Dynamical variable

A dynamical variable is a Scalar field G:RN→RG:\mathbb{R}^{N}\to\mathbb{R} that represents some physical quantity related to a dynamical system. Given an ODE or system of ODEs x˙(t)=f(x(t),t)\dot{\mathbf{x}}(t)=f(\mathbf{x}(t),t) with x∈RN\mathbf{x}\in \mathbb{R}^{N}, the variable sends x(t)\mathbf{x}(t) to G(x(t))G(\mathbf{x}(t)). Its derivative is

ddtG(x(t))=∑i=1N∂G∂xi(x(t))x˙i(t)\frac{d}{dt}G(\mathbf{x}(t))=\sum_{i=1}^{N} \frac{ \partial G }{ \partial x_{i} } (\mathbf{x}(t))\dot{\mathbf{x}}_{i}(t)