Lie derivative
Undergraduate (dynamical systems)
Consider a function $V$ and a vector field $F$, with solutions of $\dot{x} = F(x)$. The Lie derivative of $V$ along $F$ is $L_F V(x) = \nabla V(x) \cdot F(x)$. It is the rate of change of $V$ along those solutions.
Graduate (differential geometry)
Consider a function $V$ and a vector field $F$ on a manifold, where $\phi_t$ is the flow of $F$. The Lie derivative of $V$ along $F$ is
\[L_F V(x) = \frac{\mathrm{d}}{\mathrm{d}t} V(\phi_t(x)) \big\rvert_{t = 0}.\]Lie derivatives are also defined for vector fields and differential forms.
Examples
Worked examples
-
$f(x, y) = x^2 + 3y$
: $L_F f(x, y) = \nabla f(x, y) \cdot F(x, y) = -2x^2$, along the vector field $F(x, y) = (-x, 0)$
Undergraduate
$\nabla f \cdot F = 2x \cdot (-x) + 3 \cdot 0$. The direction varies from point to point.
Properties
- If $L_F V(x) \leq 0$ for all $x$, then $V$ never increases along solutions of $\dot{x} = F(x)$. This is how Lyapunov functions are checked, without solving the system.
Distinctions involving this term
- Types of multivariable derivatives: All of these package the partial derivatives of a function differently: as a vector (gradient), a matrix (Jacobian), a linear map (differential), or a single number in a chosen direction (directional derivative) or along a vector field (Lie derivative).