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.

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 Drumlin Formation Related to Inverted Melt-Water Erosional Marks

1983 ◽  
Vol 29 (103) ◽  
pp. 461-479 ◽  
Author(s):  
John Shaw
Keyword(s):  
Reynolds Number ◽  
Large Scale ◽  
Ice Sheet ◽  
Fluid Flows ◽  
Minor Elements ◽  
Water Flows ◽  
Melt Water ◽  
Northern Saskatchewan ◽  
Specific Discharge ◽  
Ice Sheet Dynamics

AbstractDrumlin forms are described from maps and air photographs of a part of the Athabasca Plains, northern Saskatchewan. Three major forms, spindle, parabolic and transverse asymmetrical are recognized. These forms, which may show superimposed minor elements, depart from classical descriptions of drumlins, but are similar to moulds of erosional marks created by separated fluid flows. Assemblages of drumlins also show characteristics similar to those of erosional marks. The form analogy between drumlins and moulds of erosional marks is carried to a conclusion that drumlins may be formed by the infilling of erosional marks created on the under-side of glaciers by separated, subglacial melt-water flows. Estimates of specific discharge are obtained by means of an expected range of Reynolds number. Geomorphological evidence is given for large-scale erosion by subglacial melt water. A discussion of the sedimentology, stratigraphy, and deformational structure of the interiors of drumlins shows that they may be explained by the erosional-mark hypothesis. This paper emphasizes the importance of melt water as a geomorphic agent and may have broad implications for ice-sheet dynamics and profiles, rates of deglaciation, and the occurrence of bedrock thrusting by ice.

A numerical investigation of excess pore pressures and continental slope stability in response to ice-sheet dynamics

2019 ◽  
Vol 500 (1) ◽  
pp. 255-266 ◽  
Author(s):  
Morelia Urlaub ◽  
Isabel Kratzke ◽  
Berit Oline Hjelstuen
Keyword(s):  
Pore Pressure ◽  
Continental Slope ◽  
Large Scale ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Excess Pore Pressure ◽  
Submarine Landslides ◽  
Pore Pressures ◽  
Ice Sheet Dynamics ◽  
Excess Pore

AbstractSubmarine landslides are common at glaciated continental margins. The onset of large-scale landslides coincides with the initiation of Northern Hemisphere glaciations in the Quaternary. This implies that processes related to glacial cycling provide favourable conditions for submarine landslides at high-latitude margins. Potential processes include glacial deposition patterns and enhanced seismicity. It is also possible that advances and retreats of ice sheets, a highly dynamic process in geological terms, makes slopes discernible to failure by modifying the stress regime. Here, we quantify this effect using 2D finite element modelling of a glaciated continental margin. Different model runs investigate the pore-pressure development in homogeneous, as well as layered, slopes during glaciation when loaded by an ice stream with one or more ice advances. Ice streams cause significant variations in excess pore pressure in the very shallow sediment sequences at the continental shelf. However, lateral fluid flow is not efficient enough to increase pore pressures significantly at the slope, where large-scale submarine slides are observed. Hence, while ice-sheet dynamics appear to favour the occurrence of shallow slides close to the shelf edge, ice sheets seem to be irrelevant for the generation of large-scale submarine landslides at the continental slope.

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.

Large-scale integrated subglacial drainage around the former Keewatin Ice Divide, Canada reveals interaction between distributed and channelised systems

2020 ◽  
Author(s):  
Emma Lewington ◽  
Stephen Livingstone ◽  
Chris Clark ◽  
Andrew Sole ◽  
Robert Storrar
Keyword(s):  
Large Scale ◽  
Water Pressure ◽  
Pressure Fluctuations ◽  
Ice Sheet ◽  
Drainage System ◽  
Spatial Location ◽  
Variable Pressure ◽  
Ice Sheet Dynamics ◽  
Subglacial Meltwater ◽  
Subglacial Drainage

<p>Despite being widely studied, subglacial meltwater landforms are typically mapped and investigated individually, thus the drainage system as a whole remains poorly understood. Here, we identify and map all visible traces of subglacial meltwater flow across the Keewatin sector of the former Laurentide Ice Sheet from the ArcticDEM, generating significant new insights into the connectedness of the drainage system.</p><p>Due to similarities in spacing, morphometry and spatial location, we suggest that the 100s-1000s m wide features often flanking and connecting sections of eskers (i.e. tunnel valleys, meltwater tracks and esker splays) are varying expressions of the same phenomena and collectively term these features ‘meltwater corridors’. Based on observations from contemporary ice masses, we propose a new formation model based on the pressure fluctuations surrounding a central conduit, in which the esker records the imprint of the central conduit and the wider meltwater corridors the interactions with the surrounding distributed drainage system, or variable pressure axis (VPA).</p><p>We suggest that the widespread aerial coverage of meltwater corridors across the Keewatin sector provides constraints on the extent of basal uncoupling induced by basal water pressure fluctuation and variations in spatial distribution and evolution of the subglacial drainage system, which have important implications for ice sheet dynamics. </p>

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.

Sign in / Sign up

Export Citation Format

Share Document