@article{gao_shi_zenkov_2023, title={Discrete Hamiltonian Variational Mechanics and Hamel's Integrators}, volume={33}, ISSN={["1432-1467"]}, DOI={10.1007/s00332-022-09875-w}, number={2}, journal={JOURNAL OF NONLINEAR SCIENCE}, author={Gao, Shan and Shi, Donghua and Zenkov, Dmitry V.}, year={2023}, month={Apr} }
@article{gao_shi_zenkov_2023, title={Discrete Hamiltonian Variational Mechanics and Hamel's Integrators (vol 33, 26, 2023)}, volume={33}, ISSN={["1432-1467"]}, DOI={10.1007/s00332-023-09890-5}, number={2}, journal={JOURNAL OF NONLINEAR SCIENCE}, author={Gao, Shan and Shi, Donghua and Zenkov, Dmitry V.}, year={2023}, month={Apr} }
@article{an_gao_shi_zenkov_2020, title={A Variational Integrator for the Chaplygin-Timoshenko Sleigh}, volume={30}, ISSN={["1432-1467"]}, DOI={10.1007/s00332-020-09611-2}, number={4}, journal={JOURNAL OF NONLINEAR SCIENCE}, author={An, Zhipeng and Gao, Shan and Shi, Donghua and Zenkov, Dmitry V.}, year={2020}, month={Aug}, pages={1381–1419} }
@article{shi_zenkov_bloch_2020, title={Hamel's Formalism for Classical Field Theories}, volume={30}, ISSN={["1432-1467"]}, DOI={10.1007/s00332-020-09609-w}, number={4}, journal={JOURNAL OF NONLINEAR SCIENCE}, author={Shi, Donghua and Zenkov, Dmitry V. and Bloch, Anthony M.}, year={2020}, month={Aug}, pages={1307–1353} }
@article{shi_berchenko-kogan_zenkov_bloch_2017, title={Hamel's Formalism for Infinite-Dimensional Mechanical Systems}, volume={27}, ISSN={["1432-1467"]}, DOI={10.1007/s00332-016-9332-7}, number={1}, journal={JOURNAL OF NONLINEAR SCIENCE}, author={Shi, Donghua and Berchenko-Kogan, Yakov and Zenkov, Dmitry V. and Bloch, Anthony M.}, year={2017}, month={Feb}, pages={241–283} }
@article{zenkov_2016, title={ON HAMEL'S EQUATIONS}, volume={43}, ISSN={["1450-5584"]}, DOI={10.2298/tam160612011z}, abstractNote={This paper reviews recent results on the extension of Hame?s formalism to infinite-dimensional mechanical systems and to variational integrators. Of a particular interest are applications to the dynamics and numerical integration of systems with velocity constraints.}, number={2}, journal={THEORETICAL AND APPLIED MECHANICS}, author={Zenkov, Dmitry V.}, year={2016}, pages={191–220} }
@article{bloch_krupka_zenkov_2015, title={The Helmholtz Conditions and the Method of Controlled Lagrangians}, ISBN={["978-94-6239-108-6"]}, DOI={10.2991/978-94-6239-109-3_1}, abstractNote={In this chapter we consider the relationship between the classical inverse problem of the calculus of variations and the method of controlled Lagrangians. The latter is a technique for deriving stabilizing feedback controls for nonlinear controlled mechanical systems. It relies on deriving a Lagrangian which describes the feedback controlled dynamics. This is a nontrivial extension to the theory of the inverse problem as it involves controls. We discuss various aspects of both subjects and illustrate the theory with examples.}, journal={INVERSE PROBLEM OF THE CALCULUS OF VARIATIONS: LOCAL AND GLOBAL THEORY}, author={Bloch, Anthony M. and Krupka, Demeter and Zenkov, Dmitry V.}, year={2015}, pages={1–29} }
@book{the inverse problem of the calculus of variations: local and global theory_2015, ISBN={9789462391086}, DOI={10.2991/978-94-6239-109-3}, publisher={Amsterdam: Atlantis Press}, year={2015} }
@article{fernandez_bloch_zenkov_2014, title={The geometry and integrability of the Suslov problem}, volume={55}, ISSN={["1089-7658"]}, DOI={10.1063/1.4901754}, abstractNote={In this paper, we discuss the integrability of a nonholonomic mechanical system—a generalized Klebsh–Tisserand case of the Suslov problem. Using the theory of Hamiltonization and the Poincaré–Hopf theorem we analyze the topology of the invariant manifolds and in particular describe their genus. We contrast the results with those for Hamiltonian systems.}, number={11}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Fernandez, Oscar E. and Bloch, Anthony M. and Zenkov, Dmitry V.}, year={2014}, month={Nov} }
@article{maruskin_bloch_marsden_zenkov_2012, title={A Fiber Bundle Approach to the Transpositional Relations in Nonholonomic Mechanics}, volume={22}, ISSN={["1432-1467"]}, DOI={10.1007/s00332-012-9144-3}, number={4}, journal={JOURNAL OF NONLINEAR SCIENCE}, author={Maruskin, Jared M. and Bloch, Anthony M. and Marsden, Jerrold E. and Zenkov, Dmitry V.}, year={2012}, month={Aug}, pages={431–461} }
