Constrained problems

This section is dedicated to the study of an equation (in x) F(x,p)=0 where one wishes to add a constraint g(x,p)=0. Hence, one is interested in solving in the couple $(x,p)$:

\[\left\{ \begin{array}{l} F(x,p)=0 \\ g(x,p)=0 \end{array}\right.\]

There are several situations where this proves useful:

1. the pseudo-arclength continuation method is such a constrained problem, see continuation for more details.
2. when the equation $F(x)$ has a continuous symmetry described by a Lie group $G$ and action $g\cdot x$ for $g\in G$. One can reduce the symmetry of the problem by considering the constrained problem:

\[\left\{ \begin{array}{l} F(x) + p\cdot T\cdot x=0 \\ \langle T\cdot x_{ref},x-x_{ref}\rangle=0 \end{array}\right.\]

where $T$ is a generator of the Lie algebra associated to $G$ and $x_{ref}$ is a reference solution. This is known as the freezing method.