Abstract
In this pedagogic article we study the geometrical structure of nonholonomic system and elucidate the relationship between Jacobi’s last multiplier (JLM) and nonholonomic systems endowed with the almost symplectic structure. In particular, we present an algorithmic way to describe how the two form and almost Poisson structure associated to nonholonomic system, studied by L. Bates and his coworkers (Rep Math Phys 42(1–2):231–247, 1998; Rep Math Phys 49(2–3):143–149, 2002; What is a completely integrable nonholonomic dynamical system, in Proceedings of the XXX symposium on mathematical physics, Toruń, 1998; Rep Math Phys 32:99–115, 1993), can be mapped to symplectic form and canonical Poisson structure using JLM. We demonstrate how JLM can be used to map an integrable nonholonomic system to a Liouville integrable system. We map the toral fibration defined by the common level sets of the integrals of a Liouville integrable Hamiltonian system with a toral fibration coming from a completely integrable nonholonomic system.
To Mahouton Norbert Hounkonnou, collaborator, colleague and friend, dedicated to his 60th birthday with admiration and gratitude
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Notes
- 1.
The physicist way of looking the constrained dynamics is different from our presentation, it is described by \( L = \frac {1}{2}\big (\dot {x}^{2} + \dot {y}^{2} + \dot {z}^{2}\big ) + \lambda \big ( \dot {z} - y\dot {x}\big ), \) where momenta are given by \(p_x = \dot {x} - \lambda y, \qquad p_y = \dot {y}, \qquad p_z = \dot {z} + \lambda , \qquad p_{\lambda }= 0, \) The usual Dirac analysis of constraints then identifies the following two constraints, ϕ 1 = p λ = 0 ϕ 2 = p z − yp x − λ(1 + y 2) primary and secondary, respectively, which are second class, {ϕ 1, ϕ 2} = (1 + y 2). It would be interesting to bridge the gap between these two methods.
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Acknowledgements
Most of the results herewith presented have been obtained in long-standing collaboration with Anindya Ghose Choudhury and Pepin Cariñena, which is gratefully acknowledged. The author wishes also to thank Gerardo Torres del Castillo, Clara Nucci, Peter Leach, Debasish Chatterji, Ravi Banavar, and Manuel de Leon for many enlightening discussions. This work was done mostly while the author was visiting IHES. He would like to express his gratitude to the members of IHES for their warm hospitality. The final part was done at IFSC, USP at Sao Carlos and the support of FAPESP is gratefully acknowledged with grant number 2016/06560-6.
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Guha, P. (2018). The Role of the Jacobi Last Multiplier in Nonholonomic Systems and Locally Conformal Symplectic Structure. In: Diagana, T., Toni, B. (eds) Mathematical Structures and Applications. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-319-97175-9_12
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