<s>
and	O
functionals	O
,	O
to	O
find	O
maxima	O
and	O
minima	O
of	O
functionals	O
:	O
mappings	B-Algorithm
from	O
a	O
set	O
of	O
functions	O
to	O
the	O
real	O
numbers	O
.	O
</s>
<s>
Functionals	O
are	O
often	O
expressed	O
as	O
definite	B-Algorithm
integrals	I-Algorithm
involving	O
functions	O
and	O
their	O
derivatives	B-Algorithm
.	O
</s>
<s>
Functions	O
that	O
maximize	O
or	O
minimize	O
functionals	O
may	O
be	O
found	O
using	O
the	O
Euler	O
–	O
Lagrange	O
equation	O
of	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
.	O
</s>
<s>
Although	O
such	O
experiments	O
are	O
relatively	O
easy	O
to	O
perform	O
,	O
their	O
mathematical	O
formulation	O
is	O
far	O
from	O
simple	O
:	O
there	O
may	O
be	O
more	O
than	O
one	O
locally	O
minimizing	O
surface	O
,	O
and	O
they	O
may	O
have	O
non-trivial	O
topology	B-Architecture
.	O
</s>
<s>
The	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
may	O
be	O
said	O
to	O
begin	O
with	O
Newton	O
's	O
minimal	O
resistance	O
problem	O
in	O
1687	O
,	O
followed	O
by	O
the	O
brachistochrone	O
curve	O
problem	O
raised	O
by	O
Johann	O
Bernoulli	O
(	O
1696	O
)	O
.	O
</s>
<s>
After	O
Euler	O
saw	O
the	O
1755	O
work	O
of	O
the	O
19-year-old	O
Lagrange	O
,	O
Euler	O
dropped	O
his	O
own	O
partly	O
geometric	O
approach	O
in	O
favor	O
of	O
Lagrange	O
's	O
purely	O
analytic	O
approach	O
and	O
renamed	O
the	O
subject	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
in	O
his	O
1756	O
lecture	O
Elementa	O
Calculi	O
Variationum	O
.	O
</s>
<s>
Marston	O
Morse	O
applied	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
in	O
what	O
is	O
now	O
called	O
Morse	O
theory	O
.	O
</s>
<s>
Lev	O
Pontryagin	O
,	O
Ralph	O
Rockafellar	O
and	O
F	O
.	O
H	O
.	O
Clarke	O
developed	O
new	O
mathematical	O
tools	O
for	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
in	O
optimal	O
control	O
theory	O
.	O
</s>
<s>
The	O
dynamic	B-Algorithm
programming	I-Algorithm
of	O
Richard	O
Bellman	O
is	O
an	O
alternative	O
to	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
.	O
</s>
<s>
The	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
is	O
concerned	O
with	O
the	O
maxima	O
or	O
minima	O
(	O
collectively	O
called	O
extrema	O
)	O
of	O
functionals	O
.	O
</s>
<s>
A	O
functional	B-Algorithm
maps	I-Algorithm
functions	O
to	O
scalars	O
,	O
so	O
functionals	O
have	O
been	O
described	O
as	O
"	O
functions	O
of	O
functions.	O
"	O
</s>
<s>
Functionals	O
have	O
extrema	O
with	O
respect	O
to	O
the	O
elements	O
of	O
a	O
given	O
function	B-Algorithm
space	I-Algorithm
defined	O
over	O
a	O
given	O
domain	B-Algorithm
.	O
</s>
<s>
For	O
a	O
function	B-Algorithm
space	I-Algorithm
of	O
continuous	O
functions	O
,	O
extrema	O
of	O
corresponding	O
functionals	O
are	O
called	O
strong	O
extrema	O
or	O
weak	O
extrema	O
,	O
depending	O
on	O
whether	O
the	O
first	B-Algorithm
derivatives	I-Algorithm
of	O
the	O
continuous	O
functions	O
are	O
respectively	O
all	O
continuous	O
or	O
not	O
.	O
</s>
<s>
Both	O
strong	O
and	O
weak	O
extrema	O
of	O
functionals	O
are	O
for	O
a	O
space	O
of	O
continuous	O
functions	O
but	O
