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ALGEBRAIC DUALITY IN ANALYTICAL MECHANICS Journal Article Mechanics, Materials Science & Engineering, 19 , 2019, ISSN: 2412-5954. |

**Authors: Kalnitsky V. S.**

**ABSTRACT. **We consider one of the possible approaches to the conceptual description of the phase-space for mechanical systems with singular configuration space. We start with the real mechanical system for which no classical mathematical models are applicable, but the behaviour of the system can be experimentally observed. The choice of the adequate mathematical model for notion of cotangent bundle over singular manifold even for 0-singularity is an open problem. The next problem is the description of ODE or PDE notions and their solutions over such space. We hope that these models still lay in the domain of geometry. One example of such approach is the algebra of cosymbols of differential operators of the smooth functions algebra on a singular manifold. This algebra has the natural structure of the Hopf algebra and the dual algebra in the classical case coincides with the cotangent bundle of a smooth manifold. Generalizing this example, we introduce the notion of a universal graded algebra for which we can define the structure of the Hopf algebra and the Poisson bracket on the dual algebra in a natural way.

**Keywords: **Hopf algebra, cotangent space, double pendulum.

**References**

[1] S.N. Burian, V.S. Kalnitsky, On the motion of one-dimensional double pendulum, AIP Conference Proceedings 1959, 030004, 2018. DOI: 10.1063/1.5034584

[2] S. N. Burian, Behaviour of the pendulum with a singular configuration space, Vestnik SPbSU. Mathematics. Mechanics. Astronomy 4(62), 541-551, 2017. DOI: 10.21638/11701/spbu01.2017.402

[3] D.N. Mishra, L.N. Mishra, Differential operators over modules and rings as a path to the generalized differential geometry, Facta Universitatis (NIS) Ser. Math. Inform. 30(5), 1–12, 2015.

[4] A. Vinogradov, V. Kalnitsky, M. Sorokina, Tangent Space of a Commutative Algebra, ESF Exploratory Workshop on Current Problems in Differential Calculus over Commutative Algebras, Secondary Calculus, and Solution Singularities of Nonlinear PDEs, Vietri sul Mare, Italy, 2011. DOI: 10.13140/RG.2.2.27684.27523

[5] J. Watts, Diffeologies, Differential Spaces, and Symplectic Geometry, PhD thesis, University of Toronto, 2012. arXiv:1208.3634 [math.DG]

[6] T.A. Batubenge, Symplectic Frölicher space of constant dimension, PhD thesis, University of Cape Town, 2004.

[7] K. Drachal, Remarks on the behaviour of the higher-order derivations on the gluing differential cpaces, Czechoslovak Mathematical Journal, 65 (140), 1137-1154, 2015. DOI: 10.1007/s10587-015-0232-z

[8] V.E. Nazaikinskii, A.Yu. Savin, B.-W. Schulze, B.Yu. Sternin, Elliptic theory on singular manifolds, Chapman & Hall/CRC, Boca Raton, FL, 2006.

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