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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?

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    $\begingroup$ In my view a constraint leads to possibly many boundary conditions. A boundary condition is one out of, possibly many, boundary conditions due to the constraint. $\endgroup$
    – anna v
    Feb 7, 2022 at 11:06

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  1. 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.

  2. 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.

  3. More generally, be aware that the words constraint and condition depend on context, and are used differently by different authors.

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I'm sure there is a lot of opinions on this, as some terminology is not generally agreed upon or consistent.

A boundary condition is usually a specific precondition for solving an equation, often facilitating going from a very general problem to a specific solvable instance.

A constraint is a condition on solutions that enable you to discard or accept solutions.

So all in all - in my view. The boundary you use in the start of a problem - the constraint in the end.

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