strong	O
extrema	O
have	O
the	O
additional	O
requirement	O
that	O
the	O
first	B-Algorithm
derivatives	I-Algorithm
of	O
the	O
functions	O
in	O
the	O
space	O
be	O
continuous	O
.	O
</s>
<s>
The	O
maxima	O
and	O
minima	O
of	O
a	O
function	O
may	O
be	O
located	O
by	O
finding	O
the	O
points	O
where	O
its	O
derivative	B-Algorithm
vanishes	O
(	O
i.e.	O
,	O
is	O
equal	O
to	O
zero	O
)	O
.	O
</s>
<s>
The	O
extrema	O
of	O
functionals	O
may	O
be	O
obtained	O
by	O
finding	O
functions	O
for	O
which	O
the	O
functional	O
derivative	B-Algorithm
is	O
equal	O
to	O
zero	O
.	O
</s>
<s>
If	O
the	O
functional	O
attains	O
a	O
local	O
minimum	O
at	O
and	O
is	O
an	O
arbitrary	O
function	O
that	O
has	O
at	O
least	O
one	O
derivative	B-Algorithm
and	O
vanishes	O
at	O
the	O
endpoints	O
and	O
then	O
for	O
any	O
number	O
close	O
to	O
0	O
,	O
</s>
<s>
According	O
to	O
the	O
fundamental	O
lemma	O
of	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
,	O
the	O
part	O
of	O
the	O
integrand	O
in	O
parentheses	O
is	O
zero	O
,	O
i.e.	O
</s>
<s>
Substituting	O
for	O
and	O
taking	O
the	O
derivative	B-Algorithm
,	O
</s>
<s>
If	O
depends	O
on	O
higher-derivatives	O
of	O
that	O
is	O
,	O
if	O
then	O
must	O
satisfy	O
the	O
Euler	O
–	O
Poisson	O
equation	O
,	O
</s>
<s>
The	O
discussion	O
thus	O
far	O
has	O
assumed	O
that	O
extremal	O
functions	O
possess	O
two	O
continuous	O
derivatives	B-Algorithm
,	O
although	O
the	O
existence	O
of	O
the	O
integral	O
requires	O
only	O
first	B-Algorithm
derivatives	I-Algorithm
of	O
trial	O
functions	O
.	O
</s>
<s>
then	O
has	O
two	O
continuous	O
derivatives	B-Algorithm
,	O
and	O
it	O
satisfies	O
the	O
Euler	O
–	O
Lagrange	O
equation	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
denotes	O
the	O
displacement	O
of	O
a	O
membrane	O
above	O
the	O
domain	B-Algorithm
in	O
the	O
plane	O
,	O
then	O
its	O
potential	O
energy	O
is	O
proportional	O
to	O
its	O
surface	O
area	O
:	O
</s>
<s>
The	O
difficulty	O
with	O
this	O
reasoning	O
is	O
the	O
assumption	O
that	O
the	O
minimizing	O
function	O
u	O
must	O
have	O
two	O
derivatives	B-Algorithm
.	O
</s>
<s>
This	O
corresponds	O
to	O
an	O
external	O
force	O
density	O
in	O
an	O
external	O
force	O
on	O
the	O
boundary	O
and	O
elastic	O
forces	O
with	O
modulus	O
acting	O
on	O
The	O
function	O
that	O
minimizes	O
the	O
potential	O
energy	O
with	O
no	O
restriction	O
on	O
its	O
boundary	O
values	O
will	O
be	O
denoted	O
by	O
Provided	O
that	O
and	O
are	O
continuous	O
,	O
regularity	O
theory	O
implies	O
that	O
the	O
minimizing	O
function	O
will	O
have	O
two	O
derivatives	B-Algorithm
.	O
</s>
<s>
If	O
these	O
forces	O
are	O
in	O
equilibrium	O
,	O
then	O
the	O
variational	B-Algorithm
problem	I-Algorithm
has	O
a	O
solution	O
,	O
but	O
it	O
is	O
not	O
unique	O
,	O
since	O
an	O
arbitrary	O
constant	O
may	O
be	O
added	O
.	O
</s>
<s>
Both	O
one-dimensional	O
and	O
multi-dimensional	O
eigenvalue	O
problems	O
can	O
be	O
formulated	O
as	O
variational	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
The	O
primary	O
variational	B-Algorithm
problem	I-Algorithm
is	O
to	O
minimize	O
