Statistical derivation of the equations of motion of second-order liquids

1971 ◽  
Vol 4 (2) ◽  
pp. 812-816
Author(s):  
V. A. Savchenko ◽  
T. N. Khazanovich
Keyword(s):  
Equations Of Motion ◽  
Second Order ◽  
Statistical Derivation

scholarly journals Higher order Hamiltonian Monte Carlo sampling for cosmological large-scale structure analysis

2021 ◽  
Vol 502 (3) ◽  
pp. 3976-3992
Author(s):  
Mónica Hernández-Sánchez ◽  
Francisco-Shu Kitaura ◽  
Metin Ata ◽  
Claudio Dalla Vecchia
Keyword(s):  
Fourth Order ◽  
Large Scale ◽  
Equations Of Motion ◽  
Large Scale Structure ◽  
Real Space ◽  
Higher Order ◽  
Second Order ◽  
Scale Structure ◽  
Integration Schemes ◽  
Reconstruction Methods

ABSTRACT We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 h−1 Mpc side and 2563 cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth order in the leap-frog scheme shortens the burn-in phase by a factor of at least ∼30. This implies that 75–90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 2563 cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only when considering lower dimensional problems, e.g. meshes with 643 cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.

A Second Order Lagrangian Model for Irregular Ocean Waves

2005 ◽  
Vol 128 (3) ◽  
pp. 177-183 ◽  
Author(s):  
Sébastien Fouques ◽  
Harald E. Krogstad ◽  
Dag Myrhaug
Keyword(s):  
Specular Reflection ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Ocean Waves ◽  
Second Order ◽  
Lagrangian Model ◽  
Lagrangian Description ◽  
Linear Wave ◽  
Irregular Solution

Synthetic aperture radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, slope, and mean curvature are studied.

Towards a Technique for Nonlinear Modal Analysis

2012 ◽  
Author(s):  
Simon A. Neild ◽  
Andrea Cammarano ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Transformation Process ◽  
Second Order ◽  
Identity Transformation ◽  
Modal Testing ◽  
System Response ◽  
Weakly Nonlinear ◽  
Nonlinear Modal Analysis

In this paper we discuss a theoretical technique for decomposing multi-degree-of-freedom weakly nonlinear systems into a simpler form — an approach which has parallels with the well know method for linear modal analysis. The key outcome is that the system resonances, both linear and nonlinear are revealed by the transformation process. For each resonance, parameters can be obtained which characterise the backbone curves, and higher harmonic components of the response. The underlying mathematical technique is based on a near identity normal form transformation. This is an established technique for analysing weakly nonlinear vibrating systems, but in this approach we use a variation of the method for systems of equations written in second-order form. This is a much more natural approach for structural dynamics where the governing equations of motion are written in this form as standard practice. In fact the first step in the method is to carry out a linear modal transformation using linear modes as would typically done for a linear system. The near identity transform is then applied as a second step in the process and one which identifies the nonlinear resonances in the system being considered. For an example system with cubic nonlinearities, we show how the resulting transformed equations can be used to obtain a time independent representation of the system response. We will discuss how the analysis can be carried out with applied forcing, and how the approximations about response frequencies, made during the near-identity transformation, affect the accuracy of the technique. In fact we show that the second-order normal form approach can actually improve the predictions of sub- and super-harmonic responses. Finally we comment on how this theoretical technique could be used as part of a modal testing approach in future work.

scholarly journals Modified Gravity Models Admitting Second Order Equations of Motion

Entropy ◽  
2015 ◽  
Vol 17 (12) ◽  
pp. 6643-6662 ◽  
Author(s):  
Aimeric Colléaux ◽  
Sergio Zerbini
Keyword(s):  
Modified Gravity ◽  
Equations Of Motion ◽  
Second Order ◽  
Gravity Models ◽  
Second Order Equations ◽  
Modified Gravity Models

The Selection of the Linearized Natural Frequency for the Second-Order Normal Form Method

2011 ◽  
Author(s):  
Zhenfang Xin ◽  
S. A. Neild ◽  
D. J. Wagg
Keyword(s):  
Normal Form ◽  
Dynamic Response ◽  
Natural Frequency ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Second Order ◽  
Nonlinear Dynamic Response ◽  
Weakly Nonlinear ◽  
Selection Of ◽  
Normal Form Technique

The normal form technique is an established method for analysing weakly nonlinear vibrating systems. It involves applying a simplifying nonlinear transform to the first-order representation of the equations of motion. In this paper we consider the normal form technique applied to a forced nonlinear system with the equations of motion expressed in second-order form. Specifically we consider the selection of the linearised natural frequencies on the accuracy of the normal form prediction of sub- and superharmonic responses. Using the second-order formulation offers specific advantages in terms of modeling lightly damped nonlinear dynamic response. In the second-order version of the normal form, one of the approximations made during the process is to assume that the linear natural frequency for each mode may be replaced with the response frequencies. Here we will show that this step, far from reducing the accuracy of the technique, does not affect the accuracy of the predicted response at the forcing frequency and actually improves the predictions of sub and superharmonic responses. To gain insight into why this is the case, we consider the Duffing oscillator. The results show that the second-order approach gives an intuitive model of the nonlinear dynamic response which can be applied to engineering applications with weakly nonlinear characteristics.

Evaluation of Parametric Vibration and Stability of Flexible Cam-Follower Systems

1994 ◽  
Vol 116 (1) ◽  
pp. 291-297 ◽  
Author(s):  
A. I. Mahyuddin ◽  
A. Midha ◽  
A. K. Bajaj
Keyword(s):  
Equations Of Motion ◽  
Second Order ◽  
Valve Train ◽  
Time Dependent ◽  
Equivalent Model ◽  
System Response ◽  
Parametric Stability ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Cam Follower

A method to study parametric stability of flexible cam-follower systems is developed. This method is applied to an automotive valve train which is modeled as a single-degree-of-freedom vibration system. The inclusion of the transverse and rotational flexibilities of the camshaft results in a system governed by a second-order, linear, ordinary differential equation with time-dependent coefficients. This class of equations, known as Hill’s equations, merits special notice in determination of the system response and stability. The analysis includes development of the equivalent model of the system, derivation of its equation of motion, and a method to evaluate its parametric stability based on Floquet theory. A closed-form numerical algorithm, developed to compute the periodic response of systems governed by second-order, linear, ordinary differential equations of motion with time-dependent coefficients, is utilized. The results of this study are presented in a companion paper in the forms of parametric stability charts and three-dimensional stability and response charts.

NONHOLONOMIC DYNAMICS OF SECOND ORDER AND THE HEISENBERG SPINNING PARTICLE

2012 ◽  
Vol 09 (07) ◽  
pp. 1220010
Author(s):  
MIRCEA CRASMAREANU ◽  
IULIAN STOLERIU
Keyword(s):  
Equations Of Motion ◽  
Second Order ◽  
Spinning Particle ◽  
Nonholonomic Dynamics

The equations of motion for the associated constrained Lagrangian to a nonholonomic Lagrangian of second order are computed. The spinning particle subject to the Heisenberg constraint is treated as example and its dynamics is completely described.

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