Euler-Lagrange equations and geometric mechanics on Lie groups with potential
Subject Areas : Statistics
Ali Suri
1
(Department of Mathematics, Faculty of Sciences, Bu-Ali Sina University, Hamedan, Iran)
Keywords: لاگرانژی, جسم صلب, معادلات اویلر-لاگرانژ, اسپری, معادلات کاماسا-هلم,
Abstract :
Abstract. Let G be a Banach Lie group modeled on the Banach space, possibly infinite dimensional, E. In this paper first we introduce Euler-Lagrange equations on the Lie group G with potential and right invariant metric. Euler-Lagrange equations are natural extensions of the geodesic equations on manifolds and Lie groups. In the second part, we study the geometry of the mechanical system of a rigid body with a fixed point in the gravitational field. This Mechanical systems is usually know as symmetric heavy top. Then we show that the extracted equations by this theory coincide with the known equations of heavy top. Finally, as an infinite dimensional example, we study the Camassa-Holm equations on Bott-Virasoro group at the presence of potential. Bott-Virasoro group is the product of the group of diffeomorphisms of the circle of Sobolev class by the real line and by a potential on a Lie group G we mean a differentiable function from G to the real line R.
[1] R. Abraham and J. Marsden, Foundations of Mechanics, 2nd ed., Addison–Wesley (1978).
[2] V. I. Arnold,: Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l’hydrodynamique des fluids parfaits, Ann. Inst. Grenoble, 16, 319–361 (1966).
[3] V.I. Arnold, Mathematical Methods of Classical Mechanics, New York, Springer-Verlag (1978).
[4] P. R. Chernoff and J. E. Marsden Properties of infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, Vol. 421, Springer-Verlag, New York (1974).
[5] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92, 102-163 (1970).
[6] J. Marsden, D. Ebin and A.E. Fischer, Diffeomorphism groups, hydrodynamics and relativity, Proceedings of the 13th Biennial Seminar of Canadian Mathematical Congress, Vol. 1, 135–279 (1972).
[7] A. Iacob, Invariant manifolds in the motion of a rigid body about a fixed point, Rev. Roum. Math. Pures. Appl., 16 (10), 1497-1521 (1971).
[8] D. Lewis, T. Ratiu, J.C. Simo and J. Marsden Heavy top: a geometric treatment, Nonlinearity, 5, 1-48 (1992).
[9] J.E. Marsden and T.S. Ratiu Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, Springer-Verlag, (1994).
[10] G. Misiolek, Shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys. 24, 203 – 208 (1998).
[11] S. Shkoller, Geometry and curvature of diffeomorphism groups with metric and mean Hydrodyna- mics, J. Funct. Anal. 160, 337–365 (1998).
[12] A. Suri, Second order time dependent tangent bundles and Geometric Mechanics, Mediterr. J. Math. 14: 154. https://doi.org/10.1007 /s00009-017-0954-2 (2017).
[13] M. Taylor, Finite and infinite dimensional Lie groups and evolution equations, Lecture notes (2003).
