Before leaving last lecture's problem -- finding the geodesics on a sphere -- behind, we mentioned some other famous variational problems: geodesics on other surfaces (including our own spacetime manifold, for understanding particle motion in general relativity); the Brachistochrone problem; Fermat's principle for optics, which states that light follows the path of least time; and minimal surfaces.
Discussed the next tasks for our study of variational problems/ the
Lagrangian approach. We will: 1) generalize our derivation of the
Euler-Lagrange equations, to extremize functionals that depend on more
than one function f(x) (necessary to do multidimensional mechanics
problems, with multiple coordinates 
); 2) consider constrained
problems, where we minimize a functional not with respect to all
possible functions f(x), but only over the subspace of functions f(x)
that all obey some particular constraint; and 3) implement this new
formalism in the context of physics problems. We already saw how
Newton's second law can be rewritten as the Euler-Lagrange equations,
which arise naturally from Hamilton's principle that the action
functional is extremized by particle motions. We will see how to
incorporate constraints into that formalism, and then how to
systematically approach physical problems, characterizing a system's
degrees of freedom, the possible sets of generalized coordinates to
describe its motions, proper (or complete) sets of generalized
coordinates, and how to obtain forces of constraint by using
overcomplete sets of generalized coordinates.
First, we generalized our approach to functionals dependent on many
functions 
. In deriving the Euler-Lagrange equation, we
calculate the first order variation of the functional I, which is
the integral over the first order variation of the integrand
F. Naively, F varies through its dependence on every possible
argument, so we can blindly write out 
using the chain rule,
obtaining a contribution due to each 
and 
(since all appear symbolically as arguments to F). We then notice
that the function 
is really related to its derivative 
,
and so the variations are really related. After noticing this we can
do the integration by parts that reduces the variation in F to a sum
of terms, each given by a single times a coefficient
which looks like our old Euler-Lagrange operator acting on
. Since the functions (and thus their variations) are all
independent, each coefficient must simultaneously vanish to get an
extremal set 
. Thus we have n different
Euler-Lagrange equations which must all be satisfied.
We then began discussing constraints, where we minimize the functional I not with respect to all possible functions f(x), but only over the subspace of functions f(x) that all obey some particular constraint. We consider now constraints of the form g(f) = 0, which depend only on f directly and not on its derivatives. These are called "holonomic" constraints, are common in physics, and are the only constraints we can systematically cope with.
Discussed how knowing how to solve such a constrained problem would allow us to solve the problem of finding geodesics on a surface -- as we did for the sphere, thinking carefully about coordinates and distances on the sphere, last lecture -- in a more general way. We could instead minimize the ordinary Cartesian path length in 3 dimensions, but restricting ourselves to paths which remain only on the surface of the sphere -- that is, paths over which the radius is constant.
We reverted to calculus, attempting to minimize some function 
subject to constraint that 
vanishes. To minimize f, its variation df must vanish. We used
the chain rule to find df. Ordinarily, because all of the variations
are independent, this means that each partial derivative
-- the coefficient to in the sum
df -- must vanish. Here, however, the are not independent:
since the constraint function g must always vanish, its variation
dg vanishes -- giving a relationship among the . This gives
us a new way to make df vanish: we can set 
,
which vanishes for the constrained problem only. Thus we get
additional minima, which are local minima among their neighbors on the
constrained surface, only because adjacent lower values for the
function are excluded by the constraint. We will consider a geometric
example next time.
--
KB