the	O
ratio	O
among	O
all	O
satisfying	O
the	O
endpoint	O
conditions	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
(	O
see	O
Gelfand	O
and	O
Fomin	O
1963	O
)	O
that	O
the	O
minimizing	O
has	O
two	O
derivatives	B-Algorithm
and	O
satisfies	O
the	O
Euler	O
–	O
Lagrange	O
equation	O
.	O
</s>
<s>
The	O
associated	O
minimizing	O
function	O
will	O
be	O
denoted	O
by	O
This	O
variational	O
characterization	O
of	O
eigenvalues	O
leads	O
to	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
:	O
choose	O
an	O
approximating	O
as	O
a	O
linear	O
combination	O
of	O
basis	O
functions	O
(	O
for	O
example	O
trigonometric	O
functions	O
)	O
and	O
carry	O
out	O
a	O
finite-dimensional	O
minimization	O
among	O
such	O
linear	O
combinations	O
.	O
</s>
<s>
The	O
variational	B-Algorithm
problem	I-Algorithm
also	O
applies	O
to	O
more	O
general	O
boundary	O
conditions	O
.	O
</s>
<s>
Hence	O
,	O
solving	O
the	O
associated	O
partial	O
differential	O
equation	O
of	O
first	O
order	O
is	O
equivalent	O
to	O
finding	O
families	O
of	O
solutions	O
of	O
the	O
variational	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
This	O
is	O
the	O
essential	O
content	O
of	O
the	O
Hamilton	O
–	O
Jacobi	O
theory	O
,	O
which	O
applies	O
to	O
more	O
general	O
variational	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
In	O
classical	O
mechanics	O
,	O
the	O
action	O
,	O
is	O
defined	O
as	O
the	O
time	B-Algorithm
integral	I-Algorithm
of	O
the	O
Lagrangian	O
,	O
The	O
Lagrangian	O
is	O
the	O
difference	O
of	O
energies	O
,	O
</s>
<s>
Further	O
applications	O
of	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
include	O
the	O
following	O
:	O
</s>
<s>
Variational	B-Algorithm
method	I-Algorithm
(	O
quantum	O
mechanics	O
)	O
,	O
one	O
way	O
of	O
finding	O
approximations	O
to	O
the	O
lowest	O
energy	O
eigenstate	O
or	O
ground	O
state	O
,	O
and	O
some	O
excited	O
states	O
;	O
</s>
<s>
Variational	O
Bayesian	O
methods	O
,	O
a	O
family	O
of	O
techniques	O
for	O
approximating	O
intractable	O
integrals	B-Algorithm
arising	O
in	O
Bayesian	O
inference	O
and	O
machine	O
learning	O
;	O
</s>
<s>
Variational	B-Algorithm
methods	I-Algorithm
in	O
general	O
relativity	O
,	O
a	O
family	O
of	O
techniques	O
using	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
to	O
solve	O
problems	O
in	O
Einstein	O
's	O
general	O
theory	O
of	O
relativity	O
;	O
</s>
<s>
Finite	B-Application
element	I-Application
method	I-Application
is	O
a	O
variational	B-Algorithm
method	I-Algorithm
for	O
finding	O
numerical	O
solutions	O
to	O
boundary-value	O
problems	O
in	O
differential	O
equations	O
;	O
</s>
<s>
Total	B-Algorithm
variation	I-Algorithm
denoising	I-Algorithm
,	O
an	O
image	B-Algorithm
processing	I-Algorithm
method	O
for	O
filtering	O
high	O
variance	O
or	O
noisy	O
signals	O
.	O
</s>
<s>
Calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
is	O
concerned	O
with	O
variations	O
of	O
functionals	O
,	O
which	O
are	O
small	O
changes	O
in	O
the	O
functional	O
's	O
value	O
due	O
to	O
small	O
changes	O
in	O
the	O
function	O
that	O
is	O
its	O
argument	O
.	O
</s>
