Covariant Hamiltonian field theory

Definition

Let $Y \to X$ be a smooth fiber bundle over an oriented $n$-dimensional manifold $X$, with fibers of dimension $k$. Use bundle coordinates $(x^\mu, y^i)$, $\mu = 1, \dots, n$, $i = 1, \dots, k$. Let $\omega = dx^1 \wedge \cdots \wedge dx^n$ and $\omega_\mu = \iota_{\partial_\mu} \omega$, the contraction of $\omega$ with $\partial / \partial x^\mu$.

Let $V Y$ be the vertical tangent bundle of $Y$.

1. The momentum phase space is the Legendre bundle $\Pi \to Y$ with fiber $V^*_y Y \otimes T_x X \otimes \Lambda^n T^*_x X$ over $y \in Y$ above $x \in X$. Its elements are $p^\mu_i \, dy^i \otimes \partial_\mu \otimes \omega$, and $(x^\mu, y^i, p^\mu_i)$ are coordinates on $\Pi$.
2. A Hamiltonian form is a differential form on $\Pi$ that reads in these coordinates $$H = p^\mu_i \, dy^i \wedge \omega_\mu - \mathcal{H}(x^\nu, y^j, p^\nu_j) \, \omega,$$ for a smooth local function $\mathcal{H}$, the Hamiltonian density. Here the first term is summed over $\mu$ and $i$.
3. For a section $x \mapsto (y^i(x), p^\mu_i(x))$ of $\Pi \to X$, its covariant Hamilton equations for $H$ are $$\partial_\mu y^i = \frac{\partial \mathcal{H}}{\partial p^\mu_i}, \qquad \sum_{\mu = 1}^n \partial_\mu p^\mu_i = -\frac{\partial \mathcal{H}}{\partial y^i}.$$

Covariant Hamiltonian field theory is the study of the fields as solutions of the covariant Hamilton equations of a Hamiltonian form.