p400_02b
TRANSCRIPT
-
7/29/2019 p400_02b
1/9
Variational Calculus
-
7/29/2019 p400_02b
2/9
Functional
Calculus operates on functions of one or more
variables.
Example: derivative to find a minimum or maximum
Some problems involve a functional.
The function of a function
Example: work defined on a path; path is a function in space
-
7/29/2019 p400_02b
3/9
Path Variation
A trajectoryy in space is a
parametric function.
y(a,x) =y(0,x) + ah(x)
Continuous variation h(x) End points h(x1) = h(x2) = 0
Define a functionfin space.
Minimize the integralJ.
Ify is variedJmust increase
x2x1
y(x)
y(a,x)
2
1
));('),((x
xdxxxyxyfJ
-
7/29/2019 p400_02b
4/9
Integral Extremum
Write the integral in
parametrized form.
May depend on y=dy/dx
Derivative on parametera
Expand with the chain rule.
Term a only appears with h
2
1
));,('),,((x
xdxxxyxyfJ aa
0
0
aaJ for all h(x)
dxy
y
fy
y
fJ x
x)(
2
1 aaa
dxdx
d
y
fx
y
fJ x
x))((
2
1
hh
a
-
7/29/2019 p400_02b
5/9
Boundary Conditions
The second term can be
evaluated with integration by
parts.
Fixed at boundaries
h(x1) =h(x2) = 0
dxxy
f
dx
dx
y
f
dxdx
d
y
f
x
x
x
x
x
x
)()()(
)(
2
1
2
1
2
1
hh
h
dxxy
f
dx
ddx
dx
d
y
f x
x
x
x)()()(
2
1
2
1
hh
2
1
2
1
)(x
x
x
xdx
dx
d
y
fdxx
y
fJ hh
a
-
7/29/2019 p400_02b
6/9
Eulers Equation
The variation h(x) can be
factored out of the integrand.
The quantity in bracketsmust vanish.
Arbitrary variation
This is Eulers equation.
General mathematical
relationship
dxxy
f
dx
dx
y
fJ x
x
2
1
)()( hha
0
y
f
dx
d
y
f
dxxy
f
dx
d
y
fJ x
x)(
2
1
ha
-
7/29/2019 p400_02b
7/9
Soap Film
y
222 dydxdA
dxyxAx
x 2
1
212
(x2,y2)
(x1,y1)
Problem
A soap film forms between
two horizontal rings that
share a common verticalaxis. Find the curve that
defines a film with the
minimum surface area.
Define a functiony.
The areaA can be found as
a surface of revolution.
-
7/29/2019 p400_02b
8/9
Euler Applied
The area is a functional of
the curve.
Define functional
Use Eulers equation to find
a differential equation.
Zero derivative implies
constant
Select constant a
The solution is a hyperbolic
function.
2
12 yxf
01
02
y
yx
dx
d
y
f
dx
d
y
f
ay
yx
2
1
22ax
ay
a
byax cosh
dxxyyfdxyxAx
x
x
x 2
1
2
1
;,122
-
7/29/2019 p400_02b
9/9
Action
The time integral of the
Lagrangian is the action.
Action is a functional
Extends to multiple coordinates
The Euler-Lagrange equations
are equivalent to finding the
least time for the action.
Multiple coordinates givemultiple equations
This is Hamiltons principle.
2
1
);,(t
tdttqqLS
0
q
L
dt
d
q
L
next
http://localhost/var/www/apps/conversion/tmp/scratch_8/p400_02c.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_8/p400_02c.ppt