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2.4.2 Legendre transformation
Consider a function
, whose argument we denote by
(the
curve in Figure 2.10 is a possible graph of
).
For simplicity we are considering the scalar case, but
the extension to
is straightforward. The
Legendre transform of
will be a new function,
, of a new
variable,
.
Figure:
Legendre transformation
|
Let
be given. Draw a line through the origin with slope
.
Take a point
at which the (directed) vertical distance
from the graph of
to this line is maximized:
|
(2.33) |
(Note that
may not exist, so the domain of
is not known a
priori. Also,
is not necessarily unique unless
is a strictly convex
function.)
Now, define
to be this maximal value of the
gap between
and
:
|
(2.34) |
We can also write this definition more symmetrically as
|
(2.35) |
where
and
are related via (2.34).
When
is differentiable,
the maximization condition (2.34) implies that the
derivative of
with respect to
must equal 0 at
:
|
(2.36) |
Geometrically, the tangent line to the graph of
at
must have slope
, i.e., it must be parallel
to the original line through the origin (see Figure 2.10). If
is convex then (2.34) and (2.37) are equivalent.
The Legendre transformation has some nice properties.
For example,
is a convex function even if
is not convex. The reason is
that
is a pointwise maximum of functions that are
affine in
, as is clear from (2.35).
Also, for convex functions
the Legendre transformation is involutive: if
is convex, then
.
Now let us return to the Hamiltonian
defined in (2.29).
We claim that it can be obtained by applying the Legendre
transformation to the Lagrangian
. More precisely, for arbitrary
fixed
and
let us consider
as a function of
. The
relation (2.37) between
and
becomes
|
(2.37) |
which corresponds to our earlier definition (2.28) of the
momentum
. Next,
(2.35) gives
|
(2.38) |
which is essentially our earlier definition (2.29) of the Hamiltonian
. But there is a
difference: in (2.29) we had
as an independent
argument of
, while in (2.39)
is a dependent variable
expressed in terms of
by the implicit relation (2.38).
In other words, the Legendre transform of
as a function of
(with
fixed) is
, which is a function of
(with
fixed)
and no longer has
as an argument.
Note that the above derivation is formal, i.e., we are ignoring the question
of whether or not (2.38) can indeed be solved for
. This issue
did not arise earlier when we were working with the Hamiltonian
.
The above approach has another, more important drawback.
Recall the observation based on (2.31) that
has a stationary point as a function of
along an optimal
curve. This property will be crucial
later; combined with the canonical equations (2.30),
it will lead us to the maximum principle.
But it only makes sense when we treat
as an independent
variable in the definition of
. On the other hand,
Hamilton and other 19th century mathematicians
did not write the Hamiltonian in this way; they followed the convention of
viewing
as a dependent variable defined implicitly
by (2.38). This is probably why it was not until the late
1950s that the maximum principle was discovered.
Next: 2.4.3 Principle of least
Up: 2.4 Hamiltonian formalism and
Previous: 2.4.1 Hamilton's canonical equations
Contents
Index
Daniel
2010-12-20