Dynamics over cocycle double cross-products
International Journal of Geometric Methods in Modern Physics, cilt.22, sa.7, 2025 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 22 Sayı: 7
- Basım Tarihi: 2025
- Doi Numarası: 10.1142/s0219887825500367
- Dergi Adı: International Journal of Geometric Methods in Modern Physics
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, Metadex, zbMATH, Civil Engineering Abstracts
- Anahtar Kelimeler: Euler-Poincaré dynamics, Lie-Poisson dynamics, Unified product
- Maltepe Üniversitesi Adresli: Evet
Özet
In this paper, we present the Euler-Lagrange and Hamilton's equations for a system whose configuration space is a unified product Lie group G = ×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 ≅ 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 m ≅ g/h 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, is the Lie algebra of G, T(n) ≅ g×2n-1-n ×γ TnG for some γ: g×2n-1-n ×g ×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(Tn-1G). Finally, we illustrate our results in the realm of the Kepler problem, and the nonlinear tokamak plasma dynamics.