My question relates to a graphical approach to identifying the nonlinear terms of an oscillator [1].
The oscillator is
\begin{equation} \ddot{x} + x + W(x,\dot{x})=0 \end{equation}
which [1] says can be written as
\begin{equation} \frac{1}{2}\mathrm{d}(x^2+\dot{x}^2) + W(x,\dot{x})\dot{x} \, \, \mathrm{d}t =0 \end{equation}
or, equivalently, as
\begin{equation} W(x,\dot{x})=-\mathrm{d}(x^2+\dot{x}^2)/2 \,\, \mathrm{d}x. \end{equation}
My questions are (i) how did the authors derive this equation and (ii) given observed values $x$ and estimated values $\dot{x}$, how does one evaluate $W(x,\dot{x})$ in a scripting language like MATLAB?
- P.J. Beek, W.J. Beek, Tools for constructing dynamical models of rhythmic movement, Human Movement Science, Volume 7, Issues 2–4, October 1988, Pages 301-342, ISSN 0167-9457, http://dx.doi.org/10.1016/0167-9457(88)90015-2. (http://www.sciencedirect.com/science/article/pii/0167945788900152)
Update 1:
Using the advice of John Sinclair, we have that \begin{align} \ddot{x} + x + W(x,\dot{x})&=0\\ \dot{x}\ddot{x} + x\dot{x} + W(x,\dot{x})\dot{x}&=0\\ \end{align} and because \begin{equation} \dot{x}\ddot{x} + x\dot{x} = \frac{\mathrm{d}(x^2+\dot{x}^2)}{2\mathrm{d}t} \end{equation} we write \begin{align} \frac{\mathrm{d}(x^2+\dot{x}^2)}{2\mathrm{d}t} + W(x,\dot{x})\dot{x} &=0\\ \frac{\mathrm{d}(x^2+\dot{x}^2)}{2} + W(x,\dot{x})\dot{x}\mathrm{d}t &=0\\ \frac{\mathrm{d}(x^2+\dot{x}^2)}{2} + W(x,\dot{x})\mathrm{d}x &=0 \end{align} from which we can get \begin{equation} W(x,\dot{x}) = -\frac{\mathrm{d}(x^2+\dot{x}^2)}{2\mathrm{d}x} \end{equation} which is the desired result.