The derivation of the equations of motion of an ideal fluid by Hamilton's principle

1955 ◽  
Vol 51 (2) ◽  
pp. 344-349 ◽  
Author(s):  
J. W. Herivel
Keyword(s):  
Variational Principle ◽  
Ideal Fluid ◽  
Rotational Motion ◽  
Equations Of Motion ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Proper Form ◽  
Entropy Conservation ◽  
Fluid Particles ◽  
New Formulation

ABSTRACTA new formulation of Hamilton's principle for the case of an ideal fluid is proposed which is claimed to be the uniquely proper form for such a system. The resulting derivation of the equations of motion on varying with respect to the position of the fluid particles is free from the difficulties encountered in previous treatments based on incorrect forms of Hamilton's principle. The treatment also differs from previous treatments in transforming to the fixed (a, b, c) space before carrying out the variation, and in allowing for the equations of mass and entropy conservation by means of undetermined multipliers. The necessity for conservation of entropy, if Hamilton's principle is to be applied, is emphasized. Finally, the method, first used by Eckart, of deriving the equations of motion for an ideal fluid by means of a variational principle of the same form as Hamilton's, but varying with respect to the velocities of the fluid particles, is extended to the general case of rotational motion.

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.

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.

A note on Hamilton's principle for perfect fluids

1970 ◽  
Vol 44 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Francis P. Bretherton
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Velocity Fields ◽  
Lagrangian Formulation ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Barotropic Fluid ◽  
Perfect Fluids ◽  
Vortex Lines ◽  
Equal Density

A derivation is given of the Eulerian equations of motion directly from the Lagrangian formulation of Hamilton's principle. The circulation round a circuit of material particles of uniform entropy appears as a constant of the motion associated with the indistinguishability of fluid elements with equal density, entropy and velocity. A discussion is given of the Lin constraint, and it is pointed out that, for a barotropic fluid, the variational principle recently suggested by Seliger & Whitham does not permit velocity fields in which the vortex lines are knotted.

scholarly journals Multisymplectic formulation of fluid dynamics using the inverse map

2007 ◽  
Vol 463 (2086) ◽  
pp. 2671-2687 ◽  
Author(s):  
C.J Cotter ◽  
D.D Holm ◽  
P.E Hydon
Keyword(s):  
Fluid Dynamics ◽  
Conservation Laws ◽  
Fluid Particle ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Numerical Integrators ◽  
Fluid Particles ◽  
Infinite Set ◽  
Circulation Theorem

We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map, the ‘back-to-labels’ map, gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamilton's principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to Kelvin's circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.

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.

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.

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.

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.

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.

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