An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes

Reports on Mathematical Physics

Volumn 60, Issue 2, 2007, Pages 175-198

  Eberard, D ^a   Maschke, B M ^a   van der Schaft, A J ^b  

^a UNIVERSITÉ DE LYON   (France)

^b UNIVERSITY OF GRONINGEN   (Netherlands)

Author keywords

contact structure; Hamiltonian systems; irreversible thermodynamics

Indexed keywords

EID: 37549022211     PISSN: 00344877     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0034-4877(07)00024-9     Document Type: Article

References (41)

1. Abraham R., and Marsden J.E. Foundations of Mechanics (1994), Addison
2. Arnold V.I. Equations Différentielles Ordinaires (1978), Mir, Moscou
3. Arnold V.I. Mathematical Methods of Classical Mechanics (1989), Springer
4. Brockett R.W. Control theory and analytical mechanics. Geometric Control Theory, Vol. 7 of Lie Groups: History, Frontiers and Applications (1977), Math. Sci. Press, Brookline
5. Carathéodory C. Untersuchungen über die Grundlagen der Thermodynamik. Math. Ann. 67 (1909)
6. Courant T.J. Dirac manifolds. Trans. American Math. Soc. 319 (1990) 631-661
7. Dalsmo M., and van der Schaft A.J. On representations and integrability of mathematical structures in energy-conserving physical systems. SIAM Journal of Control and Optimization 37 1 (1999) 54-91
8. de Groot S.R., and Mazur P. Non-equilibrium Thermodynamics (1962), North-Holland Publ., Amsterdam
9. Dorfman I. Dirac Structures and lntegrability of Nonlinear Evolution Equations (1993), Wiley
10. Eberard D., and Maschke B.M. An extension of port Hamiltonian systems to irreversible systems. Proc. IFAC NOLCOS'04. Stuttgart (2004)
11. Eberard D., Maschke B.M., and van der Schaft A.J. Conservative systems with ports on contact manifolds. Proc. IFAC World Congress. Prague (2005)
12. Eberard D., Maschke B.M., and van der Schaft A.J. Port contact systems for irreversible thermodynamical systems. Proc. 44th IEEE CDC and ECC 2005. Seville (2005)
13. Gibbs J.W. Graphical methods in the thermodynamics of fluids. Trans. Conn. Acad 2 (1873) 309-342
14. Gibbs J.W. Method of geometrical representation of the termodynamic properties of substances by means of surfaces. Trans. Conn. Acad 2 (1873) 382-404
15. Godbillon C. Géomêtrie différentielle et mécanique analytique (1969), Hermann, Paris
16. Grmela M. Reciprocity relations in thermodynamics. Physica A 309 (2002) 304-328
17. Herman R. Geometry, Physics and Systems (1973), Dekker, New-York
18. Ingarden R.S., and Jamiolkowski A. Classical Electrodynamics (1985), PWN-Polish Scientific Publisher Elsevier, Warszawa
19. Jongschaap R., and Öttinger H.C. The mathematical representation of driven thermodynamical systems. J. Non-Newtonian Fluid Mech 120 (2004) 3-9
20. Libermann P., and Marie C.-M. Symplectic Geometry and Analytical Mechanics (1987), D. Reidel Publishing Company, Dordrecht
21. Marsden J.E. Number 174 in London Mathematical Society Lecture Notes Series. Lectures on Mechanics (1992), Cambridge University Press, Cambridge
22. Maschke B.M., and van der Schaft A.J. Port controlled Hamiltonian systems: modeling origins and system theoretic properties. Proc NOLCOS'92 (1992) 282-288 Bordeaux
23. Maschke B.M., and van der Schaft A.J. Interconnected mechanical systems. Part 2: The dynamics of spatial mechanical networks. Modelling and Control of Mechanical Systems (1997), Imperial College Press, London 17-30
24. Maschke B.M., van der Schaft A.J., and Breedveld P.C. An intrinsic Hamiltonian formulation of network dynamics: Non-standard Poisson structures and gyrators. Journal of the Franklin Institute 329 (1992) 923-966
25. Maschke B.M., van der Schaft A.J., and Breedveld P.C. An intrinsic Hamiltonian formulation of the dynamics of LC-circuits. IEEE Trans. Circuit and Systems I : Fundamental Theory and Applications 42 (1995) 73-82
26. Mrugała R. Geometrical formulation of equilibrium phenomenological thermodynamics. Rep. Math. Phys 14 (1978) 419-427
27. Mrugała R. A new representation of Thermodynamic Phase Space. Bull. Polish Acad. Sci 28 (1980) 13-18
28. Mrugała R. Geometrical structure of the state space in classical statistical and phenomenological thermodynamics. Rep. Math. Phys 27 (1989) 145-159
29. Mrugała R. Continuous contact transformations in thermodynamics. Rep. Math. Phys 33 (1993) 149-154
30. Mrugała R. On a special family of thermodynamic processes and their invariants. Rep. Math. Phys 46 (2000) 461-468
31. Mrugała R., Nulton J.D., Schon J.C., and Salamon P. Contact structure in thermodynamic theory. Rep. Math. Phys 29 (1991) 109-121
32. Nijmeijer H., and van der Schaft A.J. Nonlinear Dynamical Control Systems (1990), Springer, New York
33. Ortega J.-P., and Planas-Bielsa V. Dynamics on Leibniz manifolds. J. Geom. Phys 52 (2004) 1-27
34. Ortega R., van der Schaft A.J., Maschke B.M., and Escobar G. Interconnection and damping assignment: passivity-based control of port-controlled Hamiltonian systems. Automatica 38 (2002) 585-596
35. Sandler S.I. Chemical and Engineering Thermodynamics (1999), Wiley
36. van der Schaft A.J. System Theory and Mechanics. Three Decades of Mathematical System Theory, Vol. 135 of Lect. Notes Contr. Inf. Sci. (1989), Springer, Berlin 426-452 chapter
37. van der Schaft A.J. Interconnection and Geometry. In: Polderman J.W., and Trentelman H.L. (Eds). The Mathematics of Systems and Control, From Intelligent Control to Behavioral Systems (1999), Univ. Groningen, Groningen 203-218 chapter
38. 2-Gain and Passivity Techniques in Nonlinear Control (2000), Springer, London
39. van der Schaft A.J., and Maschke B.M. The Hamiltonian formulation of energy conserving physical systems with external ports. Archiv für Elektronik und Übertragungstechnik 49 (1995) 362-371
40. van der Schaft A.J., and Maschke B.M. Interconnected mechanical systems. Part 1 : Geometry of interconnection and implicit Hamiltonian systems. Modelling and Control of Mechanical Systems (1997), Imperial College Press, London 1-16 chapter
41. Willems J.C. Dissipative dynamical systems, Part 1: General theory. Arch. Rat. Mech. Anal 45 (1972) 321-351