  The most natural first-order PDE's to be imposed on a Cayley 4-form in eight
dimensions is the condition that it is closed. As is known, this implies
integrability of the Spin(7) structure defined by the Cayley form, as well as
Ricci-flatness of the associated metric. We address the question as to what the
most natural second-order in derivatives set of conditions is. We start by
constructing the most general diffeomorphism invariant second order in
derivatives Lagrangian that is quadratic in the perturbations of the Cayley
form. We find that there is a one-parameter family of such Lagrangians. We then
describe a non-linear completion of the linear story. To this end, we
parametrise the intrinsic torsion of a Spin(7) structure by a 3-form, and show
that this 3-form is completely determined by the exterior derivative of the
Cayley form. We construct an action functional, which depends on the Cayley
4-form and and auxiliary 3-form as independent variables. There is a unique
functional whose Euler-Lagrange equation for the auxiliary 3-form states that
it is equal to the torsion 3-form. There is, however, a more general
one-parameter family of functionals that can be constructed, and we show how
the linearisation of these functionals reproduces the linear story. For any
member of our family of theories, the Euler-Lagrange equations are written only
using the operator of exterior differentiation of forms, and do not require the
knowledge of the metric-compatible Levi-Civita connection. Geometrically, there
is a preferred member in the family of Lagrangians, and we propose that its
Euler-Lagrange equations are the most natural second-order equations to be
satisfied by Cayley forms. Our construction also leads to a natural geometric
flow in the space of Cayley forms, defined as the gradient flow of our action
functional.
