By Karl Friedrich Siburg

ISBN-10: 3540219447

ISBN-13: 9783540219446

New variational equipment by way of Aubry, Mather, and Mane, chanced on within the final 20 years, gave deep perception into the dynamics of convex Lagrangian structures. This ebook exhibits how this precept of Least motion seems in a number of settings (billiards, size spectrum, Hofer geometry, glossy symplectic geometry). hence, subject matters from sleek dynamical structures and glossy symplectic geometry are associated in a brand new and infrequently dazzling method. The imperative item is Mather's minimum motion practical. the extent is for graduate scholars onwards, but additionally for researchers in any of the themes touched within the book.

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Proof. First of all, one can show that min{AL (µ) | µ ∈ ML } = min{AL (µγ ) | γ abs. cont. curve} where µγ is the measure equally distributed along some absolutely continuous curve γ; see [21]. So we will prove that −c(L) = min{AL (µγ ) | γ abs. cont. curve}. For any curve γ, we have AL+c(L) (µγ ) ≥ 0 by definition of c(L). Therefore, −c(L) ≤ min{AL (µγ ) | γ abs. cont. curve}. To prove the reversed inequality, we observe that, whenever k < c(L), there exists a curve γ with AL+k (µγ ) < 0, which implies −k ≥ min{AL (µγ ) | γ abs.

The preimage aΛ ∈ H 1 (X, R) of [λ|Λ ] ∈ H 1 (Λ, R) under this isomorphism is called the Liouville class of Λ. The next theorem, firstly, says that Λ consists of supports of minimal measures and, secondly, shows that the Liouville class of Λ is a subgradient of the minimal action. Recall that a vector v ∈ Rn is a subgradient of a function f : Rn → R at x ∈ Rn if f (y) ≥ f (x) + v, y − x for all y ∈ Rn . If we have a strict inequality for all y = x, we say that v is a subgradient with only one point of tangency.

2] we have for any pair (u− , u+ ) of conjugate functions that u+ (γ(t)) − u− (γ(s)) = u+ (γ(t)) − u+ (γ(s)) = AL+c (γ|[s,t] ) for all s ≤ t. 11) we can choose a pair (u− , u+ ) of conjugate functions for which the Peierls barrier h satisfies h(γ(t), γ(s)) = u− (γ(s)) − u+ (γ(t)). Therefore, we can estimate AL+c (γ|[s,t] )+Φc (γ(t), γ(s)) ≤ u+ (γ(t))−u− (γ(s))+h(γ(t), γ(s)) = 0. 12) It is easy to show that Φc satisfies the triangle inequality Φc (x, y) ≤ Φc (x, z) + Φc (z, y) as well as Φc (x, x) = 0 for all x ∈ Tn .

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The Principle of Least Action in Geometry and Dynamics by Karl Friedrich Siburg


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