Flow (mathematics)

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In mathematics, a flow formalizes, in mathematical terms, the general idea of "a variable that depends on time" that occurs very frequently in engineering, physics and the study of ordinary differential equations. Informally, if $x (t)$ is some coordinate of some system that behaves continuously as a function of t, then $x (t)$ is a flow. More formally, a flow is the group action of a one-parameter group on a set.

The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and the Anosov flow.

[edit] Formal definition

A flow on a set $X$ is a group action of $(\mathbb{R},+)$ on $X .$ More explicitly, a flow is a function $\varphi:X\times \mathbb{R}\rightarrow X$ with $\varphi(x,0) = x$ and that is consistent with the structure of a one-parameter group:

$\varphi(\varphi(x,t),s) = \varphi(x,s+t)$

for all $s, t$ in $\mathbb{R}$ and $x\in X.$

The set $\mathcal{O}(x,\varphi) = \{\varphi(x,t):t\in\mathbb{R}\}$ is called the orbit of $x$ by $\varphi.$

Flows are usually required to be continuous or even differentiable, when the space $X$ has some additional structure (e.g. when $X$ is a topological space or when $X = \mathbb{R}^n.$ )

It is very common in many fields, including engineering, physics and the study of differential equations, to use a notation that makes the flow implicit. Thus,

x (t)

is written for $φ(x, t)$ , and one might say that the "variable x depends on the time t". In fact, notationally, one has strict equivalence: $x(t)\equiv\phi(x,t)$ . Similarly

x 0 = x (0)

is written for $x = φ(x,0)$ , and so on.

[edit] Examples

The most common examples of flows arise from describing the solutions of the autonomous ordinary differential equation

$y' = f(y),\;\;\; y(0)=x$

as a function of the initial condition $x$ , when the equation has existence and uniqueness of solutions. That is, if the equation above has a unique solution $\psi_x:\mathbb{R}\rightarrow X$ for each $x\in X$ , then $\varphi(x,t) = \psi_x(t)$ defines a flow.