Vilasi G. Hamiltonian Dynamics
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24
The
Lagrangian Coonlanates
Among all close curves
qh
=
qh(t) joining
A
and
B,
the one for which
the action integral,
(1.33)
takes the least value, will represent the evolution of the system.
Remark
6
It is worth to remark that this formulation holds for a small part
of
the trajectory. On the whole trajectory the integral
S[q]
may have just an
extremum, not necessarily
a
minimum. However, the equations
of
the motion
can be written
by
using only the extremum condition.
The axiomatic approach
is
called the principle
of
least action
or
the Hamil-
Integrals like those in Eq.
(1.33)
are defined on
a
space of functions
ton principle.
S:F-+%,
and could be called functions, but for historical reasons, are called functionals.
A
few words on their use will be spent after a short historical comment.
1.4.1
Historical
notes
The Newton problem
The calculus of variations
was
founded simultaneously to the differential cal-
culus
(1686).
In his Philosophiae Naturalis Principia Mathematica, Newton
was
the first to propose the problem of the body with the least opposition.
It
is
a
problem involving different concrete cases,
as
it concerns the best form
of
a
body (submarine
or
a
missile, etc.) in order to suffer, from the medium,
the least opposition to its motion.
So
formulated, the problem is too difficult.
For the sake
of
simplicity, the question was first proposed for
a
body with a
form invariant for rotation about an axis parallel to the direction of the motion
(equal inertial moments in the plane orthogonal to the velocity), and only for
its head to avoid the difficult problem of vortices surrounding its tail. By king
the length and the height of the head the problem becomes:
Given two points
P
and
Q,
find the planar curve joining them and genera-
ting,
by
rotation around the planar normal
n'
at
P,
a revolution surface while
moving parallel to
6,
that suffers the least opposition from the medium.
The
Calculus
of
Variations and the Lagmnge
Equations
25
The solution clearly depends on the
opposition law.
Newton assumed that
the opposition, on
a
generic element of the surface,
was
proportiond to the
square of the projection of the velocity along
8.
By
taking an orthogonal Cartesian frame, with the x
axis
coincident with
the rotation axis and by denoting with
y
=
y(z)
the unknown curve, it turns
out that the opposition
R
is given by
where
p,
q
are the x-coordinates of the points
P,
Q
and
k
is a constant which
depends on the velocity of the surface. The problem is then to find, among all
curves
y
=
y(x) joining
P
and
Q,
the one
for
which the previous integral
takes
the least value.
The brachistochrone problem
Ten
years
later, the following problem was proposed and solved by Johann
Bernoulli:
Given
two points
P
and
Q
in
a vertical plane, find among all planar curves
joining
them, the one which the time required
for
a
particle
to
descend, without
fiction, from the origin
P
to
Q,
would be the least possible.
By
choosing
a
suitable orthogonal Cartesian frame with
a
vertical
y
axis,
the time for
a
particle to descend from
P
to
Q
along the curve
y
=
y(x)
will
be given by
where
p,
q
are the x-coordinates of the points
P,
Q
and
g
N
9.8
m/s2
is
the
modulus of the acceIeration due to gravity. Thus, the searched curve is the
one for which the above integral takes the least value.
Johann Bernoulli found,
as
a
solution, the curve whose parametric equa-
tions
are
x
=
k(8
-
sin$),
where
k
represents the ratio between the distances
of
points
P
and
Q
from the
origin.
His
method of solution
was
strongly criticized, from a mathematical
I
=
k(8
-
COS~)
,
26
The Lagrangian
Coordinates
point of view, by Newton, Leibnitz,
1'
Hospital and Jacob Bernoulli** who was
able to find the same solution by using different methods. In particular, Jacob
Bernoulli solved the problem by using
a
geometrical method,
of
larger applica-
bility, based on the principle according to which, if
a
curve is characterized by
a
property of maximum or minimum, any part
of
it, no matter how small, is
characterized by the same property, According to this principle, for instance,
if
a
curve
is
a
brachistochrone, any part of it
is
again
a
brachistochrone. It
is,
therefore, possible to replace the curve with
a
broken line in such
B,
way that
the problem reduces to find its vertex by using ordinary differential calculus
met hods.
