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Melvin Leok

Professor of Mathematics
University of California, San Diego
Department of Mathematics
University of California, San Diego
9500 Gilman Drive, Dept. 0112
La Jolla, CA 92093-0112, USA.
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mleok@math.ucsd.edu
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Announcements

Research Opportunities

Teaching

Research Interests

Selected Honors

Journal of Control and 
Optimization Journal of Nonlinear 
Science Journal of Computational Dynamics Journal of Geometric Mechanics Journal of Nonlinear 
Science

Editorial Boards

Thesis

Publications and Preprints (search the ArXiv, AMS MathSciNet) [ BibTeX ]

  1. Discrete Poincaré Lemma (with M. Desbrun, and J.E. Marsden), Appl. Numer. Math. 53 (2-4), 231-248, 2005.
    PDF | Journal Link | BibTeX ]
  2. A Lie Group Variational Integrator for the Attitude Dynamics of a Rigid Body with Applications to the 3D Pendulum (with T. Lee, and N.H. McClamroch), Proc. IEEE Conf. on Control Applications, 962-967, 2005.
    PDF | IEEE Xplore | BibTeX | AnimationChaotic Motion ]
  3. Controlled Lagrangians and Stabilization of the Discrete Cart-Pendulum System (with A.M. Bloch, J.E. Marsden, and D.V. Zenkov), Proc. IEEE Conf. on Decision and Control, 6579-6584, 2005.
    PDF | IEEE Xplore | BibTeX ]
  4. Discrete Routh Reduction (with S.M. Jalnapurkar, J.E. Marsden, and M. West), J. Phys. A: Math. Gen. 39, 5521-5544 (Geometric Integration special issue, invited paper), 2006.
    PDF | arXiv:math.NA/0508330 | Journal Link | BibTeX ]
  5. Attitude Maneuvers of a Rigid Spacecraft in a Circular Orbit (with T. Lee, and N.H. McClamroch), Proc. American Control Conf., 1742-1747, 2006.
    PDF | arXiv:math.OC/0509299 | IEEE Xplore | BibTeX | AnimationFig2(a)Fig2(b)Fig2(c)Fig3(a) ]
  6. Polyhedral Potential and Lie Group Variational Integrator Computations for the Full Two Body Problem (with E. Fahnestock, T. Lee, N.H. McClamroch, D.J. Scheeres), Proc. AIAA/AAS Astrodynamics Specialist Conf., AIAA-2006-6289, 2006.
    PDF | arXiv:math.NA/0608695 | Journal Link | BibTeX ]
  7. Optimal Control of a Rigid Body using Geometrically Exact Computations on SE(3) (with T. Lee, and N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 2710-2715, 2006.
    PDF | arXiv:math.OC/0602588 | IEEE Xplore | BibTeX | AnimationFig1(a)Fig2(a)Fig3(a)Fig4(a) ]
  8. Deterministic Global Attitude Estimation (with T. Lee, A.K. Sanyal, and N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 3174-3179, 2006.
    PDF | arXiv:math.OC/0602589 | IEEE Xplore | BibTeX ]
  9. Controlled Lagrangians and Potential Shaping for Stabilization of Discrete Mechanical Systems (with A.M. Bloch, J.E. Marsden, and D.V. Zenkov), Proc. IEEE Conf. on Decision and Control, 3333-3338, 2006.
    PDF | arXiv:math.OC/0602590 | IEEE Xplore | BibTeX ]
  10. A Discrete Variational Integrator for Optimal Control Problems in SO(3) (with A.M. Bloch, I.I. Hussein, and A.K. Sanyal), Proc. IEEE Conf. on Decision and Control, 6636-6641, 2006.
    PDF | arXiv:math.OC/0509536 | IEEE Xplore | BibTeX ]
  11. Global Attitude Estimation using Single Direction Measurements (with T. Lee, N.H. McClamroch, and A.K. Sanyal), Proc. American Control Conf., 3659-3664, 2007.
    PDF | arXiv:math.OC/0609481 | IEEE Xplore | BibTeX ]
  12. Optimal Attitude Control for a Rigid Body with Symmetry (with T. Lee, and N.H. McClamroch), Proc. American Control Conf., 1073-1078, 2007.
    PDF | arXiv:math.OC/0609482 | IEEE Xplore | BibTeX | AnimationFig2(a)Fig3(a)Fig4(a)Fig5(a) ]
