Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
classical mechanics; rigid system; Newton’s law; Riemannian geometry
classical mechanics; rigid system; Newton’s law; Riemannian geometry
Summary:
We start by formulating geometrically the Newton’s law for a classical free particle in terms of Riemannian geometry, as pattern for subsequent developments. For constrained systems we have intrinsic and extrinsic viewpoints, with respect to the environmental space. Multi–particle systems are modelled on $n$-th products of the pattern model. We apply the above scheme to discrete rigid systems. We study the splitting of the tangent and cotangent environmental space into the three components of center of mass, of relative velocities and of the orthogonal subspace. This splitting yields the classical components of linear and angular momentum (which here arise from a purely geometric construction) and, moreover, a third non standard component. The third projection yields a new explicit formula for the reaction force in the nodes of the rigid constraint.
References:
[1] Abraham R., Marsden J.: Foundations of Mechanics. Benjamin, New York, 1986.
[2] Arnol’d V. I.: Mathematical methods of classical mechanics. MIR, Moscow 1975; GTM n. 70, Springer. MR 0997295
[3] Cortizo S. F.: Classical mechanics–on the deduction of Lagrange’s equations. Rep. Math. Phys. 29, No. 1 (1991), 45–54. MR 1137498 | Zbl 0744.70024
[4] Crampin M.: Jet bundle techniques in analytical mechanics. Quaderni del CNR, GNFM, Firenze, 1995.
[5] Curtis W. D., Miller F. R.: Differentiable manifolds and theoretical physics. Academic Press, New York, 1985. MR 0793015
[6] de Leon M., Rodriguez P. R.: Methods of differential geometry in analytical mechanics. North Holland, Amsterdam, 1989. MR 1021489
[7] Gallot S., Hulin D., Lafontaine J.: Riemannian Geometry. II ed., Springer Verlag, Berlin, 1990. MR 1083149 | Zbl 0716.53001
[8] Godbillon C.: Geometrie differentielle et mechanique analytique. Hermann, Paris, 1969. MR 0242081 | Zbl 0174.24602
[9] Goldstein H.: Classical Mechanics. II ed., Addison–Wesley, London, 1980. MR 0575343 | Zbl 0491.70001
[10] Guillemin V., Sternberg S.: Symplectic techniques in physics. Cambridge Univ. Press, 1984. MR 0770935 | Zbl 0576.58012
[11] Janyška J., Modugno M., Vitolo R.: Semi–vector spaces. preprint 2005.
[12] Landau L., Lifchits E.: Mechanics. MIR, Moscow 1975.
[13] Levi–Civita T., Amaldi U.: Lezioni di Meccanica Razionale. vol. II, II ed., Zanichelli, Bologna, 1926.
[14] Libermann P. Marle C.-M.: Symplectic geometry and analytical mechanics. Reidel, Dordrecht, 1987. MR 0882548
[15] Lichnerowicz A.: Elements of tensor calculus. John Wiley & Sons, New York, 1962. MR 0149903 | Zbl 0103.38402
[16] Littlejohn R. G., Reinsch M.: Gauge fields in the separation of rotations and internal motions in the $n$–body problem. Rev. Modern Phys. 69, 1 (1997), 213–275. MR 1432649
[17] Marsden J. E., Ratiu T.: Introduction to Mechanics and Symmetry. Texts Appl. Math. 17, Springer, New York, 1995. MR 1723696
[18] Massa E., Pagani E.: Classical dynamics of non–holonomic systems: a geometric approach. Ann. Inst. H. Poincaré 55, 1 (1991), 511–544. MR 1130215 | Zbl 0731.70012
[19] Massa E., Pagani E.: Jet bundle geometry, dynamical connections and the inverse problem of Lagrangian mechanics. Ann. Inst. H. Poincaré (1993).
[20] Modugno M., Tejero Prieto C., Vitolo R.: A covariant approach to the quantisation of a rigid body. preprint 2005.
[21] Park F. C., Kim M. W.: Lie theory, Riemannian geometry, and the dynamics of coupled rigid bodies. Z. Angew. Math. Phys. 51 (2000), 820–834. MR 1788187 | Zbl 0998.70004
[22] Souriau J.-M.: Structure des systèmes dynamiques. Dunod, Paris 1969. MR 0260238
[23] Tulczyjew W. M.: An intrinsic formulation of nonrelativistic analytical mechanics and wave mechanics. J. Geom. Phys. 2, 3 (1985), 93–105. MR 0851123 | Zbl 0601.70001
[24] Vershik A. M., Faddeev L. D.: Lagrangian mechanics in invariant form. Sel. Math. Sov. 4 (1981), 339–350.
[25] Warner F. W.: Foundations of differentiable manifolds and Lie groups. Scott, Foresman and Co., Glenview, Illinois, 1971. MR 0295244 | Zbl 0241.58001
[26] Whittaker E. T.: A treatise on the analytical dynamics of particles and rigid bodies. Wiley, New York, 1936.
Search
Partner of
