Sometimes, in Lagrangian mechanics, we need to identify the constraints of the problems and using the Lagrange multiplier technique, we got to the equilibrium equations of the problem under study and do some optimization analysis. I used to select these constraints according to the boundary conditions, but my professor said they are not the same.
What is then the difference between a constraint and a boundary condition? How can we know that this is a "constraint" and that is a "boundary condition"? And when they are the same?
Best Answer
In the context of a classical mechanical system of $N$ point particles with positions ${\bf r}_1, \ldots, {\bf r}_N$, a (holonomic) constraint takes the form $$ f({\bf r}_1, \ldots, {\bf r}_N,t)~=~0. $$ The corresponding Lagrange multiplier $\lambda(t)$ becomes a function of time $t\in[t_i,t_f]$. In contrast, a boundary condition in point mechanics is an initial or a final condition.
The above has a generalization to field theory in the following sense: Constraints are typically imposed in the bulk, while boundary conditions are imposed on the boundary.
More generally, be aware that the words constraint and condition depend on context, and are used differently by different authors.