[Paper Review] Reductions of Dynamics on Second Iterated Bundles of Lie Groups

Oùgul Esen, Hasan Gümral|arXiv (Cornell University)|Mar 23, 2015

Advanced Differential Geometry Research27 references3 citations

TL;DR

This paper develops a systematic framework for Hamiltonian and Lagrangian dynamics on second iterated bundles of Lie groups using trivializations of the first kind, which preserve lifted group structures. It derives reduced dynamics—including Euler-Poincaré and Lie-Poisson equations—via symplectic and Poisson reductions on spaces like $^{1}TT^*G$, $^{1}T^*TG$, and $^{1}T^*T^*G$, enabling unified treatment of higher-order dynamics and symplectic structures on Lie group bundles.

ABSTRACT

We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent of cotangent bundle of Lie group. We present all possible Poisson, symplectic and Lagrangian reductions of spaces and corresponding dynamics on them. In particular, reduction of Lagrangian dynamics on second iterated tangent bundle includes reduction of dynamics on second order tangent bundle.

Motivation & Objective

• To develop a consistent framework for Hamiltonian and Lagrangian dynamics on second iterated bundles of Lie groups using trivializations that preserve semidirect product and group structures.
• To derive reduced dynamics—specifically Euler-Poincaré and Lie-Poisson equations—on tangent and cotangent bundles of Lie groups via symplectic and Poisson reduction.
• To unify higher-order dynamics, including second-order Euler-Lagrange equations, on $T^2G$ through trivialized structures on $^{1}TTG$ and $^{1}TT^*G$.
• To establish the role of Tulczyjew’s symplectic structure on $^{1}TT^*G$ in enabling both Hamiltonian and Lagrangian dynamics with multiple reduction pathways.
• To generalize reduction techniques to higher-order symplectic and Lagrangian systems on iterated bundles using group actions and lifted structures.

Proposed method

• Utilizes trivializations of the first kind to identify $TG$ and $T^*G$ with semidirect products $G\circledS\mathfrak{g}$ and $G\circledS\mathfrak{g}^*$, preserving group and Lie algebraic structures.
• Trivializes iterated bundles $T(G\circledS\mathfrak{g})$, $T(G\circledS\mathfrak{g}^*)$, $T^*(G\circledS\mathfrak{g})$, and $T^*(G\circledS\mathfrak{g}^*)$ as semidirect products with Lie algebras and their duals.
• Derives trivialized Euler-Lagrange and Hamilton’s equations on $^{1}TTG$, $^{1}TT^*G$, $^{1}T^*TG$, and $^{1}T^*T^*G$ using canonical symplectic forms and group actions.
• Applies symplectic and Poisson reduction via group actions of $G$, $\mathfrak{g}$, and their semidirect products to obtain reduced dynamics on quotient spaces.
• Employs Tulczyjew’s symplectic structure on $^{1}TT^*G$ with two potential one-forms $\theta_1$ and $\theta_2$ to define Hamiltonian dynamics and derive reduced equations.
• Uses the immersion $T^2G \to TTG$ to define second-order Euler-Lagrange equations on $G\circledS(\mathfrak{g}\times\mathfrak{g})$ and second-order Euler-Poincaré equations on $2\mathfrak{g}$.

Experimental results

Research questions

• RQ1 How can Hamiltonian and Lagrangian dynamics be consistently formulated on second iterated bundles of Lie groups using trivializations that preserve lifted group structures?
• RQ2 What are the reduced dynamics—specifically Euler-Poincaré and Lie-Poisson equations—on $^{1}TT^*G$, $^{1}T^*TG$, and $^{1}T^*T^*G$ via symplectic and Poisson reduction?
• RQ3 How does the Tulczyjew symplectic structure on $^{1}TT^*G$ support both Hamiltonian and Lagrangian dynamics, and what reductions arise from its group action symmetries?
• RQ4 What is the relationship between second-order dynamics on $T^2G$ and the trivialized dynamics on $^{1}TTG$ and $^{1}TT^*G$?
• RQ5 Can the framework be generalized to symplectic reduction of tangent bundles of symplectic manifolds with lifted symplectic structures?

Key findings

• Trivialized Euler-Lagrange equations on $^{1}TTG \simeq (G\circledS\mathfrak{g})\circledS(\mathfrak{g}\circledS\mathfrak{g})$ yield second-order dynamics on $T^2G$ via immersion into $TTG$.
• Reduction of dynamics on $^{1}TT^*G$ under $G$-action yields Poisson reduced space $\mathfrak{g}\circledS(\mathfrak{g}^*\times\mathfrak{g}^*)$ with symplectic leaves $\mathcal{O}_\lambda \times \mathfrak{g}\times\mathfrak{g}^*$.
• Hamilton’s equations on $^{1}T^*TG$ lead to a Lie-Poisson structure on $\mathfrak{g}\circledS(\mathfrak{g}^*\times\mathfrak{g}^*)$ distinct from the product Poisson structure on $\mathfrak{g}^*\times\mathfrak{g}^*$.
• On $^{1}T^*T^*G$, symplectic reduction by $G\circledS\mathfrak{g}^*$ yields a Lie-Poisson structure on $\mathfrak{g}^*\circledS(\mathfrak{g}^*\times\mathfrak{g})$ with symplectic leaves $\mathcal{O}_\lambda \times \mathfrak{g}\times\mathfrak{g}^*$.
• Hamiltonian dynamics on $^{1}TT^*G$ with Tulczyjew’s symplectic form admits two potential one-forms $\theta_1$ and $\theta_2$, enabling dual formulations of dynamics.
• Lagrangian reductions on $^{1}TT^*G$ via $G$, $\mathfrak{g}_1^*$, and $G\circledS\mathfrak{g}_1^*$ yield reduced equations on $\mathfrak{g}_1^*\circledS(\mathfrak{g}_2\times\mathfrak{g}_3^*)$, $G\circledS(\mathfrak{g}_2\times\mathfrak{g}_3^*)$, and $\mathfrak{g}_2\circledS\mathfrak{g}_3^*$, respectively.