Why Must Hamilton's Principle Apply to Dependent Coordinates?

[aaaa202]

Hamiltons principle can be used to derive Lagrange's equations IF our coordinates are independent. Thus you pretty much show that for independent coordinates Hamiltons principle is equivalent to Newtons laws.
However, it seems that my book also likes to think that Hamiltons principle must still be satisfied even when our coordinates are not independent - this is crucial for the introduction of lagrangian multipliers in mechanical problems, but I don't see why the statement is true. Surely we can't derive Lagranges equation now. So where does this knowledge come from?

 

[gtoguy97]

The idea behind Hamilton’s principle is to formulate the equations of motion for a system with independent coordinates by using a single variational principle. This allows us to derive Lagrange’s equations even when our coordinates are not independent. The basic idea is that if we have a set of coordinates that are not independent, then we can introduce a set of Lagrange multipliers which act as constraints on the system. This allows us to still use Hamilton’s principle to derive the equations of motion. In doing so, we must also take into account the constraints, which are imposed by the Lagrange multipliers.