Research Interests


My research interests originate from my fascination with the interplay between mathematics and physics. In particular, I study geometric mechanics, which can most easily be described as mechanics on manifolds.


A particularly interesting related field is that of nonholonomic mechanics. These are mechanical systems subjected to non-integrable constraints on the velocities of the system. Some familiar examples of nonholonomic systems are cars and bowling balls. It turns out that these systems are not Hamiltonian. However, under certain conditions they can be embedded in a larger Hamiltonian system. Much of my research to date has been focused on finding different ways to do this. Lately I have been exploring the applications of these results to questions like the integrability and quantization of nonholonomic systems.

Recent Talks


"The Poincare-Hopf Theorem in Nonholonomic Mechanics,"

ICERM Workshop on Integrability in Mechanics, Providence, RI, Jun. 2015


"The Poincare-Hopf Theorem in Nonholonomic Mechanics,"

SIAM Conference on Applications of Dynamical Systems, Snowbird, UT, May 2015


"Quantizing Nonholonomic Systems,"

Annual Meeting of Mid-Atlantic APS, University Park, PA, Oct. 2014


"The Quantum Mechanics of Nanocars,"

Alan Aspuru-Guzik chemistry group, Harvard Univ., Cambridge, MA, Jun. 2014


"A New Approach to the Integrability of the Suslov Problem," AMMCS 2013 Conference, Waterloo, Canada, Aug. 2013


"A New Approach to the Integrability of the Suslov Problem," SIAM Conference on Applications of Dynamical Systems, Snowbird, Utah, May 2013


"An Improved Integrability Theorem for Hamiltonizable Nonholonomic Systems," 3rd Iberoamerican Meeting on Geometry, Mechanics, and Control, Salemanca, Spain, Sep. 2012


"Nonholonomic Systems - Dynamics and Simulations via Time Transformations," Univ. of Mass. Amherst Applied Math Seminar, Amherst, MA, Nov. 2012



Thesis












The Hamiltonization of Nonholonomic Systems and its Applications, Ph.D. Thesis, Oscar E. Fernandez, University of Michigan (2009)

[8]    Variational Integrators for Hamiltonizable Nonholonomic Systems, J. Geometric Mechanics, 4(2) (2012), 137-163 (with Anthony M. Bloch and Peter J. Olver) [doi]






[7]     Nonholonomic Hamilton-Jacobi Theory via Chaplygin Hamiltonization, J. Geometry and Physics, 61(8) (2011), 1263-1291 (with Tomoki Ohsawa, Anthony M. Bloch, and Dmitry V. Zenkov) [doi]





[6]    The Weitzenbock Connection and Time Reparameterization in Nonholonomic Mechanics, J. Math. Physics, 52(1) 012901 (2011) (with Anthony M. Bloch) [doi]






[5]    A Generalization of Chaplygin's Reducibility Theorem, Reg. and Chaotic Dyn., 14(6) (2009) (with Tom Mestdag and Anthony M. Bloch) [doi]





[4]    Hamiltonization and geometric integration of nonholonomic systems, Proc. of the 8th Nat. Congress on Theor. and Appl. Mechanics, Brussels (Belgium) (2009) (with Tom Mestdag and Anthony M. Bloch) [doi]





[3]    Hamiltonization of Nonholonomic Systems and the Inverse Problem of the Calculus of Variations, Rep. Math. Phys. 63 (2009), 225-249 (with Anthony M. Bloch and Tom Mestdag) [doi]






[2]    The Pontryagin Maximum Principle applied to Nonholonomic Mechanics, Proceedings of the IEEE 47th Conference on Decision and Control (2008), 4306-4311(with Anthony M. Bloch and Tom Mestdag) [doi]





[1]    Equivalence of the Dynamics of Nonholonomic and Variational Nonholonomic Systems for Certain Initial Data, J. Phys A: Math. Theor. 41 (2008) (with Anthony M. Bloch) [doi]

Publications

(click on paper image to download)

[10]    The Geometry and Integrability of the Suslov Problem, J. Math. Phys. 55, 112704 (2014) (with Anthony M. Bloch and Dmitry Zenkov) [doi]


[9]   Quantizing Conditionally Variational Nonholonomic Systems, J. Phys. A: Math. Theor., 47(30) (2014) 305206 [doi]

[12]    The Entropy of Life Table: A Reappraisal,

Theor. Pop. Biology, 104 (2015), 26-45 (with Hiram Beltran-Sanchez) [doi]


[11]    Reduction of nonholonomic systems in two stages,

Nonlinearity, 28(8) (2015), 2873-2912 (with Paula Balseiro) [doi]