In theoretical physics, an analysis of flows is the study of "gauge" or "gaugelike" "symmetries" (i.e. flows the formulation of a theory is invariant under). It is generally agreed that flows indicate nothing more than a redundancy in the description of the dynamics of a system, but oftentimes, it is simpler computationally to work with a redundant description.

To be expanded

Flows in classical mechanics

flows in the action formalism

Recall that classically, the action is a functional of the configuration space and the on shell solutions are given by the variational problem of extremizing the action subject to boundary conditions on the boundary. While the boundary is often not given much mention is elementary textbooks, it turns out it is crucial in the study of flows. Suppose we have a "flow", that is the generator of a smooth one-dimensional group of transformations of the configuration space which maps on shell states to on shell states while preserving the boundary conditions. Because of the variational principle, the action for all of the configurations on the orbit (at least on shell only) are the same. Note that this is not the case for more general transformations which maps on shell to on shell states but change the boundary conditions.

Let me give a couple of examples. In a translationally symmetric theory, timelike translations are NOT flows because in general, timelike translations would change the boundary conditions even though they map on shell states to on shell states. However, let's take the case of a simple harmonic oscillator where the boundary points are at a separation of a multiple of the period from each other and the initial and final position are the same at the boundary points. For this particular example, it turns out there IS a flow. Even though this is technically a flow, this would usually not be considered a gauge symmetry because it is not local.

Flows can be given as derivations over the algebra of smooth functionals over the configuration space. If we have a flow distribution (i.e. flow valued distribution) such that the flow convolved over a local region only affects the field configuration in that region, we call the flow distribution a gauge flow.

Of course, since we're only interested in what happens on shell. We would often take the quotient by the ideal generated by the Euler-Lagrange equations, or in other words, consider the equivalence class of functionals/flows which agrees on shell.

BRST Batalin-Vilkovisky

flows in the Hamiltonian formalism

first class constraints second class constraints BRST Batalin-Vilkovisky

flows in other formalisms

Nambu mechanics

Flows in quantum mechanics