  13. Lie Group Variational Integrators for the Full Body Problem, (with T. Lee, and N.H. McClamroch), Comput. Methods Appl. Mech. Engrg. 196(29-30), 2907-2924, 2007.
    PDF | arXiv:math.NA/0508365 | Journal Link | BibTeX | AnimationFig4 ]
  14. Lie Group Variational Integrators for the Full Body Problem in Orbital Mechanics (with T. Lee, and N.H. McClamroch), Celestial Mechanics and Dynamical Astronomy 98(2), 121-144, 2007.
    PDF | Journal Link | BibTeX | AnimationFig2 ]
  15. Propagation of Uncertainty Sets for Rigid Body Attitude Flows (with N.A. Chaturvedi, T. Lee, N.H. McClamroch, and A.K. Sanyal), Proc. IEEE Conf. on Decision and Control, 2689-2694, 2007.
    PDF | arXiv:math.DS/0702737 | IEEE Xplore | BibTeX ]
  16. A Combinatorial Optimal Control Problem for Spacecraft Formation Reconfiguration (with T. Lee, and N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 5370-5375, 2007.
    PDF | arXiv:math.OC/0702738 | IEEE Xplore | BibTeX | AnimationFig2 ]
  17. Global Optimal Attitude Estimation using Uncertainty Ellipsoids (with A.K. Sanyal, T. Lee, and N.H. McClamroch), Systems and Control Letters, 57(3), 236-245, 2008.
    PDF | arXiv:math.OC/0606083 | Journal Link | BibTeX ]
  18. Time Optimal Attitude Control for a Rigid Body (with T. Lee, and N.H. McClamroch), Proc. American Control Conf., 5210-5215, 2008.
    PDF | arXiv:0709.2514 | IEEE Xplore | BibTeX | AnimationFig1(a)Fig2(a) ]
  19. Matching and stabilization of discrete mechanical systems (with A.M. Bloch, J.E. Marsden, and D.V. Zenkov), Proc. Appl. Math. Mech., 7, 1030603-1030604, 2007.
    PDF | Journal Link | BibTeX ]
  20. Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on SO(3) (with T. Lee, and N.H. McClamroch), Journal of Dynamical and Control Systems, 14(4), 465-487, 2008.
    PDF | arXiv:math.OC/0601424 | Journal Link | BibTeX | AnimationFig2(a)Fig3(a)Fig4(a)Fig5(a) ]
  21. Global Symplectic Uncertainty Propagation on SO(3) (with T. Lee, and N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 61-66, 2008.
    PDF | arXiv:0803.1515 | IEEE Xplore | BibTeX ]
  22. Discrete Control Systems (with T. Lee, and N.H. McClamroch), invited article for Springer Encyclopedia of Complexity and System Science, 2002-2019, 2009.
    PDF | arXiv:0705.3868 | Journal Link | BibTeX ]
  23. Geometric Structure-Preserving Optimal Control of the Rigid Body (with A.M. Bloch, I.I. Hussein, and A.K. Sanyal), Journal of Dynamical and Control Systems, 15(3), 307-330, 2009.
    PDF | arXiv:0712.4400 | Journal Link | BibTeX ]
  24. Lagrangian Mechanics and Variational Integrators on Two-Spheres (with T. Lee, and N.H. McClamroch), International Journal for Numerical Methods in Engineering, 79(9), 1147-1174, 2009.
    PDF | arXiv:0707.0022 | Journal Link | BibTeX | AnimationFig3.1Fig3.2Fig3.3Fig3.4Fig3.5Fig3.6Fig3.7 ]
  25. Computational Geometric Optimal Control of Rigid Bodies (with T. Lee, and N.H. McClamroch), Brockett Legacy Special Issue, Communications in Information and Systems, 8(4), 445-472, 2008.
    PDF | arXiv:0805.0639 | Journal Link | BibTeX ]
  26. Dynamics of Connected Rigid Bodies in a Perfect Fluid (with T. Lee, and N.H. McClamroch), Proc. American Control Conf., 408-413, 2009.
    PDF | arXiv:0809.1488 | IEEE Xplore | BibTeX | AnimationFig1 ]
  27. Controlled Lagrangians and Stabilization of Discrete Mechanical Systems (with A.M. Bloch, J.E. Marsden, and D.V. Zenkov), Discrete and Continuous Dynamical Systems - Series S, 3(1), 19-36, 2010.
    PDF | arXiv:0704.3875 | Journal Link | BibTeX ]
  28. Nonlinear Dynamics of the 3D Pendulum (with N.A. Chaturvedi, T. Lee, and N.H. McClamroch), Journal of Nonlinear Science, 21(1), 3-32, 2011.
