Non-Hamiltonian Actions and Lie-Algebra Cohomology of Vector Fields

Two examples of $\mathrm{Diff}^+S^1$-invariant closed two-forms obtained from forms on jet bundles, which does not admit equivariant moment maps are presented. The corresponding cohomological obstruction is computed and shown to coincide with a nontrivial Lie algebra cohomology class on $H^2(\mathfrak{X}(S^1))$

Homogeneous algebraic distributions

Let p:E→M be a vector bundle of dimension n+m and (xλ,yi), λ=1,…,n, i=1,…,m, be fibre coordinates. A vertical vector field X on E is said to be algebraic [respectively, algebraic homogeneous of degree d] if its coordinate expression is of the type X=∑mi=1Pi∂/∂yi, where Pi are polynomials [respectively, homogeneous polynomials of degree d] in coordinates yi. A vertical distribution over E is said to be algebraic [respectively, homogeneous algebraic of degree d] if all local generators are homogeneous algebraic [respectively, homogeneous algebraic of the same degree d] vector fields. It is proved that a vertical distribution locally spanned by vector fields X1,…,Xr is homogeneous algebraic of degree d if and only if an r×r matrix A=(aij), aij∈C∞(E), exists which is equal to d−1 times the identity matrix along the zero section of E, and such that [χ,Xj]=∑ri=1aijXi, j=1,…,r, where χ is the Liouville vector field

Hamiltonian structure of gauge-invariant variational problems

Let C→M be the bundle of connections of a principal bundle on M . The solutions to Hamilton–Cartan equations for a gauge-invariant Lagrangian density Λ on C satisfying a weak condition of regularity, are shown to admit an affine fibre-bundle structure over the set of solutions to Euler–Lagrange equations for Λ . This structure is also studied for the Jacobi fields and for the moduli space of extremals

Structure of diffeomorphism-invariant Lagrangians on the product bundle of metrics and linear connections

Let pC : C = CN → N be the bundle of linear connections on a smooth manifold N and let pM : M ! N be the bundle of pseudo-Riemannian metrics of a given signature (n₊ ; n-) , n₊ n- = n = dimN on N. The structure of the first-order Lagrangians defined on the bundle M x N C → N that are invariant under the natural action of the diffeomorphisms of N, is determined

Diffeomorphism-invariant covariant Hamiltonians of a Pseudo-Riemannian metric and a linear connection

\noindent Let $M\to N$ (resp.\ $C\to N$) be the fibre bundle of pseudo-Riemannian metrics of a given signature (resp.\ the bundle of linear connections) on an orientable connected manifold $N$. A geometrically defined class of first-order Ehresmann connections on the product fibre bundle $M\times_NC$ is determined such that, for every connection $\gamma$ belonging to this class and every $\mathrm{Diff}N$-invariant Lagrangian density $\Lambda$ on $J^1(M\times_NC)$, the corresponding covariant Hamiltonian $\Lambda ^\gamma$ is also $\mathrm{Diff}N$-invariant. The case of $\mathrm{Diff}N$-invariant second-order Lagrangian densities on $J^2M$ is also studied and the results obtained are then applied to Palatini and Einstein-Hilbert Lagrangians