Causality of relativistic many-particle classical dynamics theories

1992 ◽  
Vol 70 (9) ◽  
pp. 772-781 ◽  
Author(s):  
R. A. Moore ◽  
D. W. Qi ◽  
T. C. Scott
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Mathematical Structure ◽  
Fundamental Difference ◽  
Classical Dynamics ◽  
Hamilton’S Principle ◽  
Relativistic Generalization ◽  
Hamilton's Principle ◽  
Isolated Systems

At the present time, the relativistic many-particle classical dynamics theory of the Fokker–Wheeler–Feynman type for isolated systems admits to two contradictory and mutually exclusive interpretations, noncausal versus causal. This work resolves this contradiction by demonstrating definitively that the mathematical structure of this theory is inconsistent with the assumption of noncausality. The demonstration is effected by identifying the fundamental difference in the input assumptions responsible for the two interpretations, by presenting general arguments proving the deterministic nature of the equations of motion, and by giving explicit examples that not only display the deterministic nature of the equations of motion but also their causal character, namely, being solvable by a stepwise forward in time integration using solely past information. One concludes, first, that the assumption of noncausality, and related deductions, is invalid and must be rejected and, second, that one has, at last, the relativistic generalization of many-particle nonrelativistic classical dynamics based on a Hamilton's principle and with a Lagrangian structure. Thus, one can sensibly proceed to examine the physical implications.

Modal Analysis for the Active Multi-Layer Beams by Using Spectrally Formulated Exact Eigenfunctions

1999 ◽  
Author(s):  
Usik Lee ◽  
Joohong Kim
Keyword(s):  
Modal Analysis ◽  
Closed Form ◽  
Dynamic Characteristics ◽  
Equations Of Motion ◽  
Hamilton’S Principle ◽  
Analysis Method ◽  
Hamilton's Principle ◽  
Numerical Examples ◽  
Coupled Equations ◽  
Fully Coupled

Abstract In this paper, a modal analysis method (MAM) is introduced for the active multi-layer laminate beams. Two types of active multi-layer laminate beams are considered: the elastic-viscoelastic-piezoelectric three-layer beams and the elastic-piezoelectric two-layer beams. The dynamics of the multi-layer laminate beams are represented by a set of fully coupled equations of motion, derived by using Hamilton’s principle. The exact eigenfunctions are spectrally formulated and the orthogonality of eigenfunctions is derived in a closed form. The present MAM is evaluated through some numerical examples. It is shown that the dynamic characteristics obtained by the present MAM certainly converge to the exact ones obtained by SEM as the number of eigenfunctions superposed in MAM is increased. The modal analysis results are also compared with the results obtained by FEM.

scholarly journals Hamilton-type principles applied to ice-sheet dynamics: new approximations for large-scale ice-sheet flow

2010 ◽  
Vol 56 (197) ◽  
pp. 497-513 ◽  
Author(s):  
J.N. Bassis
Keyword(s):  
Variational Principle ◽  
Large Scale ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Ice Sheet ◽  
Alternative Formulation ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Ice Sheet Dynamics ◽  
Large Scale Flow

AbstractIce-sheet modelers tend to be more familiar with the Newtonian, vectorial formulation of continuum mechanics, in which the motion of an ice sheet or glacier is determined by the balance of stresses acting on the ice at any instant in time. However, there is also an equivalent and alternative formulation of mechanics where the equations of motion are instead found by invoking a variational principle, often called Hamilton’s principle. In this study, we show that a slightly modified version of Hamilton’s principle can be used to derive the equations of ice-sheet motion. Moreover, Hamilton’s principle provides a pathway in which analytic and numeric approximations can be made directly to the variational principle using the Rayleigh–Ritz method. To this end, we use the Rayleigh–Ritz method to derive a variational principle describing the large-scale flow of ice sheets that stitches the shallow-ice and shallow-shelf approximations together. Numerical examples show that the approximation yields realistic steady-state ice-sheet configurations for a variety of basal tractions and sliding laws. Small parameter expansions show that the approximation reduces to the appropriate asymptotic limits of shallow ice and shallow stream for large and small values of the basal traction number.

