Dynamics Over Homogeneous Spaces

2024/04/18 by Uçgun, Filiz Çağatay, Esen, Oğul, Sütlü, Serkan
#14M17 #37J37 #57T15 #70H03 #70H05 #70H50 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2404.12101

Abstract

We present the Euler-Lagrange and Hamilton's equations for a system whose configuration space is a unified product Lie group G=M\bowtie_γ H, for some γ:M× M → H. By reduction, then, we obtain the Euler-Lagrange type and Hamilton's type equations of the same form for the quotient space M≅ G/H, although it is not necessarily a Lie group. We observe, through further reduction, that it is possible to formulate the Euler-Poincaré type and Lie-Poisson type equations on the corresponding quotient \mathfrakm≅ \mathfrakg/\mathfrakh of Lie algebras, which is not a priori a Lie algebra. Moreover, we realize the nth order iterated tangent group T⁽ⁿ⁾G of a Lie group G as an extension of the nth order tangent group TⁿG of the same type. More precisely, \mathfrakg being the Lie algebra of G, T⁽ⁿ⁾G ≅ \mathfrakg^{× 2n-1-n} \bowtie_γTⁿG for some γ:\mathfrakg^{× 2n-1-n} × \mathfrakg^{× 2n-1-n} → TⁿG. We thus obtain the nth order Euler-Lagrange (and then the nth order Euler-Poincaré) equations over TⁿG by reduction from those on T(T^n-1G). Finally, we illustrate our results in the realm of the Kepler problem, and the non-linear tokamak plasma dynamics.

Related

• The Euler-Poincare Equations and Semidirect Products with Applications to Continuum Theories
• Tulczyjew's Triplet for Lie Groups II: Dynamics
• Reduction by symmetries of contact mechanical systems on Lie groups
• Discrete Euler-Poincaré and Lie-Poisson Equations
• Lagrangian Dynamics on Matched Pairs