The approach used by Jacob Bernoulli, who must
be
considered the founder
of
the calculus of variations, were generalized and wisely extended by Eulertt
to a large category
of
problems. He started with the classification
of
problems
in two classes:
find among all curves satisfying suitable boundary conditions, the one
for which a given integral takes an extremum
(a
minimum or
a
maxi-
mum) value.
find among all curves satisfying suitable boundary conditions for which
given integrals take assigned values
(constraints),
the one for which
another integral takes an extremum value
(isoperimetric problems).
He invented the
zsop~r~~et~c rule
which allows to reduce, at least
on
prin-
ciple, a given problem
of
the second class to
a
problem belonging to the first
one.
A
more rigorous treatment was given by Lagrange, who introduced the
concept
of
variation, which allows this kind of problems to be treated with
ordinary differential calculus methods.
"Jacob Bernoulli was born at Basilea (BBle) on December
27,
1654.
We has been, €or many
years,
a
professor of mathe~atics
at
Basilea University. Supporter of Leibnitz' scientific
ideas, he died at Basilea in
1705.
Johann Bernoulli, the brother
of
Jacob, was born
at
Bde on August
7, 1667,
and died
there on January
1,
1748.
He has been,
for
many years,
a
professor of mathematics at
Groningen from
1695
to
1705
and at B&le University, where he succeeded his brother, from
1705
to
1748.
As
an illustration
of
his character,
it
may be mentioned that he expelted his
son Daniel from his house for obtaining a prize from the
Acadernie de hnce
which
he
had
expected to receive himself.
++Leonard Euler, born in Basilea in
1707
and died in St. Petersburg in
1783,
was director
of
the Academy
of
Sciences in Berlin, and
soon
after, the Academy of Sciences in St. Petersburg.
He
was one
of
the most important and fecund mathematicians
of
all times, both in calculus
and in its physical applications.
The
Calculus
of
Variations and the Lagmnge Equations
27
Let
zt
=
uo(z)
be the curve which extremizes the integral
I[%],
among all
the curves in the plane joining two given points
P
and
Q.
In order to analyze
the properties
of
I[.],
it
is
necessary to compare the value
I[uO],
taken by
the
integral
on
the given curve
u
=
uo(z),
with the one
I[u]
corresponding
to
a
different curve. The difference
for
u(z)
-
uo(z)
is the variation, for fixed
z,
in
passing from the curve
u
=
u~(z)
to the curve
u
=
u(x).
If the two curves are
%cry
close,” this variation will be “very small,” similarly to the differential
du
that
a
given function
u
undergoes for a fittle change
dx
of
2,
but very different
in nature. In order
to
distinguish between them, and at the same time to
underline their analogy, Lagrange introduced the symbol
6u
for
the variation
and invented
a
calculus completely analogous to the ordinary one.
In
correspondence with the variation
6u
of the curve
u
=
ug(z),
the integral
I[uo]
undergoes the variation
I[.]
-
I[uol
which decomposes in different parts conformable to the decomposition given
to
a
function
f(z)
from
an
increment of
z.
The different parts are given, save
numerical factors, by succeeding differentials
df,
d2f,.
. .
.
Similarly, the different parts
of
I[u]
-
I[uo],
save numerical factors, are
called first variation, second variation,
.
,
.
of the integral
I
and denoted by
61,
Lagrange was able to show that, similarly to the case
of
a
function, the
extrema of
I
satisfy the relation
61
=
0.
In order to distinguish extrema, in
minima and maxima, Legendre analyzed the second variation and introduced
an elegant transforma~io~
of
a2
I
Ieading to necessary conditions characterizing
maxima or minima. The Legendre condition survived
a
deep criticism by
Lagrange, who observed, among other questions, that the Riccatitt
,
equation
leading to the Legendre transformation
of
the second variation
@I,
does not
always admit
a
bounded continuous solution in all the considered interval and
that the transformation does not always exist. Finally, the difficult question
was
genially solved by Jacobi
(1837)
who gave a new necessary condition.
In
1870,
~eierstrass observed that, contrary to the case
of
f~ctions, the
sign
of
the second variation does not ensure the existence
of
the maximum
or
minimum
on
the considered extrema. According to Weierstrass and Scheeffer,
ZtJacopo Riccati
was
born in Venice in
1676
and died in “reviso in
1754.