Unified Approach for Holonomic and Nonholonomic Systems Based on the Modified Hamilton’s Principle

2009 ◽  
Author(s):  
Keisuke Kamiya ◽  
Junya Morita ◽  
Yutaka Mizoguchi ◽  
Tatsuya Matsunaga
Keyword(s):  
Equations Of Motion ◽  
Time Derivative ◽  
Nonholonomic Systems ◽  
Hamilton’S Principle ◽  
Virtual Power ◽  
Unified Approach ◽  
Hamilton's Principle ◽  
Basic Principles ◽  
Principle Of Virtual Power ◽  
Holonomic And Nonholonomic Systems

As basic principles for deriving the equations of motion for dynamical systems, there are d’Alembert’s principle and the principle of virtual power. From the former Hamilton’s principle and Langage’s equations are derived, which are powerful tool for deriving the equation of motion of mechanical systems since they can give the equations of motion from the scalar energy quantities. When Hamilton’s principle is applied to nonholonomic systems, however, care has to be taken. In this paper, a unified approach for holonomic and nonholonomic systems is discussed based on the modified Hamilton’s principle. In the present approach, constraints for both of the holonomic and nonholonomic systems are expressed in terms of time derivative of the position, and their variations are treated similarly to the principle of virtual power, i.e. time and position are fixed in operation with respect to the variations. The approach is applied to a holonomic and a simple nonholonomic systems.

The Equations of Motion for the Torsional and Bending Vibrations of a Stranded Cable

1992 ◽  
Vol 59 (2S) ◽  
pp. S224-S229 ◽  
Author(s):  
Warren N. White ◽  
Srinivasan Venkatasubramanian ◽  
P. Michael Lynch ◽  
Chi-Lung D. Huang
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Hamilton’S Principle ◽  
Attached Mass ◽  
Hamilton's Principle ◽  
Bending Vibrations ◽  
Aerodynamic Loading ◽  
Simplifying Assumptions ◽  
Rotational Degrees Of Freedom

Equations of motion of a thin, stranded elastic cable with an eccentric, attached mass and subject to aerodynamic loading are derived using Hamilton’s principle. Coupling between the translational and rotational degrees of freedom owing to inertia, elasticity, and stranded geometry are considered. By invoking simplifying assumptions, the equations of motion are reduced to those obtained previously by other researchers.

Space-Discretized Verlet-Algorithm from a Variational Principle

1997 ◽  
Vol 52 (8-9) ◽  
pp. 585-587
Author(s):  
Walter Nadler ◽  
Hans H. Diebner ◽  
Otto E. Rössler
Keyword(s):  
Variational Principle ◽  
Digital Computer ◽  
Equations Of Motion ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Space And Time ◽  
Space Discretization

Abstract A form of the Verlet-algorithm for the integration of Newton’s equations of motion is derived from Hamilton's principle in discretized space and time. It allows the computation of exactly time-reversible trajectories on a digital computer, offers the possibility of systematically investigating the effects of space discretization, and provides a criterion as to when a trajectory ceases to be physical.

Free vibration analysis of a porous rotor integrated with regular patterns of circumferentially distributed functionally graded piezoelectric patches on inner and outer surfaces

2020 ◽  
Vol 32 (1) ◽  
pp. 82-103
Author(s):  
Yaser Heidari ◽  
Mohsen Irani Rahaghi ◽  
Mohammad Arefi
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
External Work ◽  
First Order ◽  
Functionally Graded Piezoelectric

This article studies dynamic characteristics of a novel porous cylindrical hollow rotor based on the first-order shear deformation theory and Hamilton’s principle. The proposed model is made from a core including aluminum with porosity integrated with an arrangement of functionally graded piezoelectric patches placed on its inner and outer surfaces with a customized circumferential orientation. The piezoelectric patches are subjected to applied electric potential as sensor and actuator. The kinematic relations are developed based on the first-order shear deformation theory. Hamilton’s principle is used to derive governing equations of motion with calculation of strain and kinetic energies and external work. Solution procedure of the partial differential equations of motion is developed using Galerkin technique for simple boundary conditions. The accuracy and trueness of this work is justified using a comprehensive comparison with previous valid references. A large parametric study is presented to show influence of significant parameters such as dimensionless geometric parameters, porosity coefficient, angular speed, inhomogeneous index, and characteristics of patches on the mode shapes, natural frequencies, and critical speeds of the structure.

