Euler-Poincaré Equation Approach

  • Gary Webb
Part of the Lecture Notes in Physics book series (LNP, volume 946)


Poincaré (1901) wrote down the Euler equations for a rigid body on so(3) in a matrix commutator form (see also Holm (2008b), Volume 2, p. 46). Arnold (1966) showed that the equations for ideal, incompressible fluid dynamics could be derived from a variational principle in which the Lagrangian consists of the fluid kinetic energy, subject to an infinite Lie group (pseudo-Lie group) constraint, associated with the Lagrangian map (the constraint is that the Lagrangian map x = φ(x0, t) for fixed t, is a differentiable (smooth) and measure preserving diffeomorphism). The group G, is known as Sdiff(R3). The variational formulation showed that when the Lagrangian l is a metric on the tangent space TG, the resultant variational equations (the Euler-Poincaré equations) are geodesic spray equations for geodesic motion on the group G with respect to the metric l. For the case of rigid body dynamics the group involved is the semi-direct product Lie group SE(3) = SO(3)ⓈR3. Euler-Poincaré variational principles have been developed by a number of authors (e.g. Marsden et al. (1984), Holm and Kupershmidt (1983a,b), Holm et al. (1998),Cendra et al. (2003), Arnold and Khesin (1998)). The geodesic spray equations for MHD were obtained by Ono (1995a,b). These equations are sometimes referred to as the Euler-Arnold equations. Araki (2015, 2017) determine the geodesic spray equations for incompressible Hall plasmas, known as XMHD (i.e. extended MHD). The curvature associated with the geodesic metric is negative for unstable flows. Holm et al. (1985) describes the use of Casimirs in stability analyses.


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© Springer International Publishing AG 2018

Authors and Affiliations

  • Gary Webb
    • 1
  1. 1.CSPARThe University of Alabama in HuntsvilleHuntsvilleUSA

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