Hamel's formalism and variational integrators on a sphere

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.

[1]  W. Hamilton.  XV. On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function , 1834, Philosophical Transactions of the Royal Society of London.

[2]  A. Bobenko,et al.  Discrete Lagrangian Reduction, Discrete Euler–Poincaré Equations, and Semidirect Products , 1999, math/9906108.

[3]  William Rowan Hamilton.  Second Essay on a General Method in Dynamics. [Abstract] , 1830 .

[4]  J. Marsden,et al.  Discrete Euler-Poincaré and Lie-Poisson equations , 1999, math/9909099.

[5]  Anthony M. Bloch,et al.  Variational structures for Hamel's equations and stabilization , 2012 .

[6]  A. Bobenko,et al.  Discrete Time Lagrangian Mechanics on Lie Groups,¶with an Application to the Lagrange Top , 1999 .

[7]  Jerrold E. Marsden,et al.  Quasivelocities and symmetries in non-holonomic systems , 2009 .

[8]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[9]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[10]  G. Hamel,et al.  Die Lagrange-Euler'schen gleichungen der Mechanik , 1903 .

[11]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[12]  C. Marle,et al.  "Sur une forme nouvelle des ´ equations de la M´ ecanique" , 2013 .

[13]  H. C. Andersen.  Rattle: A “velocity” version of the shake algorithm for molecular dynamics calculations , 1983 .

[14]  P. Ehrenfest,et al.  Die Lagrange-Eulersche Gleichungen der Mechanik , 1905 .

[15]  David Cohen.  Trigonometric schemes for stiff second-order SDEs , 2006 .

[16]  Yuri N. Fedorov,et al.  Discrete Nonholonomic LL Systems on Lie Groups , 2004, math/0409415.

[17]  Taeyoung Lee,et al.  Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2603 Lagrangian mechanics and variational integrators on two-spheres , 2022 .

[18]  A. Bloch,et al.  Nonholonomic Mechanics and Control , 2004, IEEE Transactions on Automatic Control.

[19]  J. Marsden,et al.  Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators , 1988 .

[20]  Robert I. McLachlan,et al.  Integrators for Nonholonomic Mechanical Systems , 2006, J. Nonlinear Sci..