Lecture 3
Last time we saw that we could write Newton II in a coordinate-independent way, the Euler-Lagrange equations, but that these equations involved an ad hoc definition of the partial derivative of the Lagrangian. Today we show that the E-L equations can be derived in a sensible and rigorous way from a simple physical principle, called Hamilton's principle.
Hamilton's principle states that of all possible particle motions
between an initial point 
and final point 
, a
classical particle follows the specific path such that the action S --
defined as the time integral of the Lagrangian over the motion --
is extremal (i.e., either a maximum or minimum).
We discussed why a path has extremal action if and only if the first
order variation of the action around it vanishes -- otherwise, nearby
paths would exist with both higher and lower values of the action. We
then calculated the first order variation of the action, using the
chain rule to relate the total change in the Lagrangian to the changes
in each of its arguments. We then used the fact that fluctuations in
are related to those in 
; specifically, 
. This allowed us to integrate the 
contribution by parts to get independent contributions to 
due
to variations in each coordinate 
. For an extremal path, the
first order variation must vanish for all variations in the
path ; this can be true only if the coefficient to in the integrand vanishes. Imposing this condition gives the
Euler-Lagrange equations.
We began a discussion of a feature of the Lagrangian form of mechanics which makes it such a useful prototype for field theory: that it makes the role of symmetries and conservation laws particularly clear. Because Hamilton's principle is so simple, it is simple to see when a transformation is a symmetry -- that is, when it doesn't change the path that extremizes the action.
We first noted that the Lagrangian is not unique: shifting the
Lagrangian by a total time derivative, 
doesn't change the extremal paths. This is because the shifted
Lagrangian gives an action 
, doing
the time integral of L'. When we restrict ourselves to considering
motions between a fixed pair of endpoints and 
, this shift
between S and S' is the same for each path. Thus only the overall
scale for the action looks different in the two pictures: the relative
ranking of which paths have the least and most action remains the
same. Both Lagrangians produce the same classical motions -- the
extremal paths - and so have the same E-L equations (which are true
for the extremal paths). They are equivalent descriptions of the same
physical system.
We now look for symmetries and conservation laws. Being precise, we
call a quantity Q conserved if 
. We note that some
conservation laws drop right out from the form of the E-L equations.
If 
(that is, L has no explicit dependence
on ), then E-L tells us that 
is
conserved. We call such a coordinate "cyclic", and the quantity
its "conjugate momentum".
Showed that for polar coords and a rotationally symmetric potential,
is a cyclic coordinate and its conjugate momentum, angular
momentum, is conserved.
Then claimed that such conservation laws are associated with
symmetries: coordinate transformations that leave the system's E-L
equations unchanged. Conservation laws for cyclic coordinates stem
from invariance of the Lagrangian under the transformation 
. (for polar coords, invariance under the rotation 
).
Discussed example of transformations that might be symmetries for some physical systems: space translations; time translations (which translate the coordinates indirectly through the chain rule); and space rotations about an axis.
Next time we'll see that for every such symmetry, we can construct a conserved charge Q.
-- KB