Following are overlapping properties between the Lagrangian and Hamiltonian functions.
2.
Where is Hamiltonian function ( not operator ).
3.
The " energy " of a configuration ? is given by the Hamiltonian function
4.
This moment map can be viewed as a Hamiltonian function that generates the circle action.
5.
In each case the Lagrangian and Hamiltonian functions are replaced by a single function, the Routhian.
6.
Let be a smooth family of Hamiltonian functions of and denote by the one-time map of the flow of the Hamiltonian vector field of.
7.
For instance, Rabinowitz showed that on star-shaped level sets of a Hamiltonian function on a symplectic manifold, there were always periodic orbits ( Weinstein independently proved the special case of convex level sets ).
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Examples of symplectomorphisms include the canonical transformations of classical mechanics and theoretical physics, the flow associated to any Hamiltonian function, the map on cotangent bundles induced by any diffeomorphism of manifolds, and the coadjoint action of an element of a Lie Group on a coadjoint orbit.
9.
In symplectic Floer homology, one considers the Hamiltonian vector field of a Hamiltonian function on a symplectic manifold; in symplectic field theory, contact homology, and their variants, one considers the Reeb vector field associated to a contact form, or more generally a stable Hamiltonian structure.
10.
He considered spaces of contractible loops of " M " and defined an action functional " A " " H " associated to the family of Hamiltonian functions, so that the fixed points of the Hamiltonian diffeomorphism correspond to the critical points of the action functional.
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