On the Lagrangian Method for Steady and Unsteady Flow

A new and general Lagrangian formulation of fluid motion is given in which the independent variables are three material functions and a Lagrangian time, which differs for different fluid particles and is distinct from the Eulerian time. For steady flow it requires only three independent variables - the Lagrangian time and two stream functions - in contrast with the conventional Lagrangian formulation which apparently still requires four independent variables for describing a steady flow. This places the Lagrangian formulation for steady flow on the same footing as the Eulerian. For unsteady flow, the new formulation includes the conventional formulation as a special case when the Lagrangian time is identified with the Eulerian time and when the material functions are taken to be the fluid particle's position at some given time. The distinction between the Lagrangian and Eulerian time, however, is found useful in applications, e.g., to problems involving a free boundary

On Lagrangian and Hamiltonian systems with homogeneous trajectories

Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmetry group G is a solution of the Euler-Lagrange or Hamiltonian equations. In particular, we generalize the `geodesic lemma' known in Riemannian geometry to Lagrangian and Hamiltonian systems. We present results on the existence of homogeneous trajectories of Lagrangian systems. We study Hamiltonian and Lagrangian g.o. spaces, i.e. homogeneous spaces G/H with G-invariant Lagrangian or Hamiltonian functions on which every solution of the equations of motion is homogeneous. We show that the Hamiltonian g.o. spaces are related to the functions that are invariant under the coadjoint action of G. Riemannian g.o. spaces thus correspond to special Ad*(G)-invariant functions. An Ad*(G)-invariant function that is related to a g.o. space also serves as a potential for the mapping called `geodesic graph'. As illustration we discuss the Riemannian g.o. metrics on SU(3)/SU(2).Comment: v3: some misprints correcte

Augmented Lagrangian Functions for Cone Constrained Optimization: the Existence of Global Saddle Points and Exact Penalty Property

In the article we present a general theory of augmented Lagrangian functions for cone constrained optimization problems that allows one to study almost all known augmented Lagrangians for cone constrained programs within a unified framework. We develop a new general method for proving the existence of global saddle points of augmented Lagrangian functions, called the localization principle. The localization principle unifies, generalizes and sharpens most of the known results on existence of global saddle points, and, in essence, reduces the problem of the existence of saddle points to a local analysis of optimality conditions. With the use of the localization principle we obtain first necessary and sufficient conditions for the existence of a global saddle point of an augmented Lagrangian for cone constrained minimax problems via both second and first order optimality conditions. In the second part of the paper, we present a general approach to the construction of globally exact augmented Lagrangian functions. The general approach developed in this paper allowed us not only to sharpen most of the existing results on globally exact augmented Lagrangians, but also to construct first globally exact augmented Lagrangian functions for equality constrained optimization problems, for nonlinear second order cone programs and for nonlinear semidefinite programs. These globally exact augmented Lagrangians can be utilized in order to design new superlinearly (or even quadratically) convergent optimization methods for cone constrained optimization problems.Comment: This is a preprint of an article published by Springer in Journal of Global Optimization (2018). The final authenticated version is available online at: http://dx.doi.org/10.1007/s10898-017-0603-

p-Brane Solutions in Diverse Dimensions

A generic Lagrangian, in arbitrary spacetime dimension, describing the interaction of a graviton, a dilaton and two antisymmetric tensors is considered. An isotropic p-brane solution consisting of three blocks and depending on four parameters in the Lagrangian and two arbitrary harmonic functions is obtained. For specific values of parameters in the Lagrangian the solution may be identified with previously known superstring solutions.Comment: 15 pages, latex, no figure

Statistical properties of supersonic turbulence in the Lagrangian and Eulerian frameworks

We present a systematic study of the influence of different forcing types on the statistical properties of supersonic, isothermal turbulence in both the Lagrangian and Eulerian frameworks. We analyse a series of high-resolution, hydrodynamical grid simulations with Lagrangian tracer particles and examine the effects of solenoidal (divergence-free) and compressive (curl-free) forcing on structure functions, their scaling exponents, and the probability density functions of the gas density and velocity increments. Compressively driven simulations show a significantly larger density contrast, a more intermittent behaviour, and larger fractal dimension of the most dissipative structures at the same root mean square Mach number. We show that the absolute values of Lagrangian and Eulerian structure functions of all orders in the integral range are only a function of the root mean square Mach number, but independent of the forcing. With the assumption of a Gaussian distribution for the probability density function of the velocity increments on large scales, we derive a model that describes this behaviour.Comment: 24 pages, 13 figures, Journal of Fluid Mechanics in pres

Turbulence for (and by) amateurs

Series of lectures on statistical turbulence written for amateurs but not experts. Elementary aspects and problems of turbulence in two and three dimensional Navier-Stokes equation are introduced. A few properties of scalar turbulence and transport phenomena in turbulent flows are described. Kraichnan's model of passive advection is discussed a bit more precisely. {Part 1: Approaching turbulent flows.} Navier-Stokes equation. Cascades and Kolmogorov theory. Modeling statistical turbulence. Correlation functions and scaling. {Part 2: Deeper in turbulent flows.} Turbulence in two dimensions. Dissipation and dissipative anomalies. Fokker-Planck equations. Multifractal models. {Part 3: Scalar turbulence.} Transport and Lagrangian trajectories. Kraichnan's passive scalar model. Anomalous scalings and universality. {Part 4: Lagrangian trajectories.} Richardson's law. Lagrangian flows in Kraichnan's model. Slow modes. Breakdown of Lagrangian flows. Batchelor limit. Generalized Lagrangian flows and trajectory bundles.Comment: 37 pages, 6 figures, lecture note