Anthony M. Bloch
Nonholonomic Hamilton-Jacobi Theory via Chaplygin Hamiltonization.
Wed, 02/23/2011 - 16:01 — SomNambulAuthors: Tomoki Ohsawa, Anthony M. Bloch, Oscar E. Fernandez, Dmitry V. Zenkov
Subjects: Mathematical Physics
AbstractWe develop Hamilton-Jacobi theory for Chaplygin systems, a certain class of
nonholonomic mechanical systems with symmetries, using a technique called
Hamiltonization, which transforms nonholonomic systems into Hamiltonian
systems. We give a geometric account of the Hamiltonization, identify necessary
and sufficient conditions for Hamiltonization, and apply the conventional
Hamilton-Jacobi theory to the Hamiltonized systems.Hill's Equation with Random Forcing Parameters: Determination of Growth Rates through Random Matrices.
Fri, 02/05/2010 - 16:01 — SomNambulAuthors: Anthony M. Bloch, Fred C. Adams
Subjects: Mathematical Physics
AbstractThis paper derives expressions for the growth rates for the random 2 x 2
matrices that result from solutions to the random Hill's equation. The
parameters that appear in Hill's equation include the forcing strength and
oscillation frequency. The development of the solutions to this periodic
differential equation can be described by a discrete map, where the matrix
elements are given by the principal solutions for each cycle. Variations in the
forcing strength and oscillation frequency lead to matrix elements that vary
from cycle to cycle.Nonholonomic Hamilton-Jacobi equation and Integrability.
Tue, 12/22/2009 - 16:01 — SomNambulAuthors: Tomoki Ohsawa, Anthony M. Bloch
Subjects: Mathematical Physics
AbstractWe discuss an extension of the Hamilton-Jacobi theory to nonholonomic
mechanics with a particular interest in its application to exactly integrating
the equations of motion. We give an intrinsic proof of a nonholonomic analogue
of the Hamilton--Jacobi theorem. Our intrinsic proof clarifies the difference
from the conventional Hamilton-Jacobi theory for unconstrained systems. The
proof also helps us identify a geometric meaning of the conditions on the
solutions of the Hamilton-Jacobi equation that arise from nonholonomic
constraints.Discrete Hamilton-Jacobi Theory.
Fri, 11/13/2009 - 16:01 — SomNambulAuthors: Melvin Leok, Tomoki Ohsawa, Anthony M. Bloch
Subjects: Optimization and Control
AbstractWe develop a discrete analogue of the Hamilton-Jacobi theory in the framework
of the discrete Hamiltonian mechanics. We first reinterpret the discrete
Hamilton-Jacobi equation derived by Elnatanov and Schiff in the language of
discrete mechanics. The resulting discrete Hamilton-Jacobi equation is discrete
only in time, and is shown to recover the Hamilton-Jacobi equation in the
continuous-time limit. The correspondence between discrete and continuous
Hamiltonian mechanics naturally gives rise to a discrete analogue of Jacobi's
solution to the Hamilton-Jacobi equation.
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