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Before going on to the general theory, consider a specific example step by step to motivate the general analysis.
Start with the [[action (physics)|action]] describing a [[Newtonian dynamics|Newtonian]] particle of [[mass]] {{mvar|m}} constrained to a spherical surface of radius {{mvar|R}} within a uniform [[gravitational field]] {{mvar|g}}. When one works in Lagrangian mechanics, there are several ways to implement a constraint: one can switch to generalized coordinates that manifestly solve the constraint, or one can use a Lagrange multiplier while retaining the redundant coordinates so constrained.
In this case, the particle is constrained to a sphere, therefore the natural solution would be to use angular coordinates to describe the position of the particle instead of Cartesian and solve (automatically eliminate) the constraint in that way (the first choice). For pedagogical reasons, instead, consider the problem in (redundant) Cartesian coordinates, with a Lagrange multiplier term enforcing the constraint.
The action is given by
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