# Momentum Maps and Classical Relativistic Fields. Part II: Canonical Analysis of Field Theories

@article{Gotay2004MomentumMA, title={Momentum Maps and Classical Relativistic Fields. Part II: Canonical Analysis of Field Theories}, author={Mark J. Gotay and James Allen Isenberg and Jerrold E. Marsden}, journal={arXiv: Mathematical Physics}, year={2004} }

With the covariant formulation in hand from the first paper of this series (physics/9801019), we begin in this second paper to study the canonical (or ``instantaneous'') formulation of classical field theories. The canonical formluation works with fields defined as time-evolving cross sections of bundles over a Cauchy surface, rather than as sections of bundles over spacetime as in the covariant formulation. In Chapter 5 we begin to relate these approaches to classical field theory; in…

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## References

SHOWING 1-10 OF 40 REFERENCES

### A multisymplectic framework for classical field theory and the calculus of variations II: space + time decomposition

- Mathematics
- 1991

### The initial value problem and the dynamical formulation of general relativity

- Physics
- 1979

In this chapter we discuss some of the interrelationships between the initial value problem, the canonical formalism, linearization stability and the space of gravitational degrees of freedom. In the…

### The well‐posedness of (N=1) classical supergravity

- Mathematics
- 1985

In this paper we investigate whether classical (N=1) supergravity has a well‐posed locally causal Cauchy problem. We define well‐posedness to mean that any choice of initial data (from an appropriate…

### Stress-Energy-Momentum Tensors and the Belinfante-Rosenfeld Formula

- Physics
- 1992

We present a new method of constructing a stress-energy-momentum tensor
for a classical field theory based on covariance considerations and Noether theory.
The stress-energy-momentum tensor T ^μ …

### Exactly soluble diffeomorphism invariant theories

- Physics
- 1989

A class of diffeomorphism invariant theories is described for which the Hilbert space of quantum states can be explicitly constructed. These theories can be formulated in any dimension and include…

### Closed forms on symplectic fibre bundles

- Mathematics
- 1983

A bundle of symplectic manifolds is a differentiable fibre bundle F--~ E-% B whose structure group (not necessarily a Lie group) preserves a symplectic structure on F. The vertical subbundle V=Ker…

### Stratified symplectic spaces and reduction

- Mathematics
- 1991

Let (M, w) be a Hamiltonian G-space with proper momentum map J: M -> g*. It is well-known that if zero is a regular value of J and G acts freely on the level set J '(0), then the reduced space MO =…