Symplectic integrators for systems of rigid bodies

1996 ◽  
pp. 181-191 ◽  
Author(s):  
Sebastian Reich
Keyword(s):  
Rigid Bodies ◽  
Symplectic Integrators

Anatomy of the canonical transformation

1993 ◽  
Vol 345 (1677) ◽  
pp. 577-598 ◽  
Keyword(s):  
Fluid Dynamics ◽  
Canonical Transformation ◽  
Generating Functions ◽  
Hamiltonian Structure ◽  
Rigid Bodies ◽  
Symplectic Integrators ◽  
Classical Dynamics ◽  
Canonical Transformations ◽  
Geophysical Fluid Dynamics ◽  
Dynamics Of Particles

A concise account of the structure of the canonical transformation is given, in the lowest dimensional case. This case is chosen because it offers a special clarity in several respects. In particular, the diversity of possible generating functions is illustrated by m any examples which are not available elsewhere. Many of these are of physical interest, and some of them are multivalued. These examples are used to inform a comparative study of the several different definitions of a canonical transformation to be found in the literature. The paper is pertinent to all those branches of mechanics which can be given a hamiltonian representation. These include not only the classical dynamics of particles and rigid bodies, but also some more recent studies in continuum mechanics, including geophysical fluid dynamics. An area of particular modern interest is that of symplectic integrators. These are numerical integrating algorithms which generate a solution to Hamilton’s equations via a sequence of canonical transformations, which preserve the hamiltonian structure in the numerical solution.

The Dynamics of Coupled Planar Rigid Bodies. Part 2. Bifurcations, Periodic Solutions, and Chaos

1988 ◽  
Author(s):  
Y.-G. Oh ◽  
N. Sreenath ◽  
P. S. Krishnaprasad ◽  
J. E. Marsden
Keyword(s):  
Periodic Solutions ◽  
Rigid Bodies

Multiple Impacts in Rigid Bodies

2010 ◽  
Vol 14 (14) ◽  
pp. 1-14
Author(s):  
Mohamed Gharib ◽  
Yildirim Hurmuzlu
Keyword(s):  
Rigid Bodies ◽  
Multiple Impacts

Hamilton-Jacobi Theory

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Bbgky Hierarchy ◽  
Distribution Functions ◽  
Symplectic Integrators ◽  
Partition Functions ◽  
Von Neumann ◽  
Equipartition Theorem ◽  
Recurrence Theorem ◽  
Phase Amplitude ◽  
Canonical Ensembles

This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

Virtual Work & d’Alembert’s Principle

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Euler Equations ◽  
Virtual Work ◽  
Lagrange Equation ◽  
Classical Mechanics ◽  
Rigid Bodies ◽  
Gibbs Function ◽  
Constraint Force ◽  
Appell Equations ◽  
Jourdain's Principle ◽  
D'alembert's Principle

This chapter discusses virtual work, returning to the Newtonian framework to derive the central Lagrange equation, using d’Alembert’s principle. It starts off with a discussion of generalised force, applied force and constraint force. Holonomic constraints and non-holonomic constraint equations are then investigated. The corresponding principles of Gauss (Gauss’s least constraint) and Jourdain are also documented and compared to d’Alembert’s approach before being generalised into the Mangeron–Deleanu principle. Kane’s equations are derived from Jourdain’s principle. The chapter closes with a detailed covering of the Gibbs–Appell equations as the most general equations in classical mechanics. Their reduction to Hamilton’s principle is examined and they are used to derive the Euler equations for rigid bodies. The chapter also discusses Hertz’s least curvature, the Gibbs function and Euler equations.

Introductory Rotational Dynamics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Angular Momentum ◽  
Angular Displacement ◽  
Rigid Bodies ◽  
Rotational Dynamics ◽  
Circular Motion ◽  
Uniform Rotation ◽  
Chemical Systems ◽  
Inertial Frames ◽  
Rotating Frames ◽  
Special Case

This chapter discusses the importance of circular motion and rotations, whose applications to chemical systems are plentiful. Circular motion is the book’s first example of a special case of motion using the laws developed in previous chapters. The chapter begins with the basic definitions of circular motion; as uniform rotation around a principle axis is much easier to consider, it is the focus of this chapter and is used to develop some key ideas. The chapter discusses angular displacement, angular velocity, angular momentum, torque, rigid bodies, orbital and spin momenta, inertia tensors and non-inertial frames and explores fictitious forces as well as transformations in rotating frames.

scholarly journals Modelling of speleothems failure in the Hotton cave (Belgium). Is the failure earthquake induced?

2001 ◽  
Vol 80 (3-4) ◽  
pp. 315-321 ◽  
Author(s):  
J.F. Cadorin ◽  
D. Jongmans ◽  
A. Plumier ◽  
T. Camelbeeck ◽  
S. Delaby ◽  
...  
Keyword(s):  
Natural Frequencies ◽  
Quantitative Information ◽  
Rigid Bodies ◽  
Tensile Failure ◽  
Future Research ◽  
Geometrical Parameters ◽  
Laboratory Measurements ◽  
Ground Acceleration ◽  
Strong Ground

AbstractTo provide quantitative information on the ground acceleration necessary to break speleothems, laboratory measurements on samples of stalagmite have been performed to study their failure in bending. Due to their high natural frequencies, speleothems can be considered as rigid bodies to seismic strong ground motion. Using this simple hypothesis and the determined mechanical properties (a minimum value of 0.4 MPa for the tensile failure stress has been considered), modelling indicates that horizontal acceleration ranging from 0.3 m/s2 to 100 m/s2 (0.03 to 10g) are necessary to break 35 broken speleothems of the Hotton cave for which the geometrical parameters have been determined. Thus, at the present time, a strong discrepancy exists between the peak accelerations observed during earthquakes and most of the calculated values necessary to break speleothems. One of the future research efforts will be to understand the reasons of the defined behaviour. It appears fundamental to perform measurements on in situ speleothems.

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