A reduced order model for marine vehicle dynamics is the simple linear spring-mass-damper system. However, the various terms in the equation of motion differ in detail from their mechanical counterparts. The usual balance between mechanical inertial, damping, and stiffness loads with external forcing is maintained, but now includes additional effects reflecting the presence of the fluid. Individual coefficient matrices correspond to the mass of the platform plus the mass of the water being accelerated; the linear damping coefficient of the system due to viscous effects and the generation of radiating waves due to platform motion; a linear restoring force/moment coefficient due to hydrostatic pressure and/or mooring lines; and an external exciting force/moment due to incident waves, wind, tow lines, etc. Ideal fluid theory is introduced to model the hydrodynamic forces implicit in the marine system’s equations of motion. The purpose is not to give a detailed derivation of basic hydrodynamics, but rather to describe the assumptions necessary to apply the useful ideal, potential theory and understanding when the theory will be successful and, equally important, when it will not.

This chapter uses the ideas of hydrodynamics introduced in the last chapter to formulate the hydrodynamic theory of the flocking problem (i.e., the “Toner–Tu” equations).

This chapter “derives” the scaling laws found in the previous chapter for incompressible flocks, using a simple heuristic argument which gives some physical insight into the mechanism, and its essentially nonequilibrium nature.

I present a purely dynamical derivation of the Mermin–Wagner–Hohenberg theorem, and compare it with the standard equilibrium derivation. This also provides an opportunity to introduce diffusion equations and gradient expansions, both of which play a large role in what follows.

This chapter deals with incompressible flocks in spatial dimensions d > 2, for which the dynamical renormalization group yields exact scaling laws.

I introduce, and describe in detail, the dynamical renormalization group, using the KPZ equation as an example. In addition to spelling out the mechanics of the technique in great detail, I also emphasize its philosophical importance, as the answer to Einstein’s famous question “Why is the universe intelligible?” and its role as a guide to the formulation of hydrodynamic theories.

This chapter treats incompressible flocks in two dimensions, and shows that they map onto both equilibrium two-dimensional smectics, and our old friend the KPZ equation (albeit in one dimension), as well as a peculiar type of constrained magnet. Exact scaling laws are again found, this time by exploiting these mappings.

This chapter applies the dynamical renormalization group introduced in Chapter 4 to the flocking problem, and uses it to show that nonlinear terms in the dynamics are “relevant,” and change the dynamics in precisely the way needed to circumvent the Mermin–Wagner–Hohenberg theorem.

I introduce the problem of “dry active matter” more precisely, describing the symmetries (both underlying, and broken) of the state I wish to consider, and also discuss how shocking it is that such systems can exhibit long-ranged order – that is, all move together – even in d = 2.

The final chapter treats “Malthusian” flocks; that is, flocks with birth and death. Here, a full dynamical renormalization group calculation must be done; specifically, it can only be done using a d = 4-epsilon expansion.