Discrete mechanics—A general treatment

LaBudde, Robert A; Greenspan, Donald

doi:10.1016/0021-9991(74)90081-3

Cited by 75 publications

(41 citation statements)

References 8 publications

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“…This indicates that our restriction of the general constraint (13) to the condition (14) merely ensures that the modal energies evolve in a manner consistent with the Euler discretization of the energy equations. Below we will give derivations of integrators for other systems based on this idea.…”

Section: Discussionmentioning

confidence: 91%

Exactly conservative integrators

Shadwick¹,

Bowman²,

Morrison³

1995

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Traditional numerical discretizations of conservative systems generically yield an arti cial secular drift of any nonlinear invariants. In this work we present an explicit nontraditional algorithm that exactly conserves these invariants. We illustrate the general method by applying it to the threewave truncation of the Euler equations, the Lotka{Volterra predator{prey model, and the Kepler problem. This method is discussed in the context of symplectic (phase space conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.

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Section: Discussionmentioning

confidence: 91%

Exactly conservative integrators

Shadwick¹,

Bowman²,

Morrison³

1995

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“…Greenspan [46,67] appears to be one of the first to stress the importance that conserved quantities play a crucial role in developing time stepping algorithms. The conserved quantities include the linear and angular momenta, and the total energy.…”

Section: Energy-momentum Conserving Scheme For Discrete Systemsmentioning

confidence: 99%

An Overview and Recent Advances in Vector and Scalar Formalisms: Space/Time Discretizations in Computational Dynamics—A Unified Approach

Tamma

Har

Zhou

et al. 2011

Arch Computat Methods Eng

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An overview of recent advances in computational dynamics for modeling and simulation is described. The targeted objectives are towards a wide variety of science and engineering applications in particle and continuum dynamics of structures and materials which fall in this class. Starting with the supposition that in the beginning, the well known Newton's law of motion for N-body systems is given, and is a statement of the principle of balance of linear momentum, subsequently, using this as a landmark, firstly, the principal relations to various other distinctly different fundamental principles which are of primary interest here are established. Likewise, for continuum dynamics of structures, via the principle of balance of linear momentum, analogous developments are also established. Consequently, these distinctly different fundamental principles are shown to serve as the starting point for the various developments. Stemming from three distinctly different fundamental principles, we present recent advances in N-body dynamical systems, and also continuous-body dynamical systems with focus on numerical aspects in space/time discretization. The fundamental principles are the following:the Principle of Virtual Work in Dynamics, Hamilton's Principle and as an alternate (due to inconsistencies associated with Hamilton's principle), Hamilton's Law of Varying Action, and the Principle of Balance of Mechanical Energy. Both vector and scalar formalisms are described in detail with particular focus towards general numerical discretizations in space and/or time for N-body and continuumelastodynamics applications which are encountered in a wide class of holonomic-scleronomic problems. The formulations include the classical Newtonian mechanics framework with vector formalism, and new scalar formalisms with descriptive functions such as the Lagrangian, the Hamiltonian, and the Total Mechanical Energy to readily enable numerical discretizations. The concepts emanating from the present developments and distinctly different fundamental principles inherently: (1) can independently be shown to yield the strong form of the governing mathematical model equations of motion that are continuous in space and/or time together with the natural boundary conditions; the various frameworks for the case of holonomic-scleronomic systems with the mentioned limitations are indeed all equivalent, (2) can explain naturally how the weak statement of the Bubnov-Galerkin weighted-residual form that is customarily employed for discretization with vector formalism arises for both space and time, and (3) can circumvent relying upon traditional practices of conducting numerical discretizations starting either from balance of linear momentum (Newton's second law) involving Cauchy's equations of motion (governing equations) arising from continuum mechanics or via (1) and (2) above if one chooses this option. The concepts instead provide new avenues for numerical space/time discretization for continuum-dynamical systems or time discretization for N-body system...

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“…The current availability of high-speed computer technology has motivated the study of compatible, arithmetic models (Cadzow, 1970;Greenspan, 1974;Mehta, 1967;Miller et al, 1972;Pasta and Ulam, 1959;LaBudde and Greenspan, 1974). Recently, for example (Greenspan, 1974s;LaBudde and Greenspan, 1974), it has been shown that symmetry and all the conservation laws of Newtonian dynamics have an arithmetic basis.…”

Section: Introductionmentioning

confidence: 98%

“…Recently, for example (Greenspan, 1974s;LaBudde and Greenspan, 1974), it has been shown that symmetry and all the conservation laws of Newtonian dynamics have an arithmetic basis. The aim of the present paper is to show that special relativity also has an arithmetic basis in that symmetry, conservation of linear momentum, conservation of energy, the Einstein rest energy equation, and the direct relationship between the displacement vector and the momentum-energy vector can all be deduced using only arithmetic formulas for basic physical quantities.…”

Section: Introductionmentioning

confidence: 99%

The arithmetic basis of special relativity

Greenspan

1976

Int J Theor Phys

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Under relatively general particle and rocket frame motions, it is shown that, for special relativity, the basic concepts can be formulated and the basic properties deduced using only arithmetic. Particular attention is directed toward velocity, acceleration, proper time, momentum, energy, and 4-vectors in both space-time and Minkowski space, and to relativistic generalizations of Newton's second law. The resulting mathematical simplification is not only completely compatible with modern computer technology, but it yields dynamical equations that can be solved directly by such computers. Particular applications of the numerical equations, which are either Lorentz invariant or are directly related to Lorentz-invariant formulas, are made to the study of a relativistic harmonic oscillator and to the motion of an electric particle in a magnetic field.

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Discrete mechanics—A general treatment

Cited by 75 publications

References 8 publications

Exactly conservative integrators

Exactly conservative integrators

An Overview and Recent Advances in Vector and Scalar Formalisms: Space/Time Discretizations in Computational Dynamics—A Unified Approach

The arithmetic basis of special relativity

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