@inproceedings{zenkov_leok_bloch_2012, title={Hamel's formalism and variational integrators on a sphere}, DOI={10.1109/cdc.2012.6426779}, abstractNote={This paper discusses Hamel's formalism and its applications to structure-preserving integration of mechanical systems. It utilizes redundant coordinates in order to eliminate multiple charts on the configuration space as well as nonphysical artificial singularities induced by local coordinates, while keeping the minimal possible degree of redundancy and avoiding integration of differential-algebraic equations.}, booktitle={2012 ieee 51st annual conference on decision and control (cdc)}, author={Zenkov, D. V. and Leok, M. and Bloch, A. M.}, year={2012}, pages={7504–7510} }
@article{ohsawa_fernandez_bloch_zenkov_2011, title={Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization}, volume={61}, ISSN={["1879-1662"]}, DOI={10.1016/j.geomphys.2011.02.015}, abstractNote={We 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. We show, under a certain sufficient condition for Hamiltonization, that the solutions to the Hamilton-Jacobi equation associated with the Hamiltonized system also solve the nonholonomic Hamilton-Jacobi equation associated with the original Chaplygin system. The results are illustrated through several examples.}, number={8}, journal={JOURNAL OF GEOMETRY AND PHYSICS}, author={Ohsawa, Tomoki and Fernandez, Oscar E. and Bloch, Anthony M. and Zenkov, Dmitry V.}, year={2011}, month={Aug}, pages={1263–1291} }
@article{bloch_marsden_zenkov_2009, title={Quasivelocities and symmetries in non-holonomic systems}, volume={24}, ISSN={["1468-9375"]}, DOI={10.1080/14689360802609344}, abstractNote={This article is concerned with the theory of quasivelocities for non-holonomic systems. The equations of non-holonomic mechanics are derived using the Lagrange-d'Alembert principle written in an arbitrary configuration-dependent frame. The article also shows how quasivelocities may be used in the formulation of non-holonomic systems with symmetry. In particular, the use of quasivelocities in the analysis of symmetry that leads to unusual momentum conservation laws is investigated, as is the applications of these conservation laws and discrete symmetries to the qualitative analysis of non-holonomic dynamics. The relationship between asymptotic dynamics and discrete symmetries of the system is also elucidated.}, number={2}, journal={DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL}, author={Bloch, Anthony M. and Marsden, Jerrold E. and Zenkov, Dmitry V.}, year={2009}, pages={187–222} }
@article{bloch_leok_marsden_zenkov_2007, title={Matching and stabilization of discrete mechanical systems}, volume={7}, ISSN={1617-7061 1617-7061}, url={http://dx.doi.org/10.1002/pamm.200700390}, DOI={10.1002/pamm.200700390}, abstractNote={Controlled Lagrangian and matching techniques are developed for the stabilization of equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. Specifically, a nonconservative force that is necessary for matching in the discrete setting is introduced. The paper also discusses digital and model predictive controllers.}, number={1}, journal={PAMM}, publisher={Wiley}, author={Bloch, Anthony M. and Leok, Melvin and Marsden, Jerrold E. and Zenkov, Dmitry V.}, year={2007}, month={Dec}, pages={1030603–1030604} }
@article{fedorov_zenkov_2005, title={Discrete nonholonomic LL systems on Lie groups}, volume={18}, ISSN={["1361-6544"]}, DOI={10.1088/0951-7715/18/5/017}, abstractNote={This paper applies the recently developed theory of discrete nonholonomic mechanics to the study of discrete nonholonomic left-invariant dynamics on Lie groups. The theory is illustrated with the discrete versions of two classical nonholonomic systems, the Suslov top and the Chaplygin sleigh. The preservation of the reduced energy by the discrete flow is observed and the discrete momentum conservation is discussed.}, number={5}, journal={NONLINEARITY}, author={Fedorov, YN and Zenkov, DV}, year={2005}, month={Sep}, pages={2211–2241} }
@article{zenkov_bloch_2003, title={Invariant measures of nonholonomic flows with internal degrees of freedom}, volume={16}, ISSN={["1361-6544"]}, DOI={10.1088/0951-7715/16/5/313}, abstractNote={In this paper we study measure preserving flows associated with nonholonomic systems with internal degrees of freedom. Our approach reveals geometric reasons for the existence of measures in the form of an integral invariant with smooth density that depends on the internal configuration of the system. Mathematics Subject Classification: 37J15, 37J60, 70F25}, number={5}, journal={NONLINEARITY}, author={Zenkov, DV and Bloch, AM}, year={2003}, month={Sep}, pages={1793–1807} }