Consider a constrained ODE system: \begin{align} \dot{\bf x} &= \bf f(t,\bf x), \\ st. 0 &= \bf g(t,\bf x). \end{align} I wish to combine these into a single equation using a Lagrange multiplier, however I am unsure how to do it. I talked with my supervisor the other day about it and he quickly scribbled the following down: \begin{equation} \dot{\bf x} + (\nabla_\bf x \bf g(\bf x))^\top \lambda = \bf f(t,\bf x). \end{equation} However, it was a last moment thing so it might not be entirely correct, and I don't really see how he would get that. But then again I don't exactly see how I can apply the Lagrange method in the first place since it requires a function you wish to maximize/minimize, whereas I have $\bf f(t,\bf x) - \dot{\bf x} = 0$.
Does anyone have any advice on how to approach this problem?
Based on the information Carlo Beenakker gave me I have attempted the following:
I tried following the method outlined in: https://ep.liu.se/ecp/084/010/ecp13084010.pdf section 6.
If I understand this correctly I want to turn my constrained ODE into an optimization under constraint problem, which leads to a lagrangian equation.
So I consider equation (63-64): \begin{equation} \mathcal{F}({\bf x}) = \int_{t_0}^{t_1} L(t,{\bf x}, {\bf \dot{x}}) dt \end{equation} subject to the constraints \begin{equation} \phi({\bf x}) = \int_{t_0}^{t_1} h(t,{\bf x}) dt = 0 \end{equation}
I can see that the first equation is similar to my original equation with ${\bf f}(t,{\bf x}) = L(t,{\bf x}, \bf \dot{x})$, and the constraints I guess are related as such: \begin{equation} h(t,{\bf x}) = \frac{d}{dt} g(t,{\bf x}) \end{equation}
This then leads me to equation 67: \begin{equation} \frac{\partial L}{\partial {\bf x}} - \frac{d}{dt} \left(\frac{\partial L}{\partial {\bf \dot{x}}} \right) = \nabla_{\bf x} h({\bf x}) \lambda \label{eq:1} \end{equation} However, if I use ${\bf f}(t,{\bf x}) = L(t,{\bf x}, \bf \dot{x})$, and insert it, I get: \begin{equation} \frac{\partial f(t,{\bf x})}{\partial {\bf x}} - \frac{d}{dt} \left(\frac{\partial {\bf \dot{x}}}{\partial {\bf \dot{x}}} \right) = \frac{\partial f(t,{\bf x})}{\partial {\bf x}} = \nabla_{\bf x} h({\bf x}) \lambda \end{equation} Which doesn't look right. So I'm guessing I did something wrong or there is something I have forgotten. Anyone that can give me a hint as to what I'm doing wrong?
