Abstract
This paper investigates an algebraic structure and Poisson theory of single degree of freedom non-material volumes. The equations of motion are proposed in a contravariant algebraic form, and an algebraic product is determined. A consistent algebraic structure and a Lie algebra structure are proposed, and a proposition is obtained. The Poisson theory of the non-material volume is established, and five theorems are derived. Three examples are given to illustrate the application of the method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Irschik, H., Holl, H.J.: The equations of Lagrange written for a non-material volume. Acta Mech. 153, 231–248 (2002)
Cveticanin, L.: Dynamics of Bodies with Time-Variable Mass. Springer, Seelisberg (2016)
Irschik, H., Holl, H.J.: Mechanics of variable-mass systems—part 1: balance of mass and linear momentum. Appl. Mech. Rev. 57, 145–160 (2004)
Casetta, L.: The inverse problem of Lagrangian mechanics for a non-material volume. Acta Mech. 226, 1–15 (2015)
Casetta, L., Pesce, C.P.: The generalized Hamilton’s principle for a non-material volume. Acta Mech. 224, 919–924 (2013)
Casetta, L., Pesce, C.P.: The inverse problem of Lagrangian mechanics for Meshchersky’s equation. Acta Mech. 225, 1607–1623 (2014)
Irschik, H., Holl, H.J.: Lagrange’s equations for open systems, derived via the method of fictitious particles, and written in the Lagrange description of continuum mechanics. Acta Mech. 226, 63–79 (2015)
Casetta, L., Irschik, H., Pesce, C.P.: A generalization of Noether’s theorem for a non-material volume. Z. Angew. Math. Mech. 96, 696–706 (2016)
Casetta, L.: Poisson brackets formulation for the dynamics of a position-dependent mass particle. Acta Mech. 228, 4491–4496 (2017)
Mei, F.X.: Poisson’s theory of Birkhoffian system. Chin. Sci. Bull. 41, 641–645 (1996)
Mei, F.X.: The algebraic structure and Poisson’s theory for the equations of motion of non-holonomic systems. J. Appl. Math. Mech. 62, 155–158 (1998)
Fu, J.L., Chen, X.W., Luo, S.K.: Algebraic structures and Poisson integrals of relativistic dynamical equations for rotational systems. Appl. Math. Mech. 20, 1266–1275 (1999)
Luo, S.K., Chen, X.W., Guo, Y.X.: Algebraic structure and Poisson integrals of rotational relativistic Birkhoff system. Chin. Phys. 51, 523–528 (2002)
Liu, H.J., Tang, Y.F., Fu, J.L.: Algebraic structure and Poisson’s theory of mechanico-electrical systems. Chin. Phys. 15, 1653–1662 (2006)
Xia, L.L.: Poisson theory and inverse problem in a controllable mechanical system. Chin. Phys. Lett. 28, 120202 (2011)
Zhang, Y.: Poisson theory and integration method of Birkhoffian systems in the event space. Chin. Phys. B 19, 080301 (2010)
Zhang, Y., Shang, M.: Poisson theory and integration method for a dynamical system of relative motion. Chin. Phys. B 20, 024501 (2011)
Fu, J.L., Xie, F.P., Guo, Y.X.: Algebraic structure and Poissons integral theory of \(f (R)\) cosmology. Int. J. Theor. Phys. 51, 35–48 (2012)
Colarusso, M., Lau, M.: Lie–Poisson theory for direct limit Lie algebras. J. Pure Appl. Algebra 220, 1489–1516 (2016)
Benini, M., Schenkel, A.: Poisson algebras for non-linear field theories in the cahiers topos. Ann. Henri Poincaré 18, 1435–1464 (2017)
Ershkov, S.V.: A Riccati-type solution of Euler–Poisson equations of rigid body rotation over the fixed point. Acta Mech. 228, 2719–2723 (2017)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 11702119, 51609110, 11502071), the Natural Science Foundation of Jiangsu Province (No. BK20170565) and the Innovation Foundation of Jiangsu University of Science and Technology (1012931609, 1014801501-6).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jiang, WA., Liu, K., Xia, ZW. et al. Algebraic structure and Poisson brackets of single degree of freedom non-material volumes. Acta Mech 229, 2299–2306 (2018). https://doi.org/10.1007/s00707-018-2119-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-018-2119-1