By Marco Mazzucchelli

Lagrangian platforms represent a vital and outdated type in dynamics. Their beginning dates again to the tip of the eighteenth century, with Joseph-Louis Lagrange’s reformulation of classical mechanics. the most characteristic of Lagrangian dynamics is its variational taste: orbits are extremal issues of an motion practical. the advance of severe element conception within the 20th century supplied a strong equipment to enquire lifestyles and multiplicity questions for orbits of Lagrangian structures. This monograph offers a latest account of the applying of severe aspect thought, and extra particularly Morse idea, to Lagrangian dynamics, with specific emphasis towards lifestyles and multiplicity of periodic orbits of non-autonomous and time-periodic systems.

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**Example text**

We denote by detC (M ) : Sp(2m) ∩ O(2m) → S 1 ⊂ C the complex determinant function, that is detC X Y −Y X := det(X + iY ). We deﬁne the rotation function as the composition ρ := detC ◦ r : Sp(2m) → S 1 . From our discussion above and the properties of the unitary group, it readily follows that this map induces an isomorphism π1 (ρ) between the fundamental groups of Sp(2m) and S 1 . Let P be the space of continuous paths Ψ : [0, 1] → Sp(2m) such that Ψ(0) = I. This space is the disjoint union of the subspaces P ∗ and P 0 given by those Ψ such that Ψ(1) ∈ Sp∗ (2m) and Ψ(1) ∈ Sp0 (2m) respectively.

Fm be the standard symplectic basis of R2m , which means that e1 , . . , em is an orthonormal basis of Rm × {0} and f1 , . . , fm is an orthonormal basis of Vm = {0} × Rm with fj = J0 ej for each j ∈ {1, . . , m}. For each t ∈ R/Z, if we set ˜ fj ), f˜j := φ(t, fj ) = φ(t, ˜ ·) ◦ J0 ej = −J f˜j , e˜j := φ(t, ej ) = −J ◦ φ(t, it is straightforward to verify that e˜1 , . . , e˜m , f˜1 , . . , f˜m is a symplectic basis of the tangent space TΓ(t) T∗ M , which means ω(˜ ej , e˜h ) = ω(f˜j , f˜h ) = 0, ω(˜ ej , f˜h ) = 1 0 j = h, j = h.

Then the function z → indz (γ) is locally constant on S 1 \ {z1 , . . , zr } and lower semi-continuous on the whole of S 1 . , indzj (γ) ≤ lim indz (γ) ≤ indzj (γ) + nulzj (γ). z→zj± [1] [1] Proof. 3). We denote by σz its spectrum. By continuity, for each interval (α, β] ⊂ R such that α and β do not belong to σz , there is a neighborhood of z in S 1 such that, for each z in this neighborhood, α and β do not belong to σz and moreover indz,λ (γ) = λ∈(α,β) indz ,λ (γ). λ∈(α,β) Assume that z ∈ S 1 is not a Poincar´e point of γ, namely 0 does not belong to σz .