Elementary Thermo-mechanical Systems and Higher Order Constraints

• Cendra Hernán ^[1] ; Grillo, Sergio ^[3] ; Palacios, Amaya Maximiliano ^[2]
  1. [1] Universidad Nacional del Sur

     Universidad Nacional del Sur

     Argentina

     
  2. [2] Universidad Nacional del Comahue

     Universidad Nacional del Comahue

     Argentina

     
  3. [3] Instituto Balseiro - Comisión Nacional de Energía Atómica (Argentina)

  Mostrar afiliaciones +
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00371-8
• Enlaces
  • Texto completo
• Resumen

  • In this paper we study a class of physical systems that combine a finite number of mechanical and thermodynamic observables. We call them elementary thermo-mechanical systems (ETMS). We introduce these systems by means of simple examples, and we obtain their (time) evolution equations by using, essentially, the Newton’s laws and the First Law of Thermodynamics only. We show that such equations are similar to those defining certain constrained mechanical systems. From that, and addressing the main aim of the paper, we give a general definition of ETMS, in a variational formalism, as a particular subclass of the Lagrangian higher order constrained systems.

• Referencias bibliográficas
  • 1. Abraham, R., Marsden, J.E.: Foundations of Mechanics. The Benjamin/Cummings Publishing Company, San Francisco (1978)
  • 2. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis and Aplications. Addison-Wesley, London (1983)
  • 3. Amaya, M.P.: Introducción a los sistemas termo-mecánicos desde una perspectiva simpléctica. Ph.D. thesis, Universidad Nacional del Sur,...
  • 4. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)
  • 5. Bloch, A.M., Marsden, J.E., Zenkov, D.V.: Nonholonomic dynamics. Not. Am. Math. Soc. 52, 324–333 (2005)
  • 6. Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, New York (1985)
  • 7. Callen, H.B.: Thermodynamics and an Introduction to Thermostatistics, 2nd edn. Wiley, New York (1985)
  • 8. Cendra, H., Ferraro, S., Grillo, S.: Lagrangian reduction of generalized nonholonomic systems. J. Geom. Phys. 58, 1271–1290 (2008)
  • 9. Cendra, H.: Generalized nonholonomic mechanics, servomechanisms and related brackets. J. Math. Phys. 47, 022902 (2006)
  • 10. Cendra, H., Grillo, S.: Lagrangian systems with higher order constraints. J. Math. Phys. 48, 052904 (2007)
  • 11. Cendra, H., Ibort, A., de León, M., Martín de Diego, D.: A generalization of Chetaev’s principle for a class of higher order nonholonomic...
  • 12. Crampin, M., Sarlet, W., Cantrijn, F.: Higher-order differential equations and higher-order Lagrangian mechanics. Math. Proc. Camb. Philos....
  • 13. de León, M., Rodrigues, P.R.: Generalized Classical Mechanics and Field Theory. North-Holland, Amsterdam (1985)
  • 14. Dobronravov, V.V.: The fundamentals of the mechanics of nonholonomic systems. Vysshaya Shkola 271 (1970)
  • 15. Eberard, D., Maschke, B.M., van der Schaft, A.J.: An extension of hamiltonian systems to the thermodynamic phase space: towards a geometry...
  • 16. Fermi, E.: Thermodynamics. Dover Publications, Mineola (1956)
  • 17. Gay-Balmaz, F., Yoshimura, H.: A lagrangian variational formulation for nonequilibrium thermodynamics. Part I: discrete systems. J. Geom....
  • 18. Gay-Balmaz, F., Yoshimura, H.: Dirac structures in nonequilibrium thermodynamics. J. Math. Phys. 58, 012701 (2018)
  • 19. Goldstein, H., Poole, C., Safko, P.: Classical Mechanics, 3rd edn. Wiley, New York (2000)
  • 20. Grillo, S.: Higher order constrained Hamiltonian systems. J. Math. Phys. 50, 082901 (2009)
  • 21. Grillo, S., Zuccalli, M.: Variational reduction of Lagrangian systems with general constraints. J. Geom. Mech. 4(1), 49–88 (2012)
  • 22. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Wiley, New York (1963)
  • 23. Marsden, J.E., Ratiu, T.: Introduction to Mechanics and Symmetry. Springer, New York (1994)
  • 24. Moran, M.J., Shapiro, H.N., Boettner, D.D., Bailey, M.B.: Fundamentals of Engineering Thermodynamics, 7th edn. Wiley, New York (2011)
  • 25. Mrugala, R.: On contact and metric structures on thermodynamic spaces. Departmental Bulletin Paper, pp. 167–181 (2000)
  • 26. Mrugala, R., Nulton, J.D., Schön, J.C., Salamon, P.: Contact structure in thermodynamic theory. Rep. Math. Phys. 29, 109–121 (1991)
  • 27. Neimark, J.I., Fufaev, N.A.: Dynamics of Non-holonomic Systems. Translations of Mathematical Monographs, vol. 33. American Mathematical...
  • 28. Planck, M.: Treatise on Thermodynamics. Dover Publications, Mineola (1917)
  • 29. Sniatycki, J., Tulczyjew, W.M.: Generating forms of Lagrangian submanifolds. Indiana Univ. Math. J. ´ 22(3), 267 (1972)
  • 30. Stueckelberg, E.C.G., Scheurer, P.B.: Thermocinétique phénoménologique galiléenne. Birkhäuser, Basel (1974)
  • 31. Tulczyjew, W.M.: The Legendre transformation. Ann. l’Inst. Henri Poincaré Sect. A XXVII, 101 (1977)
  • 32. Tulczyjew, W.M., Urba ´nski, P.: A slow and careful Legendre transformation for singular Lagrangians. Acta Phys. Pol. Ser. B 30(10), 2909–2978...
  • 33. Zemansky, M.: Heat and Thermodynamics: An Intermediate Textbook, 5th edn. McGraw-Hill, New York (1967)