    PDF | arXiv:0707.1196 | Journal Link | BibTeX ]
  29. Variational and Geometric Structures of Discrete Dirac Mechanics (with T. Ohsawa), Foundations of Computational Mathematics, 11(5), 529-562, 2011.
    PDF | arXiv:0810.0740 | Journal Link | BibTeX ]
  30. Dynamics of a 3D Elastic String Pendulum (with T. Lee and N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 3347-3352, 2009.
    PDF | arXiv:0903.0332 | IEEE Xplore | BibTeX | AnimationFig2 ]
  31. Computational Dynamics of a 3D Elastic String Pendulum Attached to a Rigid Body and an Inertially Fixed Reel Mechanism (with T. Lee, N.H. McClamroch), Nonlinear Dynamics, 64(1-2), 97-115, 2011.
    PDF | arXiv:0909.2083 | Journal Link | BibTeX | AnimationFig2(a)Fig3(a)Fig4(a) ]
  32. Computational Geometric Optimal Control of Connected Rigid Bodies in a Perfect Fluid (with T. Lee, N.H. McClamroch), Proc. American Control Conf., 5985-5990, 2010.
    PDF | arXiv:0909.3871 | BibTeX | AnimationFig2(a)Fig3(a) ]
  33. Discrete Hamilton-Jacobi Theory (with A.M. Bloch, T. Ohsawa), SIAM Journal on Control and Optimization, 49(4), 1829-1856, 2011.
    PDF | arXiv:0911.2258 | Journal Link | BibTeX ]
  34. Discrete Hamiltonian Variational Integrators (with J. Zhang), IMA Journal of Numerical Analysis, 31(4), 1497-1532, 2011.
    PDF | arXiv:1001.1408 | Journal Link | BibTeX ]
  35. Geometric Tracking Control of a Quadrotor UAV on SE(3) (with T. Lee, N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 5420-5425, 2010.
    PDF | arXiv:1003.2005 | IEEE Xplore | BibTeX ]
  36. Discrete Hamilton-Jacobi Theory and Discrete Optimal Control (with T. Ohsawa, A.M. Bloch), Proc. IEEE Conf. on Decision and Control, 5438-5443, 2010.
    PDF | IEEE Xplore | BibTeX ]
  37. Stokes-Dirac Structures through Reduction of Infinite-Dimensional Dirac Structures (with J. Vankerschaver, H. Yoshimura, J.E. Marsden), Proc. IEEE Conf. on Decision and Control, 6265-6270, 2010.
    PDF | arXiv:1010.2547 | IEEE Xplore | BibTeX ]
  38. Discrete Dirac Structures and Implicit Discrete Lagrangian and Hamiltonian Systems (with T. Ohsawa), XVIII International Fall Workshop on Geometry and Physics, 91-102, AIP Conference Proceedings 1260, 2010.
    PDF | Journal Link | BibTeX ]
  39. Geometric Numerical Integration of Complex Dynamics of Tethered Spacecraft (with T. Lee, N. H. McClamroch), Proc. American Control Conf., 1885-1891, 2011.
    PDF | arXiv:1010.1724 | BibTeX ]
  40. Nonlinear Robust Tracking Control of a Quadrotor UAV on SE(3) (with T. Lee, N. H. McClamroch), Asian Journal of Control, 15(3), 1-18, 2013,
    PDF | Journal Link | BibTeX ]
  41. Prolongation-Collocation Variational Integrators (with T. Shingel), IMA Journal of Numerical Analysis, 32(3), 1194-1216, 2012.
    PDF | arXiv:1101.1995 | Journal Link | BibTeX ]
  42. On the Geometry of Multi-Dirac Structures and Gerstenhaber Algebras (with J. Vankerschaver, H. Yoshimura), Journal of Geometry and Physics, 61(8), 1415-1425, 2011. 2011.
    PDF | arXiv:1102.2835 | Journal Link | BibTeX ]
  43. General Techniques for Constructing Variational Integrators (with T. Shingel), Frontiers of Mathematics in China (Special issue on computational mathematics, invited paper), 7(2), 273-303, 2012.
    PDF | arXiv:1102.2685 | Journal Link | BibTeX ]
  44. Stable Manifolds of Saddle Points for Pendulum Dynamics on S^2 and SO(3) (with T. Lee, N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 3915-3921, 2011.
    PDF | arXiv:1103.2822 | BibTeX ]
  45. Variational Integrators, invited article for Springer Encyclopedia of Applied and Computational Mathematics, in press, 2012.
    PDF | BibTeX ]
