Lie bracket
Consider two continuously differentiable vector fields $f$ and $g$ on $\reals^n$, with Jacobian matrices $\frac{\partial f}{\partial x}$ and $\frac{\partial g}{\partial x}$. The Lie bracket of $f$ and $g$ is the vector field $[f, g]$ defined by
\[[f, g](x) = \frac{\partial g}{\partial x}(x) \, f(x) - \frac{\partial f}{\partial x}(x) \, g(x).\]A Lie bracket on a vector space $V$ is a bilinear map $[\cdot, \cdot] : V \times V \to V$ such that, for all $x, y, z \in V$,
- $[x, y] = -[y, x]$, and
- $\big[x,~[y, z]\big] + \big[y,~[z, x]\big] + \big[z,~[x, y]\big] = 0$ (the Jacobi identity).
A vector space with a Lie bracket is called a Lie algebra.
Properties
- On a manifold, the Lie bracket of smooth vector fields $X$ and $Y$, viewed as derivations of the smooth functions, is $[X, Y] f = X (Y f) - Y (X f)$. On $\reals^n$, this agrees with the formula $[f, g] = \frac{\partial g}{\partial x} f - \frac{\partial f}{\partial x} g$.
- The Lie bracket $[X, Y]$ is the Lie derivative of $Y$ along $X$.
- The Lie bracket measures how much two flows fail to commute: moving along $f$, then $g$, then $-f$, then $-g$, each for time $t$, moves a point $x$ to \[x + t^2 [f, g](x) + O(t^3).\] This is why Lie brackets are useful for determing which directions a nonlinear control system can move in.
Notes
Sign conventions differ. Some texts define the Lie bracket of vector fields with the opposite sign, $\frac{\partial f}{\partial x} g - \frac{\partial g}{\partial x} f$. For matrices, the Lie bracket is the commutator $[A, B] = A B - B A$, but for the linear vector fields $f(x) = A x$ and $g(x) = B x$, the formula above gives ${[f, g]}(x) = (B A - A B) x$, with the opposite order.