Modeling of Nonlinear Oscillations for Viscoelastic Moving Belt Using Generalized Hamilton’s Principle

2006 ◽  
Vol 129 (1) ◽  
pp. 128-132 ◽  
Author(s):  
L. H. Chen ◽  
W. Zhang ◽  
Y. Q. Liu
Keyword(s):  
Nonlinear Oscillations ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Elastic Strain Energy ◽  
Plane Motion ◽  
Hamilton’S Principle ◽  
Axial Deformation ◽  
Governing Equations ◽  
Hamilton's Principle

In this paper, the nonlinear governing equations of motion for viscoelastic moving belt are established by using the generalized Hamilton’s principle for the first time. Two kinds of viscoelastic constitutive laws are adopted to describe the relation between the stress and strain for viscoelastic materials. Moreover, the correct forms of elastic strain energy, kinetic energy, and the virtual work performed by both external and viscous dissipative forces are given for the viscoelastic moving belt. Using the generalized Hamilton’s principle, the nonlinear governing equations of three-dimensional motion are established for the viscoelastic moving belt. Neglecting the axial deformation, the governing equations of in-plane motion and transverse nonlinear oscillations are also derived for the viscoelastic moving belt. Comparing the nonlinear governing equations of motion obtained here with those obtained by using the Newton’s second law, it is observed that the former completely agree with the latter.

Hamilton's Principle for Material and Nonmaterial Control Volumes Using Lagrangian and Eulerian Description of Motion

2019 ◽  
Vol 71 (1) ◽  
Author(s):  
Andreas Steinboeck ◽  
Martin Saxinger ◽  
Andreas Kugi
Keyword(s):  
Particle Motion ◽  
Equations Of Motion ◽  
Theoretical Perspective ◽  
Standard Form ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Virtual Displacement ◽  
Eulerian Description ◽  
Axially Moving ◽  
Control Volumes

The standard form of Hamilton's principle is only applicable to material control volumes. There exist specialized formulations of Hamilton's principle that are tailored to nonmaterial (open) control volumes. In case of continuous mechanical systems, these formulations contain extra terms for the virtual shift of kinetic energy and the net transport of a product of the virtual displacement and the momentum across the system boundaries. This raises the theoretically and practically relevant question whether there is also a virtual shift of potential energy across the boundary of open systems. To answer this question from a theoretical perspective, we derive various formulations of Hamilton's principle applicable to material and nonmaterial control volumes. We explore the roots and consequences of (virtual) transport terms if nonmaterial control volumes are considered and show that these transport terms can be derived by Reynolds transport theorem. The equations are deduced for both the Lagrangian and the Eulerian description of the particle motion. This reveals that the (virtual) transport terms have a different form depending on the respective description of the particle motion. To demonstrate the practical relevance of these results, we solve an example problem where the obtained formulations of Hamilton's principle are used to deduce the equations of motion of an axially moving elastic tension bar.

Derivation of First-Order Difference Equations for Dynamical Systems by Direct Application of Hamilton’s Principle

1970 ◽  
Vol 37 (2) ◽  
pp. 276-278 ◽  
Author(s):  
J. M. Vance ◽  
A. Sitchin
Keyword(s):  
Difference Equations ◽  
Equations Of Motion ◽  
Lagrangian Multiplier ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Order Difference ◽  
First Order ◽  
Finite Difference Equations ◽  
Lagrangian Multiplier Method ◽  
Dynamics Problems

In dynamics problems where the equations of motion are eventually reduced to finite-difference equations for numerical integration on a digital computer, an auxiliary condition exists that permits the application of the Lagrangian multiplier method to Hamilton’s principle in order to obtain directly a set of first-order difference equations. These equations are equivalent to Hamilton’s canonical equations and are derived without the necessity to obtain the Hamiltonian or take time derivatives.

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