He
was
an
aristocrat
who
studied mathematics privately.
His
famous equation
is
contained in
Acta
Eruditomn
(Lipsia,
1722).
621,
*
*.
.
the reason
for
the difference lies in the complexity
of
a
curve with respect
to
a
point. Two curves, although “very close,” can strongly differ by their
tangents at close points and the integral
I
also contains the derivative
u‘
which
characterizes such tangents. Weierstrass was able to give sufficient conditions
for the existence of the minimum
or
the maximum in terms of the sign of a
particular function which today is called the
Weierstmss function.
Weierstrass
also
criticized the representation
y
=
U(S)
for
a
curve, since
this representation meets
at
most
in
a
point, the parallels
to
the vertical
axis,
and then
it
greatly restricts the field of possible solutions.
He
developed its
theory by using parametric representations
and replacing the integral
I
by
At the same time, Darboux found sufficient conditions for the minimum
in the
geodesic
problem by introducing curvilinear coordinates, which allows
us to write the considered integrals in
a
particularly simple form,
where
the
minimum properties turn out to be evident.
Very important contributions were then given by Kneser, Lindeberg,
Gauss,
Ostrogradsky, Delauney, Clebsch, Schwarz, Volterra and Hilbert
.
For
more
details, see
for
instance, Ref.
53.
1.4.2
A
digression on the variation
methods
in problems
with
fixed
boundaries
Continuous
functionals
A
f~ctional
Flu]
will be called
con~~nuous,
if to
a
“small change” of
~(z),
there corresponds a “small change” in the functional
F[u].
With previous definition, only a few functionals will be continuous, since
in general the function in the integrals will depend on the derivatives
of
u.
For
instance, in the action integral, first derivatives are also included.
Thus,
it
is natural to give the following definitions of
close
curves:
Two
curves
u(x)
and
u(x)
are
of
zero-order
proximity
close,
if
the
absolute
value
of
their diflerence
Iu(z)
-
.(.)I
is
small
The
Calculus
of
Variations and the
Lagrange
Equations
29
Two
curves
~(x)
and v(x) are
of
first-order proximity close,
if
the absolute
va~ues
of
the d~#e~nces
{u(x)
-
v(x)l
and
{a'(.)
-
v'(z)l
are
~~~12.
Two
curves
U(X)
and
v(x)
are
of
kth-order proximity close,
if
the absolute
values
of
the di#erence~,
/u(x)
-
v(x)t,
lu'(x)
-
v'(x)/,
.
.
.
,
1dk)'(z)
-
v(~)(%)/,
are
small
It
is
then possible to define the notion of distance
CT
between two curves.
By
assuming that
u
and
v
have continuous derivatives up to order
k,
the
distance
of
order
k:
is
defined as
and then close-lying curves would be curves with
a
small
distance.
De~ivatiwes
in
a
vector space
1.
Strong derivative or Frechet derivative.
Let
U
and
V
be two normed vector
spaces
and
F
a
map from an open subset
A
of
U
into
V.
F:AG
U
+V.
The map
F
will be called
difierentiable at point
u
E
U,
if
a linear-bounded
operator
F'
exists, such that
F(u
+
h)
-
F(u)
=
F'h
+
b(~,
h)
,
where
a
is
~nfinitesima1 with respect to the distance in
V
given by the norm:
The reader will recognize, by identifying
U
with
Sn
and
V
with
a,
the usual
definition of the differentiability for
a
numerical function
of
n
real variables.
The linear part of the increment
F'h,
is
called the
strong diflerential
(or
the
hchet di#e~nt%ul)
of
the map
F
at point
u
and the operator
F'
is
called
the
strong de~wative
(or
the
Frechet de~vat~ve)
of the map
F
at point
u.
It
is easy to see that, if the map
F
is
differentiab~e, the corresponding derivative
30
The
Lagmngian
Coordinates
is
uniquely defined. Furthermore, the theorem on the derivative
of
composed
maps can be easily proven.”
2.
Weak derivative
or
Gateaux derivative.
If
G
denotes
a
map from an open
subset
of
U
into
V,
the limit
d
.
G(u
+
th)
-
G(u)
t
DG(u,h)
=
-G(u
+
th)(t=o
=
ltm
dt
t40
3
where the convergence is considered only with respect to the norm of the vector
space
V,
is called the
weak differential
or the
Gateaux differential
of
G
at
point
u.