  46. The Hamilton-Pontryagin Principle and Multi-Dirac Structures for Classical Field Theories (with J. Vankerschaver, H. Yoshimura), Journal of Mathematical Physics, 53(7), 072903 (25 pages), 2012.
    PDF | arXiv:1207.2814 | Journal Link | BibTeX ]
  47. Nonlinear Robust Tracking Control of a Quadrotor UAV on SE(3) (with T. Lee, N.H. McClamroch), Proc. American Control Conf., 4649-4654, 2012.
    PDF | arXiv:1109.4457 | BibTeX ]
  48. Hamilton--Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints (with T. Ohsawa, D. Sosa), Journal of Mathematical Physics, 53(7), 072905 (29 pages), 2012.
    PDF | arXiv:1109.6056 | Journal Link | BibTeX ]
  49. Generating Functionals and Lagrangian PDEs (with C. Liao, J. Vankerschaver), Journal of Mathematical Physics, 54(8), 082901 (22 pages), 2013.
    PDF | arXiv:1111.0280 | Journal Link | BibTeX ]
  50. Dynamics and Control of a Chain Pendulum on a Cart (with T. Lee and N. H. McClamroch), Proc. IEEE Conf. on Decision and Control, 2502-2508, 2012.
    PDF | arXiv:1211.4604 | BibTeX ]
  51. Hamel's Formalism and Variational Integrators on a Sphere (with A.M. Bloch, and D.V. Zenkov), Proc. IEEE Conf. on Decision and Control, 7504-7510, 2012.
    PDF | arXiv:1211.4607 | BibTeX ]
  52. High-Fidelity Numerical Simulation of Complex Dynamics of Tethered Spacecraft (with T. Lee, and N.H. McClamroch), Acta Astronautica, 99, 215-230, 2014.
    PDF | arXiv | Journal Link | BibTeX ]
  53. Dirac Structures and Hamilton-Jacobi Theory for Lagrangian Mechanics on Lie Algebroids (with D. Sosa), Journal of Geometric Mechanics, 4(4), 421-442, 2012.
    PDF | arXiv:1211.4561 | BibTeX ]
  54. A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects (with J. Vankerschaver), Journal of Nonlinear Science, 24(1), 1-37, 2014.
    PDF | arXiv:1211.4560 | Journal Link | BibTeX ]
  55. Spectral Variational Integrators (with J. Hall), Numerische Mathematik, Online First, 2014.
    PDF | arXiv:1211.4534 | Journal Link | BibTeX ]
  56. Symplectic Semiclassical Wave Packet Dynamics (with T. Ohsawa), Journal of Physica A, 46(40), 405201 (28 pages), 2013.
    PDF | arXiv:1302.1139 | Journal Link | BibTeX ]
  57. Coordinate-free Formulation for Dynamics and Control of a Chain Pendulum on a Cart (with T. Lee, and N.H. McClamroch), under revision, 2013.
    PDF | BibTeX ]
  58. Lie Group Spectral Variational Integrators (with J. Hall), Foundations of Computational Mathematics, submitted, 2014.
    PDF | arXiv:1402.3327 | BibTeX ]
  59. Geometric Representations of Whitney Forms and their Generalization to Minkowski Spacetime (with J. Salamon, and J. Moody), under revision, 2014.
    PDF | arXiv:1402.7109 | BibTeX ]
  60. Space-Time Finite-Element Exterior Calculus and Variational Discretizations of Gauge Field Theories (with J. Salamon, and J. Moody), Proc. Mathematical Theory of Networks and Systems, 743-747, 2014.
    PDF | BibTeX ]
  61. A Novel Variational Formulation for Thermoelastic Problems (with Z. Ebrahimzadeh, and M. Mahzoon), Communications in Nonlinear Science and Numerical Simulation, accepted, 2014.
    PDF | Journal Link | BibTeX ]
  62. Generalized Galerkin Variational Integrators, preprint, 2004.
    PDF | arXiv:math.NA/0508360 | BibTeX ]
  63. A Discrete Theory of Connections on Principal Bundles (with J.E. Marsden, and A.D. Weinstein), preprint, 2004.
    PDF | arXiv:math.DG/0508338 | BibTeX ]
  64. Discrete Exterior Calculus (with M. Desbrun, A.N. Hirani, and J.E. Marsden), preprint, 2003.
    PDF | arXiv:math.DG/0508341 | BibTeX ]
  65. Estimating the Attractor Dimension of the Equatorial Weather System, Acta Phys. Pol. A 85, S27-S35, 1994.
    PDF | Journal Link | BibTeX ]

