Advertisement

The General Equations of Analytical Dynamics

  • V. V. Rumyantsev
Part of the International Centre for Mechanical Sciences book series (CISM, volume 387)

Abstract

Poincaré’s remarkable idea [1], [2] to represent motion’s equations of holonomic systems in terms of a certain transitive Lie group of infinitesimal transformations was extended by Chetayev [3]–[6] to the cases of rheonomic constraints and dependent variables, when transformation group is intransitive one. Chetayev transformed Poincaré’s equations to canonical form and elaborated the theory of their integration. These fine results were developed in a number of papers [7]–[23]. Our lectures represent an introduction in the theory of generalized Poincaré’s and Chetayev’s equations based on closed systems of transformations. These equations include both the motion’s equations in independent and dependent, holonomic and non-holonomic coordinates for holonomic and non-holonomic systems and in this sense are the general equations of analytical dynamics.

Keywords

Rigid Body Analytical Dynamics General Equation Virtual Displacement Canonical Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    Poincaré, H.: Sur une forme nouvelle des équations de la mécanique, C.R.Acad.Sci.Paris, 132 (1901), 369–371.MATHGoogle Scholar
  2. [2]
    Poincaré, H.: Sur la precession des corps deformables, Bull. astronomique, 27 (1910), 321–356.Google Scholar
  3. [3]
    Chetayev, N.: Sur les equations de Poincaré, C.R.Acad.Sci.Paris, 185 (1927), 1577–1578.Google Scholar
  4. [4]
    Chetayev, N.: Sur les equations de Poincaré, Dokl.Acad.Nauk SSSR, 7 (1928), 103–104.Google Scholar
  5. [5]
    Chetayev, N.: On Poincaré’s equations. Prikl.Mat.Mech., 5, 2 (1941), 253–262.Google Scholar
  6. [6]
    Chetayev, N.G.: Poincaré’s equations, in: Theoretical Mechanics (Ed. V.Rumyantsev and K.Yakimova), Mir Publishers, Moscow, 1989, 316–355.Google Scholar
  7. [7]
    Shurova, K.E.: A variation of Poincaré’s equations, Prikl.Mat.Mech, 17, 1 (1953), 123–124; 18, 3 (1954), 384.Google Scholar
  8. [8]
    Shurova, K.E.: The principal invariant of equations in variations, Vestnik MGIJ, Ser. Mat. Mekh. Astron. Fiz, 3 (1958), 47–49.Google Scholar
  9. [9]
    Rumyantsev, V.V.: The equations of motion of a solid with a cavity filled with liquid, Prikl.Mat.Mekh, 19, 1 (1955), 3–12.MathSciNetMATHGoogle Scholar
  10. [10]
    Aminov, M.Sh.: The construction of groups of possible displacements, in: Proceedings of the Interuniversity Conference on the Applied Theory of Stability and Analytic Mechanics, KAI, Kazan’, 1962, 21–30.Google Scholar
  11. [11]
    Bogoyavlenskii, A.A.: Cyclic displacements for the generalized area integral, Prild.Mat.Mekh., 25, 4 (1961), 774–777.MathSciNetGoogle Scholar
  12. [121.
    Bogoyavlenskii, A.A.: Theorems of the interaction between the parts of a mechanical system, Prikl.Mat.Mekh., 30, 1 (1966), 203–208.Google Scholar
  13. [13]
    Bogoyavlenskii, A.A.: The properties of possible displacements for theorems of the interaction between the parts of a mechanical system, Prikl.Mat.Mekh, 31, 2 (1967), 377–384.Google Scholar
  14. [14]
    Fam Guen: The equations of motion of non-holonomic mechanical systems in Poincaré-Chetayev variables, Prikl.Mat.Mekh, 31, 2 (1967), 253–259.MATHGoogle Scholar
  15. [15]
    Fain Guen: The equations of motion of non-holonomic mechanical systems in Poincaré-Chetayev variables, Prikl.Mat.Mekh, 32, 5 (1968), 804–814.Google Scholar
  16. [16]
    Fam Guen: One form of the equations of motion of mechanical systems, Prikl.Mat.Mekh, 33, 2 (1969), 397–402.Google Scholar
  17. [17]
    Markhashov, L.M.: The Poincaré and Poincaré-Chetayev equations, Prikl.Mat.Mekh, 49, 1 (1985), 43–55.MathSciNetGoogle Scholar
  18. [18]
    Markhashov, L.M.: A generalization of the canonical forni of Poinca.ré’s equations, Prikl.Mat.Mekh., 51, 1 (1987), 157–160.MathSciNetGoogle Scholar
  19. [19]
