2.4.3 Principle of least action and conservation laws

Newton's second law of motion in the three-dimensional space can be written as the vector equation

where is the vector of coordinates, is the velocity vector, and is the potential; consequently, is the momentum and is the force. Note that we are only considering situations where the force is conservative, i.e., corresponds to the negative gradient of some potential function. Planar motion is obtained as a special case by dropping the -coordinate.

It turns out that there is a direct relationship between (2.40) and the Euler-Lagrange equation. This is difficult to see right now because the notation in (2.40) is very different from the one we have been using. So, let us modify our earlier notation to better match (2.40). First, let us write instead of for the independent variable. Second, let us write instead of for the dependent variable. Then also becomes and we have . In the new notation, the Euler-Lagrange equation becomes

Note that since , we are referring to the multiple-degrees-of-freedom version (2.21) of the Euler-Lagrange equation, with .

Some remarks on the above change of notation
are in order (as it is also a preview of things to come).The change from
to
is conceptually significant, because
it implies that the curves are parameterized by *time* and thus describe
some dynamic behavior (e.g.,
trajectories of a moving object). In problems such as Dido's problem or catenary
problem, where there is no motion with respect to time, this notation would
not be justified.
Other variational problems, such as the brachistochrone problem, do indeed
deal with paths of a moving particle. (We did not, however, explicitly
use time when formulating the brachistochrone problem, and
it would not make sense to just relabel
as
in our earlier formulation of that problem.
Instead,
we would need to reparameterize the
-trajectories with respect to time,
which yields a different Lagrangian; we will do this
in Section 3.2.)
In mechanics, as well as in control theory, time is
the default independent variable. Accordingly,
when we come to the optimal control part of the book, we will
make this change of notation permanent. For now, we adopt it just temporarily
while we discuss applications of calculus of variations in mechanics. As
for the (time-)dependent
coordinate variables, it is natural
to denote them by
and
for planar curves
and by
,
and
for spatial curves. In
general, the choice of a label
for the coordinate vector
is just a matter of preference and convention; the mechanics
literature typically favors
or
, while
in control theory it is customary to use
.

Let us now compare (2.41) with (2.40). Is there a choice of the Lagrangian that would make these two equations the same? The answer is yes, and the reader will have no difficulty in seeing that the following Lagrangian does the job:

which is the difference between the kinetic energy and the potential energy. We conclude that Newton's equations of motion can be recovered from a path optimization problem. This important result is known as

which is called the

For example, if the potential is 0, then the trajectories are the extremals of , which are straight lines. We saw in Example 2.2 that straight lines arise as extremals when the Lagrangian is the arclength; the kinetic energy gives the same extremals. In the presence of gravity, the paths along which the action integral is minimized can be viewed as ``straight lines" (shortest paths, or geodesics) in a curved space whose metric is determined by gravitational forces. This view of mechanics forms the basis for Einstein's theory of general relativity.

Observe the difference between the law that the derivative of the momentum equals the force and the principle that the action integral is minimized. The former condition holds pointwise in time, while the latter is a statement about the entire trajectory. However, their relation is not surprising, because if the action integral is minimized then every small piece of the trajectory must also deliver minimal action. In the limit as the length of the piece approaches 0, we recover the differential statement. We already discussed this point in the general context of the Euler-Lagrange equation (see page ).

Now, what is the physical meaning of the Hamiltonian? Substituting for in (2.28)-(2.29) and using (2.42), we have

which is the total energy (kinetic plus potential). We see that the Hamiltonian not only enables a convenient rewriting of the Euler-Lagrange equation in the form of the canonical equations (2.30), but also has a very clear mechanical interpretation.

Recall that in
Section 2.3.4 we studied two special cases for
which we found conserved quantities, i.e., functions that remain constant
along extremals. We now revisit these two *conservation laws*--as well
as another related case--in the context
of the principle of least action, which permits us to see their physical meaning.

CONSERVATION OF ENERGY. In a conservative system, the potential is fixed and does not change with time. Since the kinetic energy does not explicitly depend on time either, we have . In other words, the Lagrangian is invariant under time shifting. In our old notation, this corresponds to the ``no " case from Section 2.3.4, and we know that in this case the Hamiltonian is conserved. We also just saw that the Hamiltonian is the total energy of the system. Therefore, this is nothing but the well-known principle of conservation of energy.

CONSERVATION OF MOMENTUM. Suppose that no force
is acting on the system (i.e., the system is closed). Since the force
is given by
, this implies that
must be constant. The kinetic
energy
depends on
but not on
. Thus the Lagrangian
does not
explicitly depend on
, which means that it is invariant
under
parallel translations.
This situation corresponds to the
``no
" case from Section 2.3.4,
where we saw that the *momentum*,
in the present
notation,
is conserved. A more general statement is that for
each coordinate
that does not appear in
, the
corresponding component
of the momentum is conserved.

CONSERVATION OF ANGULAR MOMENTUM.
Consider a planar motion in a central field; in polar coordinates
, this is defined by the property that
, i.e., the potential depends only on the radius and not on the
angle. This means that
no *torque* is acting on the system, making the Lagrangian
invariant under
rotations. Now we can use the fact
(noted at the end of Section 2.3.3)
that the Euler-Lagrange equation looks the same in all coordinate systems. In particular,
in polar coordinates we have

and the analogous equation for (which we do not need here). Arguing exactly as before, we can show that in the present ``no " case the corresponding component of the momentum, , is conserved. Converting the kinetic energy from Cartesian to polar coordinates, we have

Thus the conserved quantity, in polar and Cartesian coordinates, is

These are familiar expressions for the

We remark that all of the above examples are special instances of a general result known as Noether's theorem, which says that invariance of the action integral under some transformation (e.g., time shift, translation, rotation) implies the existence of a conserved quantity.