Multisymplectic geometry, variational integrators, and nonlinear PDEs

This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether's theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization schemes.Comment: LaTeX2E, 52 pages, 11 figures, to appear in Comm. Math. Phy

The Jacobi-Maupertuis Principle in Variational Integrators

In this paper, we develop a hybrid variational integrator based on the Jacobi-Maupertuis Principle of Least Action. The Jacobi-Maupertuis principle states that for a mechanical system with total energy E and potential energy V(q), the curve traced out by the system on a constant energy surface minimizes the action given by ∫√[2(E-V(q))] ds where ds is the line element on the constant energy surface with respect to the kinetic energy of the system. The key feature is that the principle is a parametrization independent geodesic problem. We show that this principle can be combined with traditional variational integrators and can be used to efficiently handle high velocity regions where small time steps would otherwise be required. This is done by switching between the Hamilton principle and the Jacobi-Maupertuis principle depending upon the kinetic energy of the system. We demonstrate our technique for the Kepler problem and discuss some ongoing and future work in studying the energy and momentum behavior of the resulting integrator

Model reduction for analysis of cascading failures in power systems

In this paper, we apply a principal-orthogonal decomposition based method to the model reduction of a hybrid, nonlinear model of a power network. The results demonstrate that the sequence of fault events can be evaluated and predicted without necessarily simulating the whole system

Geometric discrete analogues of tangent bundles and constrained Lagrangian systems

Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vectors. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete tangent bundles as phase spaces for discretizations of the variational principles of Lagrangian systems, up to the generality of nonholonomic mechanical systems with nonlinear constraints. We obtain a self-contained and transparent development, where regularity, equations of motion, symmetry and momentum, and structure preservation, all have natural expressions.Comment: Typos corrected. New abstract. Diagrams added. Some additional information and a conclusions section adde

Optimal Pollution Mitigation in Monterey Bay Based on Coastal Radar Data and Nonlinear Dynamics

The article of record as published may be found at http://dx.doi.org/10.1021/es0630691High-frequency (HF) radar technology produces detailed velocity maps near the surface of estuaries and bays. The use of velocity data in environmental prediction, nonetheless, remains unexplored. In this paper, we uncover a striking flow structure in coastal radar observations of Monterey Bay, along the California coastline. This complex structure governs the spread of organic contaminants, such as agricultural runoff which is a typical source of pollution in the bay. We show that a HF radar-based pollution release scheme using this flow structure reduces the impact of pollution on the coastal environment in the bay. We predict the motion of the Lagrangian flow structures from finite-time Lyapunov exponents of the coastal HF velocity data. From this prediction, we obtain optimal release times, at which pollution leaves the bay most efficiently.Office of Naval Research grant N00014-01-1-0208ASAP MURI N00014-02-1-0826Office of Naval Research grant N00014-01-1-0208ASAP MURI N00014-02-1-082

Effects of quality grade, aging period, blade tenderization, and degree of doneness on tenderness of inside round steaks

We used 162 inside rounds to determine the influence of different quality grades, postmortem aging periods, blade tenderization passes, and degree of doneness on thawing and cooking losses and Warner-Bratzler Shear force (WBS, tenderness). Select (SEL), Choice (CHO), and Certified Angus Beef™ (CAB) inside rounds were aged for 7, 14, or 21 days and not tenderized (0X) or blade tenderized one (1X) or two (2X) times. Steaks from each inside round were assigned randomly to final endpoint cooking temperatures of 150, 160, and 170°F. Percentage of thawing loss was higher (P<.05) for steaks aged 7 days than steaks aged 14 and 21 days. For CHO steaks only, cooking loss was higher (P<.05) for the 2X group compared to the 0X and 1X groups. Steaks aged 14 and 21 days had lower (P<.05) WBS than steaks aged 7 days. Cooking loss and WBS were higher (P<.05) with each increase in endpoint cooking temperature. Postmortem aging (14 or 21 days) and lower endpoint cooking temperatures were the most effective methods to improve WBS of inside round steaks

Stability transitions for axisymmetric relative equilibria of Euclidean symmetric Hamiltonian systems

In the presence of noncompact symmetry, the stability of relative equilibria under momentum-preserving perturbations does not generally imply robust stability under momentum-changing perturbations. For axisymmetric relative equilibria of Hamiltonian systems with Euclidean symmetry, we investigate different mechanisms of stability: stability by energy-momentum confinement, KAM, and Nekhoroshev stability, and we explain the transitions between these. We apply our results to the Kirchhoff model for the motion of an axisymmetric underwater vehicle, and we numerically study dissipation induced instability of KAM stable relative equilibria for this system.Comment: Minor revisions. Typographical errors correcte