    Markhashov, L.M.: On a remark of Poincaré, Prikl.Mat.Mekh., 51, 5 (1987), 724–734.MathSciNetGoogle Scholar
  20. [20]
    Markhashov, L.M.: Particular solutions of the equations of motion and their stability, Izv.Acad.Nauk SSSR, MTT, 6 (1987), 26–32.Google Scholar
  21. [21]
    Rumyantsev, V.V.: On the Poincaré–Chetayev equations, Prikl. Mat. Mekh., 58, 3 (1994), 3–16.MathSciNetGoogle Scholar
  22. [22]
    Rumyantsev, V.V.: On the Poincaré–Chetayev equations„ Dokl. Ross. Acad. Nauk, 338, 1 (1994), 51–53.MathSciNetGoogle Scholar
  23. [23]
    Rumyantsev, V.V.: General equations of analytical dynamics, Prikl.Mat.Mekh., 60, 6 (1996), 929–940.MathSciNetGoogle Scholar
  24. [24]
    Eisenhart, L.P.: Continous Groups of Transformations, Princeton University, 1933.Google Scholar
  25. [25]
    Lur’ye, A.I.: Analytical Mechanics, Fizmatgiz, Moscow, 1961.Google Scholar
  26. [26]
    de la Vallé-Poissin, S.J.: Cours d’Analyse Infinitésimale, vol. 2, Gauthier-Villars, Paris, 1912.Google Scholar
  27. [27]
    Arnold, V.I.: Mathematical Methods of Classical Mechanics, Nauka, Moscow, 1989.CrossRefGoogle Scholar
  28. [28]
    Maggi, G.A.: Da alcune nuove forma della equazioni della dinamica applicabile ai sistemi auolonomi, Atti della Reale Acad. Naz. dei Lincei.Rend.Cl.fis. e math. Ser. 5. Vo1.10. Fasc. 2 (1901), 287–291.Google Scholar
  29. [29]
    Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, Cambridge, 1927.MATHGoogle Scholar
  30. [30]
    Boltzmann, L.: Uber die Form der Lagrange’schen gleichungen für nicht holonome, generalisierte koordinaten, Sitzungsb. der Wiener Acad. der wissenschaften. Math.-Naturwiss. K1., Bd. 140, 11.1–2 (1902), 1603–1614.Google Scholar
  31. [31]
    Hamel, G.: Die Lagrange-Eulerschen gleichungen der mechanik, Z.Math, und Phys., Bd. 50 (1904), 1–57.MATHGoogle Scholar
  32. [32]
    Zeiliger, D.N.: Motion’s theory of a similarity transformated body. Kazan’, 1892.Google Scholar
  33. [33]
    Chaplygin, S.A.: On a possible generalization of the areas theorem with application to the rolling of balls, in: Collected Works, Vol 1, Gostekhizdat, Moscow (1948), 26–56.Google Scholar
  34. [34]
    Rumyantsev, V.V.: Parametric examination of dynamical non-holonomic systems and two problems of dynamics, in: Differential Equations: Qualitative Theory, Vol.2, Amsterdam North-Holland (1987), 883–919.Google Scholar
  35. [35]
    Jacobi, C.G.J.: Vorlesungen über Dynamik. Verlag von G.Reimer, Berlin, 1884.Google Scholar
  36. [36]
    Suslov, G.K.: Theoretical Mechanics, Gostekhizdat, Moscow, 1946.Google Scholar
  37. [37]
    Marsden, J.E.: Lectures on Mechanics. London Math.Soc. Lecture Note Series, 174. Cambridge University Press, Cambridge, 1992.Google Scholar
  38. [38]
    Arnol’d, V.I., Kozlov, V.V. and A.I.Neishtadt: Mathematical Aspects of Classical and Celestial Mechanics, in: Itogi Nauki i Tekhniki, Ser. Sovr. Problemy Matematiki. Fundamental’nye Napravleniya, 3, VINITI, Moscow, 1985.Google Scholar
  39. [39]
    Pars, L.A.: A Treatise on Analytical Dynamics. Heinemann, London, 1964.Google Scholar
  40. [40]
    Neimark, Yu.I. and N.A.Fufayev: Dynamics of Non-holonomic Systems. Nauka, Moscow, 1967.Google Scholar
  41. [41]
    Voronets, P.V.: On the equations of motion for non-holonomic systems. Mat. Sbornik, 22, 4 (1902), 659–686.Google Scholar
  42. [42]
    Chaplygin, S.A.: On the motion of a heavy body of rotation upon the horizontal plane, in: Collected Works, Vol. 1, Gostekhizdat, Moscow, (1948), 57–75.Google Scholar
  43. [43]
    Volterra, V.: Sopra una classe di equazioni dinamiche. Atti della R.Accacl.delle Sci.di Torino, t. 33 (1897), 451–475.Google Scholar
  44. [44]
    Kane, T.R.: Dynamics of non-holonomic systems. Trans. ASME. Ser. E. J.Appl.Mech., 25, 4 (1961), 574–578.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1998

Authors and Affiliations

  • V. V. Rumyantsev
    • 1
  1. 1.Russian Academy of SciencesMoscowRussia

Personalised recommendations