An Extension of the New Einstein Geometry by Michal A.D., Botsford J.L.

By Michal A.D., Botsford J.L.

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It is clear that ω is closed. If X = a i ∂x i + b ∂ξ , and Y = v ∂xi + i wi ∂ξ∂ i , then ω(X, Y ) = a · w − b · v. By setting v = w we obtain a = b, and by setting w = 0 we obtain a = b = 0. The next example shows that we can always formulate Hamilton’s equations on the cotangent bundle. This motivates the name for X H . 11 (Hamilton’s equations). Suppose M is a manifold, H is a function T ∗ M → R, and XH ∈ X (T ∗ M \ {0}) is the corresponding Hamiltonian vector field. If (xi , ξi ) are standard coordinates for T ∗ M \ {0}, then XH = ∂H ∂ ∂H ∂ − i .

12 (Hilbert 1-form). Let F be a Finsler norm on M . Then the Hilbert 1-form η ∈ Ω1 (T M \ {0}) is defined as η|y = −gij (y)y i dxj |y , y ∈ T M \ {0}. The next proposition shows that η is globally defined. 13. If L : T M → T ∗ M is the Legendre transformation induced by a Finsler norm, then L ∗ θ = η, dη is a symplectic form for T M \ {0}, and L : T M \ {0} → T ∗ M \ {0} is a symplectic mapping. Proof. The first claim follows directly from the definitions by expanding the left hand side. Since dθ is non-degenerate, it follows that dη is nondegenerate.

In the above, ι is the contraction mapping ι X : Ωr M → Ωr−1 M defined by (ιX ω)(·) = ω(X, ·). To see that XH is well defined, let us consider the mapping X → ω(X, ·). By non-degeneracy, it is injective, and by the rank-nullity theorem, it is surjective, so the Hamiltonian vector field X H is uniquely determined. 4 (Conservation of energy). Suppose (M, ω) is a symplectic manifold, XH is the Hamiltonian vector field XH ∈ X (M ) corresponding to a function H : M → R, and c : I → M is an integral curve of X H .

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