2.10 Invariance of the Euler Equation
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2.10 Invariance of the Euler Equation
It is observed from higher mathematics that when the function y = f (u) has derivative and u is an independent variable, the differential is dy = f
(u)du, if u is an
intermediate variable and is a function u = ϕ(x) with derivative, and ϕ
(x) = 0,
then there is
dy
dx
= f
(u)ϕ
(x)
(2.10.1)
Thus
dy = f
(u)ϕ
(x)dx
(2.10.2)
But then
du = ϕ
(x)dx
(2.10.3)
Therefore
dy = f
(u)du
(2.10.4)
This shows that whether u is the independent variable or intermediate variable,
the differential form of the function y = f (u) is the same, this property is called the
invariance of differential form.
The Euler equation has also a similar invariance. Consider the simplest functional
J [y] =
x 1
x 0
F(x, y, y
)dx
(2.10.5)
The Euler equation is
F y −
d
dx
F y = 0
(2.10.6)
Let x be the function of ξ , then
x = x(ξ ), y = y(x(ξ )) = y(ξ )
(2.10.7)
When x = x 0 , ξ = ξ 0 , when x = x 1 , ξ = ξ 1 , and let
F 1
ξ, y,
dy
dξ
= F
x(ξ ), y,
dy/dξ
dx/dξ
dx
dξ
= G
ξ, y,
dy
dξ
dx
dξ
(2.10.8)
then there is