The Euler-Lagrange Cohomology Groups on Symplectic Manifolds
نویسندگان
چکیده
The definition and properties of the Euler-Lagrange cohomology groups H EL , 1 6 k 6 n, on a symplectic manifold (M2n, ω) are given and studied. For k = 1 and k = n, they are isomorphic to the corresponding de Rham cohomology groups H1 dR(M 2n) and H dR (M 2n), respectively. The other Euler-Lagrange cohomology groups are different from either the de Rham cohomology groups or the harmonic cohomology groups on (M2n, ω), in general. The general volume-preserving equations on (M2n, ω) are also presented from cohomological point of view. In the special cases, these equations become the ordinary canonical equations in the Hamilton mechanics. Therefore, the Hamilton mechanics has been generalized via the cohomology. Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected]
منابع مشابه
Symplectic , Multisymplectic Structures and Euler - Lagrange Cohomology
We study the Euler-Lagrange cohomology and explore the symplectic or multisym-plectic geometry and their preserving properties in classical mechanism and classical field theory in Lagrangian and Hamiltonian formalism in each case respectively. By virtue of the Euler-Lagrange cohomology that is nontrivial in the configuration space, we show that the symplectic or multisymplectic geometry and rel...
متن کاملDifference Discrete Variational Principle , Euler - Lagrange Cohomology and Symplectic , Multisymplectic Structures
We study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncomutative differential geometry. By virtue of this variational principle, we get the difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of the classical ...
متن کاملEuler-Lagrange equations and geometric mechanics on Lie groups with potential
Abstract. Let G be a Banach Lie group modeled on the Banach space, possibly infinite dimensional, E. In this paper first we introduce Euler-Lagrange equations on the Lie group G with potential and right invariant metric. Euler-Lagrange equations are natural extensions of the geodesic equations on manifolds and Lie groups. In the second part, we study the geometry of the mechanical system of a r...
متن کاملOn Symplectic and Multisymplectic Srtuctures and Their Discrete Versions in Lagrangian Formalism
We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively. We also explore their certain difference discrete counterparts in the relevant regularly discretized finite and infinite dimensional Lagrangian systems by means of the difference discrete variasional...
متن کاملThe Euler-Lagrange Cohomology and General Volume-Preserving Systems
We briefly introduce the conception on Euler-Lagrange cohomology groups on a symplectic manifold (M2n, ω) and systematically present the general form of volumepreserving equations on the manifold from the cohomological point of view. It is shown that for every volume-preserving flow generated by these equations there is an important 2-form that plays the analog role with the Hamiltonian in the ...
متن کامل