[Paper Review] Locally Lagrangian Symplectic and Poisson Manifolds

Izu Vaisman|ArXiv.org|Aug 14, 2000

Geometry and complex manifolds10 references3 citations

TL;DR

This paper introduces locally Lagrangian symplectic (l.L.s.) and Poisson (l.L.P.) manifolds, generalizing symplectic structures from Lagrangian mechanics to global, possibly compact manifolds via local Lagrangian functions. It establishes a coordinate-free characterization of l.L.s. manifolds using a tangent structure S and shows that symplectic leaves of l.L.P. manifolds inherit locally Lagrangian symplectic structures, with applications to tangent bundles and fibered products of tangent bundles.

ABSTRACT

We discuss symplectic manifolds where, locally, the structure is that encountered in Lagrangian dynamics. Exemples and characteristic properties are given. Then, we refer to the computation of the Maslov classes of a Lagrangian submanifold. Finally, we indicate the generalization of this type of structures to Poisson manifolds.

Motivation & Objective

• To define and characterize locally Lagrangian symplectic (l.L.s.) manifolds as a generalization of global Lagrangian symplectic structures on tangent bundles.
• To extend the framework to Poisson manifolds by introducing locally Lagrangian Poisson (l.L.P.) structures compatible with a tangent structure S.
• To show that symplectic leaves of l.L.P. manifolds are themselves locally Lagrangian symplectic manifolds.
• To provide a coordinate-free characterization of l.L.s. manifolds using the compatibility condition ω(X,SY) = ω(Y,SX) for all vector fields X,Y.
• To investigate the existence of globally Lagrangian Poisson (g.L.P.) structures on fibered products of tangent bundles, particularly T^{(2)}N.

Proposed method

• Define a locally Lagrangian symplectic manifold as a manifold M with a tangent structure S and a symplectic form ω locally expressible as ω = d(θ_L) with θ_L = dL∘S for local Lagrangian functions L on M.
• Use the Nijenhuis tensor condition N_S = 0 to ensure integrability of the tangent structure S, which defines a vertical foliation V = im S = ker S.
• Characterize l.L.s. manifolds via the condition ω(X,SY) = ω(Y,SX) for all X,Y ∈ Γ(TM), which ensures that the vertical distribution is Lagrangian with respect to ω.
• Construct examples using quotient spaces of R^{2n} under Z^n × Z^n actions, such as the torus T^{2n}, to obtain compact l.L.s. manifolds from global Lagrangians.
• Generalize to Poisson structures by requiring compatibility of S with a Poisson bivector P, leading to the conditions (4.1)–(4.4), including P(dq^i, du^j) = P(dq^j, du^i).
• Show that fibered products M = TN ×_f TN admit l.L.P. structures when P is of the form P = P^{ij} ∂/∂q^i ∧ ∂/∂u^j + 1/2 A^{ij} ∂/∂u^i ∧ ∂/∂u^j with P^{ij} symmetric and A^{ij} antisymmetric.

Experimental results

Research questions

• RQ1 What conditions must a symplectic manifold satisfy to be locally modeled on the symplectic structure of Lagrangian mechanics on a tangent bundle?
• RQ2 How can the notion of a globally Lagrangian symplectic structure be generalized to non-exact symplectic forms on compact manifolds?
• RQ3 What is the role of the tangent structure S in characterizing locally Lagrangian symplectic manifolds?
• RQ4 How do symplectic leaves of a Poisson manifold inherit locally Lagrangian symplectic structures when equipped with a compatible tangent structure?
• RQ5 Under what conditions does a Poisson bivector field on a tangent bundle TN induce a globally Lagrangian Poisson structure?

Key findings

• A symplectic manifold M is locally Lagrangian if and only if its symplectic form ω satisfies ω(X,SY) = ω(Y,SX) for all vector fields X,Y, which ensures that the vertical foliation defined by S is Lagrangian.
• The symplectic leaves of a locally Lagrangian Poisson manifold are themselves locally Lagrangian symplectic manifolds, inheriting the structure from the ambient Poisson structure.
• A compact example of a locally Lagrangian symplectic manifold is the torus T^{2n}, constructed as a quotient of R^{2n} by a lattice action, where the Lagrangian function transforms under shifts by closed 1-forms and functions.
• The fibered product M = TN ×_f TN admits a locally Lagrangian Poisson structure when the Poisson bivector P is of the form P = P^{ij} ∂/∂q^i ∧ ∂/∂u^j + 1/2 A^{ij} ∂/∂u^i ∧ ∂/∂u^j with P^{ij} symmetric and A^{ij} antisymmetric.
• The symplectic structure on each leaf of the Poisson structure on M = TN ×_f TN is globally given by Λ_{ij}(x) dz^i ∧ dy^j, with a global Lagrangian function L = Λ_{ij}(x) z^i y^j, making (M,P,S) a globally Lagrangian Poisson manifold.
• On TN, a Poisson bivector P induces a l.L.P. structure only if it is zero-related (i.e., P(df,dg)=0 for all f,g) and satisfies P(dq^i,du^j) = P(dq^j,du^i), leading to the form (4.14) and the rank condition rank P = 2 rank(P^{ij}).