Clebsch-Lagrange variational principle and geometric constraint analysis of relativistic field theories

For example, we may take T^*Q to be TQ and the pairing given by a Riemannian structure on Q. In applications, the fiber of TQ is often a space of mappings so that a convenient choice of the cotangent bundle consists of regular distributions inside the space of all distributions.

More generally, one could give up the assumption of the existence of a local addition and instead work in the diffeological category.²³ 

That is, $Tγ(t0)Cγ̇(t0)=0$⁠.

Here, we have used the identity $(g⋅(q,p))ωω=g⋅(q,p)$⁠, which in infinite dimensions requires additional assumptions of functional-analytic nature. For a precise formulation of the bifurcation lemma in an infinite-dimensional context, see Ref. 12.

To be more precise, the reduction by stages theorem Ref. 26 (Theorem 4.2.2) is formulated for the free and proper action of a finite-dimensional group on a finite-dimensional phase space. Nonetheless, the peculiarities of our infinite-dimensional setting can be handled easily due to the simple form of the first reduction.

This one-to-one correspondence will be used throughout the text without further notice.

Although this convention is a bit nonstandard, the consistent use of dual-valued forms has the advantage that dual objects are clearly marked, which will be helpful later to identify them as points in the cotangent bundle (as opposed to elements of the tangent bundle).

The diamond operator often occurs in the study of Lie–Poisson systems, and this is where we borrowed the notation from. Note that in a purely algebraic setting, the diamond product boils down to a map $⋄:F×F*→g*$ dual to the Lie algebra action. Hence, it is a momentum map for the G-action on T^*F. For example, if we consider the action of G = SO(3) on $F=R3$ and identify F^* with $R3$⁠, then the diamond product becomes the classical cross product.

We refer to Ref. 10 (Chap. 20) for a detailed discussion of boundary terms within the ADM formalism and to Ref. 24 for a new canonical description of the gravitational field dynamics in a finite volume with boundary. Clearly, our approach can be adapted to include boundary terms as well.