Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-19T16:56:40.891Z Has data issue: false hasContentIssue false
  • English
  • Français

An action principle for the Vlasov equation and associated Lie perturbation equations. Part 1. The Vlasov—Poisson system

Published online by Cambridge University Press:  13 March 2009

Jonas Larsson
Jonas Larsson
Affiliation:
Department of Plasma Physics, Umeå University, S-90187 Umeå, Sweden, and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

A new action principle determining the dynamics of the Vlasov–Poisson system is presented (the Vlasov–Maxwell system will be considered in Part 2). The particle distribution function is explicitly a field to be varied in the action principle, in which only fundamentally Eulerian variables and fields appear. The Euler–Lagrange equations contain not only the Vlasov–Poisson system but also equations associated with a Lie perturbation calculation on the Vlasov equation. These equations greatly simplify the extensive algebra in the small-amplitude expansion. As an example, a general, manifestly Manley–Rowesymmetric, expression for resonant three-wave interaction is derived. The new action principle seems ideally suited for the derivation of action principles for reduced dynamics by the use of various averaging transformations (such as guiding-centre, oscillation-centre or gyro-centre transformations). It is also a powerful starting point for the application of field-theoretical methods. For example, the recently found Hermitian structure of the linearized equations is given a very simple and instructive derivation, and so is the well-known Hamiltonian bracket structure of the Vlasov–Poisson system.

Type
Research Article
Information
Journal of Plasma Physics , Volume 48 , Issue 1 , August 1992 , pp. 13 - 35
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Azizov, T. Ta. & Iokhvidov, I. S. 1989 Linear Operators in Spaces with an Indefinite Metric. Wiley.Google Scholar
Brizard, A. 1989 J. Plasma Phys. 41, 541.CrossRefGoogle Scholar
Cary, J. R. 1981 Phys. Rep. 79, 131.CrossRefGoogle Scholar
Cary, J. R. & Kaufman, A. N. 1981 Phys. Fluids 24, 1238.CrossRefGoogle Scholar
Cary, J. R. & Littlejohn, R. G. 1983 Ann. Phys. (NY) 151, 1.CrossRefGoogle Scholar
Deprit, A. 1969 Cel. Mech. 1, 12.CrossRefGoogle Scholar
Dewar, R. L. 1973 Phys. Fluids 16, 1102.CrossRefGoogle Scholar
Dewar, R. L. 1976 J. Phys. A 9, 2043.Google Scholar
Dirac, P. A. M. 1964 Lectures in Quantum Mechanics. Yeshiva University, New York.Google Scholar
Dougherty, J. P. 1974 J. Plasma Phys. 11, 331.CrossRefGoogle Scholar
Dragt, A. J. & Finn, J. M. 1976 J. Math. Phys. 17, 2215.CrossRefGoogle Scholar
Dragt, A. J. & Finn, J. M. 1979 J. Math. Phys. 20, 2649.CrossRefGoogle Scholar
Gibbons, J. 1981 Physica 3D, 503.Google Scholar
Iwinski, R. & Turski, L. A. 1976 Lett. Appl. Engng Sci. 4, 179.Google Scholar
Kaufman, A. N. 1991 Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (ed. Rozmus, W. & Tuszynski, J. A.). World Scientific.Google Scholar
Kaufman, A. N. & Boghosian, B. M. 1984 Contemp. Math. 28, 169.CrossRefGoogle Scholar
Larsson, J. 1982 J. Plasma Phys. 28, 215.CrossRefGoogle Scholar
Larsson, J. 1991 Phys. Rev. Lett. 66, 1466.CrossRefGoogle Scholar
Littlejohn, R. G. 1982 J. Math. Phys. 23, 742.CrossRefGoogle Scholar
Littlejohn, R. G. 1983 J. Plasma Phys. 29, 111.CrossRefGoogle Scholar
Low, F. E. 1958 Proc. R. Soc. Lond. A 248, 282.Google Scholar
Morrison, P. J. 1980 Phys. Lett. 80A, 383.CrossRefGoogle Scholar
Nayfeh, A. 1973 Perturbation Methods, §5.7. Wiley.Google Scholar
Pfirsch, D. 1984 Z. Naturforsch. 39a, 1.CrossRefGoogle Scholar
Pfirsch, D. & Morrison, P. J. 1991 Phys. Fluids B 3, 271.CrossRefGoogle Scholar
Schutz, B. 1980 Geometrical Methods of Mathematical Physics. Cambridge University Press.CrossRefGoogle Scholar
Similon, P. L. 1985 Phys. Lett. 112A, 33.CrossRefGoogle Scholar
Ye, H. & Morrison, P. J. 1992 Phys. Fluids B (in press).Google Scholar
Ye, H., Morrison, P. J. & Crawford, J. D. 1991 Phys. Lett. 156A, 96.CrossRefGoogle Scholar