On the low Lagrangian formulation of Vlasov-Poisson equations

Year 2023, Volume: 52 Issue: 4, 983 - 994, 15.08.2023

Derya Çoksak Er

https://doi.org/10.15672/hujms.1189672

Abstract

In this work, two problems related with the Low Lagrangian formulation of the Vlasov-Poisson equations are solved. The first problem is related to the space on which the Low Lagrangian is defined. It is shown that the Low Lagrangian is defined on the tangent bundle of the densities of configuration space. The second problem is related to the assumptions which are called Low constraints. It is shown that Low constraints amount to the fact that the Low Lagrangian is invariant under a group action.

Keywords

low Lagrangian, Vlasov-Poisson equations, group actions on Lagrangians

References

• [1] R. Abraham, J.E Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications, Springer-Verlag, 1988.
• [2] H. Cendra, D.D. Holm, M.J.W. Hoyle and J.E. Marsden, The Maxwell-Vlasov equations in Euler-Poincare form, J. Math. Phys. 39, 3138-3157, 1998.
• [3] K. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Dover, 1977.
• [4] H. Gümral, Geometry of plasma dynamics I. Group of canonical diffeomorphisms, Journal of Mathematical Physics 51 (8), 501-523, 2010.
• [5] Z.R. Iwinski and L.A. Turski, Canonical theories of systems interacting electromagnetically, Lett. Appl. Eng. Sci. 4, 179-191, 1976.
• [6] J.H. Jeans, The stability of spherical nebula, Philos. Trans. R. Soc. London 199, 1-53, 1902.
• [7] J. Larsson, An action principle for the Vlasov equation and associated Lie perturbation equations. Part 1. The Vlasov-Poisson system, Journal of Plasma Physics 48, 13- 35,1992.
• [8] F.E. Low, A Lagrangian formulation of the Boltzmann-Vlasov equation for plasmas, Proc. R. Soc. London Ser. A 248, 282-287, 1958.
• [9] J.E. Marsden and T. Ratiu, Introduction to Symmetry and Mechanics, Springer, 1994.
• [10] P.J. Olver, Applications of Lie groups to Differential Equations, Springer Science and Business Media LLC, 1986.
• [11] H. Poincaré, Théorie des tourbillions, Gauthier-Villars, 1890.
• [12] H. Poincaré, Sur la stabilité de l’équilibre des figures piriformes affectées par une masse fluide en rotation, Philos. Trans. R. Soc. London Ser. A, 333-3 73, 1901.
• [13] A.A. Vlasov, On vibration properties of electron gas, J. Exp. Theor. Phys. 8 (3), 291-318, 1938.
• [14] A.A. Vlasov, The vibrational properties of an electron gas, Soviet Physics Uspekhi 10 (6), 721-733, 1968.
• [15] H. Ye and P.J. Morrison, Action principles for the Vlasov equation, Phys. Fluids B 4, 771-776, 1992.