The Gateaux differential can be nonlinear with respect to
h.
In
the
case
it
is
linear, for example,
in
the case a bounded linear operator
G,
exists, such
that
DG[u,
h)
=
G,h
,
the operator
G,
is called the
weak derivative
(or the
Gateaux derivative)
of
G
at point
u.
Let
us
observe that,
in
general,
for
the weak derivatives, the
theorem
on
the derivative of a composed map
is
not true.
It
is
easy to see that, if the strong derivative
of
a map
F
exists, then the
weak derivative also exists and the
two
derivatives coincide.
As
a
matter
of
fact,
if
F
is
strongly differentiab~e, then
F(u
4-
th)
-
F(U)
=
F’(u)(th)
4-
u(u,
th)
=
tF’(u)(h)
+
a(u,t)
,
so
that
The converse is not true, and generally never true in the finite dimensional
spaces,
as
the folIowing example well shows.
Example
3
The function
The
Colculus
of
Varlations
and
the
Lagrunge Equations
31
is
cont~~~uus on
g2.
At
the point
(0,
0),
its weak d~fferent~al exists:
Nevertheless, the weak differential
is
not the linear part
of
the increment
of
the
~un~tio~ at the point
(O,O),
since
The result of the previous exercise is not surprising, just like in the finite
dimensional case, the existence of partial derivatives of
a
given function does
not ensure its differentiabilit~, which can be ensured by the continuity
of
the
partial derivatives.
In
this context, it
is
important the following theorem's proof can be found,
for instance, in Ref.
28.
Theorem
1.1
If
the
weak: de~vat~ve
Fu
of
the
map
F
exists
in
a
ne~gh~or-
hood
of
a point
uo
and represents
a
continuous operator at
UO,
then the strong
derivative
of
F
also
exists
at
uo
and coincides with the weak one.
3,
The
gradient
of
a
functional.
Let
U
be
a
vector space
of
numerical functions
of
a
real variable, namely
u(z),
endowed with
a
scalar product
(.
,
.)
and let
F
be
a
~nctiona~ defined in
U,
F
:
u
E
U
4
F[u]
E
8.
The gradient of
F,
or
the
functional derivative
of
F,
with respect to the scalar
product
is
the function of
U,
denoted by
G
=
SF,/du,
defined by the relation
6F
E
-F[u+~p]I~=o
d
=
(g,~)
.
dc
(1.34)
The
second functional derivative of
F
is
given
by
the weak derivative
Gu
of
G,
which in this case, can be
also
defined
as
32
The
Lagrungian
Coordinates
Of
course, the operator
G,
is
symmetric with respect to
(.
,
.);
i.e.
G,=G,+,
since
Example
4
Let
U
be
the space
of
all
Cw
functions
u(x),
defined
on
the
interval
I
=
]a,
b(
of
the real axis
x
and going to zero at
a
and b, together with
all their
x
derivatives, and let
F[u]
be a functional,
6
~[u]
=
1
f(u,
ux,
Urx,
*
*
unx,.
.
.)dx
1
where
f
is a given function
of
u
and
of
its
x
derivatives, which we are denoting
with
u,,
tizx,.
. .
,
u,,,
.
. .
.
According to the definition
of Eq.
(1.34),
we
have
and integrating
by
parts all the terms, except the first,
Therefore,
so
that
The
Calcdvs
of
Variations and
the
Lagmnge Equations
33
In this way,
in
the simple case
of
a
functional
depending
only on
u,
6
FI.1
=
Jd
f
(UW,
the func~~o~al derivative
is
given
&y
SF
af
su
au'
-=-
The functional derivative of
a
functional depending only on
u
and
ux,
b
Flu1
=
1
f(u,
ux)dx,
is given by
For insta~ce, for square inte~a~$e functions, we have
and the Gateaux derivatives
of
are the following operators
d
dx
'
1,
-ax,,
2u+axx,
ax
FE
-
which are s~m~etric wzt~ ~s~ct to the
L2
scalar ~ro~uct.
Example
5
The ~nction
us
cannot be the gradient, with respect to the
Lz
scalar pmhct,
of
any functional, since its Gateaux derivative
is
the skewsym-
metric operator
a,.