On the Reciprocal Form of Hamilton’s Principle

1973 ◽  
Vol 40 (1) ◽  
pp. 93-100 ◽  
Author(s):  
Z. M. Elias
Keyword(s):  
Variational Principle ◽  
Continuous Systems ◽  
Sufficient Condition ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Energy Principle ◽  
Spurious Solutions ◽  
Complementary Energy Principle ◽  
Necessary And Sufficient ◽  
Geometric Compatibility

A complementary energy principle for dynamic analysis due to Toupin is critically examined. It is found that the variational principle is a necessary but not a sufficient condition for geometric compatibility and that consequently it allows the occurrence of spurious solutions. A necessary and sufficient condition of compatibility is obtained through the reciprocal form of Hamilton’s principle which is derived for discrete and continuous systems. Additional terms appearing in the derived principle insure that spurious solutions cannot occur. The derived variational principle can be expressed in terms of stresses and velocities or in terms of impulses.

Application of Hamilton’s Principle to Large Deformation and Flow Problems

1979 ◽  
Vol 46 (2) ◽  
pp. 285-290 ◽  
Author(s):  
S. Dost ◽  
B. Tabarrok
Keyword(s):  
Large Deformation ◽  
Equations Of Motion ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Complementary Energy ◽  
Energy Principle ◽  
Flow Problems ◽  
Side Constraints ◽  
Complementary Energy Principle ◽  
Stationary Conditions

Application of Hamilton’s principle to large deformation and flow problems is examined with an emphasis on derivation of the Eulerian equations of motion as stationary conditions of the functional. It is shown that Eulerian variations can be admitted in Hamilton’s functional providing certain side constraints are imposed. Finally the Complementary Energy principle is briefly examined and difficulties in admitting Eulerian variations in this principle are discussed.

Forces of reaction and neighbouring Hamilton's principle in the tracking control of manipulators via a sliding scheme

Robotica ◽  
1993 ◽  
Vol 11 (3) ◽  
pp. 227-232 ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
Variational Principle ◽  
Tracking Control ◽  
Taylor Expansion ◽  
Perturbation Approach ◽  
Hamilton’S Principle ◽  
Sliding Surface ◽  
Hamilton's Principle ◽  
Control Effort ◽  
Sliding Surfaces ◽  
Control Schemes

SUMMARYIt is shown that if one comes back to the formulation of the Hamilton's variational principle, it is then possible to obtain new viewpoints on the tracking control of robot manipulators. First, the Lagrange multiplier associated to the sliding surface can be interpretated in terms of control effort and/or forces of reaction of the mechanical system. Secondly, one can use the Taylor expansion of the mechanical Lagrangian, combined with a neighbour- ing Hamilton's principle, to obtain control schemes via sliding surfaces. Thirdly, a perturbation approach combined with the neighbouring Hamilton's principle provides results on the robustness of the control.

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.

Encounterless stability of stellar systems

1966 ◽  
Vol 25 ◽  
pp. 78-89
Author(s):  
D Lynden-Bell
Keyword(s):  
Density Distribution ◽  
Gas Dynamics ◽  
Stellar System ◽  
Stellar Systems ◽  
Sufficient Condition ◽  
Energy Principle ◽  
The Stability ◽  
Necessary And Sufficient

An energy principle which gives a sufficient condition for the stability of a stellar system is derived. There is a similar energy principle in self-gravitating gas dynamics which is necessary and sufficient. Comparison enables one to infer the stability of a stellar system from that of a gaseous system with the same density distribution.

Hamilton’s Principle for External Viscous Fluid-Structure Interaction

2000 ◽  
Author(s):  
Haym Benaroya ◽  
Timothy Wei
Keyword(s):  
Variational Principle ◽  
Equation Of Motion ◽  
External Flow ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Structure Interaction ◽  
Future Paper ◽  
The Subject ◽  
End Times ◽  
Transport Theorem

Abstract In this paper, Hamilton’s principle is extended so as to be able to model external flow-structure interaction. This is accomplished by using Reynold’s Transport theorem. In this form, Hamilton’s principle is hybrid in the sense that it has an analytical part as well as a part that depends on experimentally derived functions. Examples are presented. A discussion on implications and extensions is extensive. In this work, the general theory is developed for the case where the configuration is not prescribed at the end times of the variational principle. This leads to a single governing equation of motion. This limitation can be removed by prescribing the end times, as is usual. This is outlined in the present paper, and will be the subject of a future paper.

Some Problems in the Theory of Nonconservative Elasticity

1986 ◽  
Vol 10 (1) ◽  
pp. 28-33
Author(s):  
Qi-hao Zhang ◽  
Dian-kui Liu
Keyword(s):  
Finite Element Analysis ◽  
Variational Principle ◽  
Variational Principles ◽  
Theory Of Elasticity ◽  
Complementary Energy ◽  
Element Analysis ◽  
Energy Principle ◽  
Nonconservative Systems ◽  
Potential Energy Principle ◽  
Complementary Energy Principle

This study develops the general quasi-variational principles for nonconservative problems in the theory of elasticity such as the quasi-potential energy principle, the quasi-complementary energy principle, the generalized quasi-variational principle and quasi-Hamilton principle. The application of these quasi-variational principles to finite element analysis is also discussed and illustrated with some examples. The total variational principle for nonconservative systems of two variables is also studied.

Hamiltonian Fluid Dynamics

1998 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Variational Principle ◽  
Conservation Laws ◽  
Variational Principles ◽  
Symmetry And Conservation Laws ◽  
The Body ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Symmetry Properties

In this final chapter, we return to the subject of the first: the fundamental principles of fluid mechanics. In chapter 1, we derived the equations of fluid motion from Hamilton’s principle of stationary action, emphasizing its logical simplicity and the resulting close correspondence between mechanics and thermodynamics. Now we explore the Hamiltonian approach more fully, discovering its other advantages. The most important of these advantages arise from the correspondence between the symmetry properties of the Lagrangian and the conservation laws of the resulting dynamical equations. Therefore, we begin with a very brief introduction to symmetry and conservation laws. Noether’s theorem applies to the equations that arise from variational principles like Hamilton’s principle. According to Noether’s theorem : If a variational principle is invariant to a continuous transformation of its dependent and independent variables, then the equations arising from the variational principle possess a divergence-form conservation law. The invariance property is also called a symmetry property. Thus Noether’s theorem connects symmetry properties and conservation laws. We shall neither state nor prove the general form of Noether’s theorem; to do so would require a lengthy digression on continuous groups. Instead we illustrate the connection between symmetry and conservation laws with a series of increasingly complex and important examples. These examples convey the flavor of the general theory. Our first example is very simple. Consider a body of mass m moving in one dimension. The body is attached to the end of a spring with spring-constant K. Let x(t) be the displacement of the body from its location when the spring is unstretched.

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.

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