All Hamiltonian balanced models, regardless of accuracy - if derived by constraining a parent dynamics supporting inertia-gravity waves and Lighthill radiation - have a characteristic general property that may be called `velocity splitting', recognition of which is important for a full understanding of the dynamics. Imposing any balance condition as a constraint on the parent dynamics splits the parent velocity field, into just two velocity fields in the simplest set of cases. The first, u^P say, is the velocity in the ordinary sense, the velocity with which material particles move. The second, u^C say, is the constraint velocity defining the balanced model's `slow manifold', a prescribed functional of the mass configuration alone; u^C is also the velocity that enters the simplest forms of the model's PV, energy and other conserved quantities, which quantities, when evaluated with u^C, are given by the same formulae as in the parent dynamics. The model's conserved PV, in particular, is always the Rossby-Ertel PV when expressed in terms of the `constraint vorticity' zeta^C, i.e. the absolute vorticity evaluated with u^C , and furthermore is always given by the same (real) Jacobian determinants involving X as in Hoskins' semigeostrophy, in cases where canonical coordinates X are known. Semigeostrophy itself is more complicated, from this viewpoint, exhibiting double splitting in the sense of having two slow manifolds and three distinct velocity fields , u^P, u^C(H) and u^C(omega) , of which the second enters into energy conservation via the parent Hamiltonian and the third into PV conservation via the parent symplectic structure.

There is a fundamental integral equation that governs the difference-velocity field or `velocity split'
u^S = u^P - u^C and defines the balanced model dynamics including any boundary conditions additional to those used in defining the configuration space. The formulation in terms of u^S gives a mathematical description that is manifestly independent of reference-frame rotation rate. The dynamical effects of rotation enter solely through the functional or functionals, for instance through zeta^C. This greatly simplifies the formulation of variable-Coriolis-parameter models, and brings other conceptual and formal simplifications such as the ability to accommodate Galilean-invariant u^C. The integral equation takes an especially simple form for singly-split models. Solving the equation gives u^S as a functional of the mass configuration alone, defining, in turn, how particles move, and showing that u^S is generally nonzero though small for an accurate model. Any norm ||u^S|| provides a natural intrinsic measure of model accuracy, suggesting minimization of such norms as an approach to finding `optimal' Hamiltonian balanced models. The nonvanishing of ||u^S|| is connected with the existence of Lighthill radiation and the nonexistence of a parent slow manifold.