatives. A useful formula relating the generalized fractional derivatives is proved,…