We generalized the Lagrange multiplier method for constraints to
variational calculus problems. Constraints 
restrict
us to variations 
that keep the constraints true. The
variations are thus interdependent again, with us considering varied
functions over a smaller space of possible functions 
.
The exact same analysis tells us that our vector of Euler-Lagrange
coefficients (each coefficient multiplies a variation in
the integrand of the functional variation 
) need no longer
have components that each vanish identically. Again the variation of
the integrand need vanish only for the subspace of varied
functions that obey the constraints. Thus the variation
of the integrand may be a linear combination 
, which vanishes only on the surface obeying the
constraint. Thus we obtain equations of motion which can be obtained
by a slight generalization of the Lagrange multiplier method: minimize
the action for an altered Lagrangian, 
. Euler-Lagrange equations with respect to the 
reproduce the correct linear dependence, and with respect to the
enforce the constraint.
We then returned to physics with enough mathematical background to solve our problems. Mechanics can be recast in a elegant variational form, Hamilton's principle, which states that classical motions always follow paths that extremize the action. We specify a system entirely by its Lagrangian L (ordinarily L = T-V), an object which transforms under coordinate transformations like a scalar. This means that, unlike vectors, whose components all change under translations and rotations, the Lagrangian is a physical quantity that always stays the same. It is thus a simpler and more intrinsic object in which to encode the laws governing a system's motion. Newton's second law, being a vector equation, instead varies strongly with coordinate transformations.
All our variational methods lift to this one variational problem of
extremizing the action. In particular we can solve constrained
problems. The effect of adding Lagrange multipliers to enforce the
constraint is equivalent to adding an additional piece 
to the system's potential. We thus find that
enforcing constraints requires the action of a force of constraint,
given by 
The constraint force
affects the system's Euler-Lagrange equations though the
additional generalized force terms 
--
KB