Papers in Preparation

  1. Dirac Structures in Lagrangian Mechanics on Lie Groupoids (with D. Sosa).
  2. Discrete Augmented Variational Principles in Mechanics and Control (with T. Ohsawa).
  3. On the Decomposition of Forces (with D.E. Chang, J.E. Marsden, T.S. Ratiu).

Upcoming and Recent Invited Presentations

Professional Service

Education

Upcoming Conferences (Confirmed)

Interesting Upcoming Conferences

Recent Conferences Attended

Other Interesting Conferences

Erdös Number

  1. Erdös, P.; Graham, R. L.; Montgomery, P.; Rothschild, B. L.; Spencer, J.; Straus, E. G.; Euclidean Ramsey theorems. I., J. Combinatorial Theory Ser. A, 14 (1973), 341-363.
  2. Golubitsky, M. A.; Rothschild, B. L.; Primitive subalgebras of exceptional Lie algebras. Bull. Amer. Math. Soc. 77 1971 983-986.
  3. Golubitsky, M. A.; Marsden, J. E.; The Morse lemma in infinite dimensions via singularity theory. SIAM J. Math. Anal. 14 (1983), no. 6, 1037-1044.
  4. Desbrun, M.; Leok, M.; Marsden, J. E.; Discrete Poincaré Lemma, Appl. Numer. Math. 53(2-4), 231-248, 2005.

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Biographical Sketch

Melvin Leok is a tenured professor of mathematics at the University of California, San Diego, where his research is supported in part by grants from the National Science Foundation in applied and computational mathematics, including a Faculty Early Career Development (CAREER) award. He serves on the editorial boards of the Journal of Nonlinear Science, the SIAM Journal on Control and Optimization, the LMS Journal of Computation and Mathematics, the Journal of Geometric Mechanics, and the Journal of Computational Dynamics.

Prior to joining UCSD, he was a tenure-track assistant professor of mathematics at Purdue University, a visiting assistant professor of control and dynamical systems at the California Institute of Technology, and a T.H. Hildebrandt research assistant professor of mathematics at the University of Michigan, Ann Arbor. At Purdue, he was a nominee for the Packard Fellowship for Science and Engineering, and at Michigan, he received a Horace H. Rackham Faculty Fellowship and Grant, and a Margaret and Herman Sokol Spring/Summer Research Grant.

He received his B.S. with honors and M.S. in Mathematics in 2000, and his Ph.D. in Control and Dynamical Systems with a minor in Applied and Computational Mathematics under the direction of Jerrold Marsden in 2004, all from the California Institute of Technology.

His primary research interests are in computational geometric mechanics, computational geometric control theory, discrete geometry, and structure-preserving numerical schemes, and particularly how these subjects relate to systems with symmetry and multiscale systems.

He was the recipient of the SciCADE New Talent Prize in 2007 for his work on Lie Group and Homogeneous Variational Integrators, and the SIAM Student Paper Prize, and the Leslie Fox Prize (second prize) in Numerical Analysis, both in 2003, for his work on Foundations of Computational Geometric Mechanics. While a doctoral student at Caltech, he held a Poincaré Fellowship (2000-2004), a Josephine de Kármán Fellowship (2003-2004), an International Fellowship from the Agency for Science, Technology, and Research (2002-2004), a Tau Beta Pi Fellowship (2000-2001), and a Tan Kah Kee Foundation Postgraduate Scholarship (2000).

As a Caltech undergraduate, he received the Loke Cheng-Kim Foundation Scholarship (1996-2000), the Carnation Scholarship (1998-2000), the Herbert J. Ryser Scholarship (1999), the E.T. Bell Undergraduate Mathematics Research Prize (1999), and the Jack E. Froehlich Memorial Award (1999).

SciCADE New Talent Prize SIAM Student Paper Prize Leslie Fox Prize
SciCADE New Talent Award SIAM Student Paper Prize Leslie Fox Prize

Recommended Books Amazon Store

Mechanics 
and 
Symmetry. Nonholonomic 
Mechanics 
and Control. Geometric 
Control of 
Mechanical Systems. Geometric 
Numerical Integration. Simulating 
Hamiltonian Dynamics. Mathematical Methods of Classical Mechanics. Topological Methods in Hydrodynamics. Classical Dynamics: A Contemporary Approach. Feynman 
Lectures on Physics. Foundations of Mechanics, Part I Computational Electromagnetism.

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