82 4. POINT AND CONTACT TRANSFORMATIONS
Substituting (4.75) and (4.76) into (4.74) show these are satisfied if D 1; hence, (4.75) is a
contact transformation.
Example 4.8 Consider the Legendre transformation
X D p; Y D q; U D xp C yq u: (4.77)
Calculating the first derivatives, we find
P D X; Q D Y: (4.78)
Substituting (4.77) and (4.78) into (4.74) shows these are satisfied if D 1; hence, (4.77) is a
contact transformation.
Example 4.9 Consider the Ampere transformation
X D x; Y D q; U D yq u: (4.79)
Calculating the first derivatives, we find
P D p; Q D y: (4.80)
Substituting (4.79) and (4.80) into (4.74) shows these are satisfied if D 1; thus, (4.79) is a
contact transformation.
Example 4.10 Consider
X D x Cq; Y D y Cp; U D u C pq: (4.81)
Calculating the first derivatives we find
P D p; Q D q: (4.82)
Substituting (4.81) and (4.82) into (4.74) shows these are satisfied if D 1; thus, (4.81) is a
contact transformation.
4.3 PLATEAU PROBLEM
Consider the surface area of u D u.x; y/ on some region R
SA D
ZZ
R
q
1 C u
2
x
C u
2
y
dA: (4.83)
For this surface to be a minimum, then the Euler-Lagrange equation from the calculus of vari-
ations is
@
@x
@L
@u
x
C
@
@y
@L
@u
y
@L
@u
D 0 (4.84)
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