6.1 Parametric Forms of Curves and Homogeneous Condition
367
F(kx 1 , kx 2 , . . . , kx m ) = k
n F(x 1 , x 2 , . . . , x m )
(6.1.9)
and has the first continuous derivative, then there is
m
i=1
x i F x i (x 1 , x 2 , . . . , x m ) = n F(x 1 , x 2 , . . . , x m )
(6.1.10)
Proof Taking the partial derivative with respect to k on the both sides of the expression (6.1.9), then let k = 1, the expression (6.1.10) can be obtained. Quod erat
demonstrandum.
Theorem 6.1.1 is called the Euler(’s) homogeneous function theorem. This
theorem can be extended to the more general case.
Theorem 6.1.2 If the integrand F(x(t), y(t), ˙
x(t), ˙
y(t)) of the functional (6.1.8)
does not explicitly contain t, and it is the first order homogeneous function about
˙
x(t), ˙
y(t), then the form of the functional has nothing to do with the choice of
parameters.
Proof Because F(x(t), y(t), ˙
x(t), ˙
y(t)) is the first order homogeneous function
about ˙
x(t), ˙
y(t), there is
F(x(t), y(t), k ˙
x(t), k ˙
y(t)) = k F(x(t), y(t), ˙
x(t), ˙
y(t)) (k = 0)
(6.1.11)
Introducing a new parameter τ , and let t = ϕ(τ ), with ˙
ϕ(τ ) = 0, substituting
them into the expression (6.1.1), we get
x = x(τ ), y = y(τ ) (τ 0 τ τ 1 )
(6.1.12)
and there are
⎧
⎪ ⎨
⎪ ⎩
dx
dτ
=
dx
dt
dt
dτ
= ˙
x ˙
ϕ(τ )
dy
dτ
=
dy
dt
dt
dτ
= ˙
y ˙
ϕ(τ )
(6.1.13)
Substituting the expressions (6.1.12) and expressions (6.1.13) into the functional
(6.1.8), we obtain
J [x(t), y(t)] =
t 1
t 0
F(x(t), y(t), ˙
x(t), ˙
y(t))dt
=
τ 1
τ 0
F
x(τ ), y(τ ),
˙
x(τ )
˙
ϕ(τ )
,
˙
y(τ )
˙
ϕ(τ )
˙
ϕ(τ )dτ