Note on the Poisson structure of the damped oscillator
The damped harmonic oscillator is one of the most studied systems with respect to the problem of quantizing dissipative systems. Recently Chandrasekar et al. [J. Math. Phys. 48, 032701 (2007)] applied...
The damped harmonic oscillator is one of the most studied systems with respect to the problem of quantizing dissipative systems. Recently Chandrasekar et al. [J. Math. Phys. 48, 032701 (2007)] applied...
Dual Lindstedt series and Kolmogorov–Arnol'd–Moser theorem
We prove that there exists a Lindstedt series that holds when a Hamiltonian is driven by a perturbation going to infinity. This series appears to be dual to a standard Lindstedt series as it can be ob...
We prove that there exists a Lindstedt series that holds when a Hamiltonian is driven by a perturbation going to infinity. This series appears to be dual to a standard Lindstedt series as it can be ob...
On continuum dynamics
J. Math. Phys. 50, 102903 (2009); doi:10.1063/1.3215979
Published 22 October 2009
You are not logged in to this journal. Log in
The theory of continuous dynamical systems is developed with an intrinsic geometric approach based on the action principle formulated in the velocity-time manifold. By endowing the finite dimensional Riemannian ambient manifold with a connection, an induced connection is naturally defined in the infinite dimensional configuration manifold of maps. The motion is shown to be governed, in the Banach configuration manifold, by a generalized Lagrange law and, in the ambient manifold, by a generalized Euler law which is independent of the Banach topology of the configuration manifold. Extended versions of Euler–Poincaré law, Euler classical laws and d'Alembert law are also derived as special cases. Stress fields in the body are introduced as Lagrange's multipliers of the rigidity constraint on virtual velocities, dual to the Lie derivative of the metric. No special assumptions are made so that any constitutive behaviors can be modeled.
©2009 American Institute of Physics
| History: | Received 17 October 2008; accepted 7 August 2009; published 22 October 2009 |
| Permalink: |
http://link.aip.org/link/?JMAPAQ/50/102903/1 |
| BUY THIS ARTICLE | (US$24) |
|
|
Connotea
CiteULike
del.icio.us
BibSonomy
REFERENCES (36)
For access to fully linked references, you need to log in.
For access to fully linked references, you need to Log in.
- Y. Choquet-Bruhat and C. DeWitt-Morette, Analysis, Manifolds and Physics (North-Holland, New York, 1982).
- R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 3rd ed. (Springer-Verlag, New York, 2002).
- G. Romano, http://wpage.unina.it/romano/GR_Site/Lecture_notes_files/Mech.pdf.
- G. Romano, M. Diaco, and R. Barretta, in Mathematical Physics Models and Engineering Sciences (Liguori, Napoli, 2008), pp. 439–453.
- G. Romano, R. Barretta, and M. Diaco,
Int. J. Non-Linear Mech. 44, 689 (2009) . - G. Romano, R. Barretta, and A. Barretta,
Rep. Math. Phys. 63, 331 (2009) . - H. Goldschmidt and Sh. Sternberg, Ann. Inst. Fourier 23, 203 (1973).
- R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed. (Benjamin/Cummings, Reading, MA, 1978).
- L. Euler, Opera Omnia, edited by C. Carathéodory (Fussli, Zürich, 1952), pp. 298–308.
- Ju. I. Ne
mark and N. A. Fufaev, Dynamics of Nonholonomic Systems (American Mathematical Society, Providence, RI, 1972). - V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, in Dynamical Systems III, Encyclopedia of Mathematical Sciences, edited by V. I. Arnold (Springer-Verlag, New York, 1988).
- A. M. Vershik and V. Ya. Gershkovich, Nonholonomic Dynamical Systems, Geometry of Distributions and Variational Problems, Dynamical Systems VII (Springer-Verlag, New York, 1994).
- A. D. Lewis and R. M. Murray,
Int. J. Non-Linear Mech. 30, 793 (1995) . - F. Cardin and M. Favretti,
J. Geom. Phys. 18, 295 (1996) . - E. Massa and E. Pagani,
Ann. Inst. Henri Poincare 66, 1 (1997) . - G. Zampieri,
J. Differ. Equations 163, 335 (2000) . - E. Massa, E. Pagani, and S. Vignolo, J. Math. Phys. 44, 1709 (2003).
- I. Kupka and W. M. Oliva,
J. Differ. Equations 169, 169 (2001) . - J. Vankerschaver, E. Cantijn, M. de León, and D. Martín de Diego,
Rep. Math. Phys. 56, 387 (2005) . - D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series No. 142 (Cambridge University Press, Cambridge, 1989).
- V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1989) (translated from Mir, Moscow, 1979).
- I. Kolar, P. W. Michor, and J. Slovak, Natural Operations in Differential Geometry (Springer-Verlag, New York, 1993).
- W. M. Oliva, Geometric Mechanics, Lecture Notes in Mathematics 1798 (Springer-Verlag, Berlin, 2002).
- R. S. Palais,
Proc. Am. Math. Soc. 5, 902 (1954) . - P. W. Michor, http://www.mat.univie.ac.at/~michor/dgbook.pdf.
- R. S. Palais, Foundations of Global Non-Linear Analysis (Benjamin, New York, 1968).
- J. E. Marsden and T. J. R. Hughes, Mathematical Foundation of Elasticity (Prentice-Hall, Englewood Cliffs, NJ, 1983).
- H. Yoshimura and J. E. Marsden,
J. Geom. Phys. 57, 209 (2006) . - J. G. Papastavridis,
J. Appl. Mech. 64, 985 (1997) . - G. Terra and M. H. Kobayashi,
J. Math. Pures Appl. 83, 629 (2004) . - H. I. Eliasson, J. Diff. Geom. 1, 169 (1967).
- K. Yosida, Functional Analysis, 4th ed. (Springer-Verlag, New York, 1974).
- G. Fichera, Handbuch der Physik (Springer-Verlag, Berlin, 1972), Vol. VI/a.
- G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics (Springer-Verlag, New York, 1976), translated from Les Inéquations en Méchanique et en Physique (Dunod, Paris, 1972).
- G. Romano, Proceedings of the Symposium on Trends in Applications of Mathematics to Mechanics—STAMM 2000, 9–14 July (National University of Ireland, Galway, 2000).
- G. Romano and M. Diaco, in New Trends in Mathematical Physics (World Scientific, Singapore, 2004), pp. 193–204.
