Lagrangian submanifold

In symplectic geometry , the mathematical formalization of Hamiltonian mechanics , the maximum isotropic submanifolds of symplectic manifolds are called Lagrangian submanifolds .

Their importance comes from the fact that questions about periodic orbits of Hamiltonian systems can be translated into questions about the intersections of Lagrangian submanifolds. Let namely be the time-1-mapping of a Hamiltonian flow, then lies on a 1-periodic orbit if and only if belongs to the intersection of the Lagrangian graph of and . ${\displaystyle \phi :=\phi _{H}^{1}}$ ${\displaystyle x}$ ${\displaystyle (x,x)}$ ${\displaystyle \Gamma _{\phi }\cap \Gamma _{id}}$ ${\displaystyle \phi }$ ${\displaystyle id}$

definition

A submanifold of a symplectic manifold is called isotropic if ${\displaystyle L}$ ${\displaystyle (M,\omega )}$

${\displaystyle \omega \mid _{TL}=0}$

holds, that is, if the restriction of the symplectic form to the tangent space of vanishes. The inequality holds for the dimension of an isotropic submanifold . ${\displaystyle L}$ ${\displaystyle \dim(L)\leq {\tfrac {1}{2}}\dim(M)}$

A Lagrangian submanifold of a 2n-dimensional symplectic manifold is an n-dimensional isotropic submanifold , i.e. an isotropic submanifold of maximum dimension. ${\displaystyle (M,\omega )}$ ${\displaystyle L\subset M}$

Examples

• In the symplectic , every curve is a Lagrangian submanifold. ${\displaystyle (\mathbb {R} ^{2},\omega =dxdy)}$
• In the symplectic , the one corresponding to the coordinates is a Lagrangian submanifold. ${\displaystyle (\mathbb {R} ^{2n},\omega =dx_{1}dy_{1}+\ldots +dx_{n}dy_{n})}$ ${\displaystyle (x_{1},\ldots ,x_{n})}$ ${\displaystyle \mathbb {R} ^{n}}$
• The zero cut in the symplectic cotangent bundle is a Lagrangian submanifold.
• Be a symplectic manifold. The function graph of a mapping is a Lagrangian submanifold of if and only if is a symplectomorphism . ${\displaystyle (M,\omega )}$ ${\displaystyle \Gamma _{f}}$ ${\displaystyle f\colon M\to M}$ ${\displaystyle (M\times M,\omega \oplus \omega )}$ ${\displaystyle f}$
• Arnold-Liouville's theorem : For an integrable system given on a -dimensional symplectic manifold by functions with vanishing Poisson brackets , the level surfaces are Lagrangian submanifolds. ${\displaystyle 2n}$ ${\displaystyle H_{1},\ldots ,H_{n}}$ ${\displaystyle \left\{H_{1}=c_{1},\ldots ,H_{n}=c_{n}\right\}}$

literature

• VI Arnold : Mathematical Methods of Classical Mechanics (= Graduate Texts in Mathematics . Vol. 60). 2nd edition, Springer, New York NY et al. 1989, ISBN 0-387-96890-3 .