Necessary and Sufficient Condition for Integral Parametric Functional to be Independent of Parametric Representation

Theorem
Let $x=x(t)$ and $y=y(t)$ be real functions.

Let $J[x,~y]$ be a functional of the form

$\displaystyle J[x,~y]=\int_{t_0}^{t_1}\Phi\left( t,~x,~y,~\dot{x},~\dot{y} \right)\mathrm{d}{t}$

where $\dot{y}=\frac{\mathrm{d}{y}}{\mathrm{d}{t}}$.

Then $J[x,~y]$ depends only on the curve in the xy-plane defined by the parametric equations $x=x(t)$, $y=y(t)$

and not on the choice of the parametric representation of the curve if and only if the integrand $\Phi$ does not involve $t$ explicitely

and is a positive-homogeneous of dregree $1$ in $\dot{x}$ and $\dot{y}$.

Proof
Suppose that in the functional


 * $\displaystyle \mathscr{J}[y]=\int_{x_0}^{x_1}F\left(x, y, y'\right)\mathrm{d}{x}$

the argument $y$ stands for a curve which is given in a parametric form.

In other words, the curve is described by $\left(y(t),~x(t)\right)$ rather than $\left(y(x),~x\right)$

Then the functional can be rewritten as

The function on the RHS does not involve $t$ explicitely.

Suppose, it is positive-homogeneous of degree 1 in $\dot{x}(t)$ and $\dot{y}(t)$:

$\Phi\left(x,~y~,~\lambda\dot{x},~\lambda\dot{y}\right)=\lambda\Phi\left(x,~y,~\dot{x},~\dot{y} \right)$ for every $\lambda>0$.

Now we will show that the value of such functional depends only on the curve in the xy-plane defined by the parametric equaions $x=x(t)$ and $y=y(t)$,

and not on the functions $x(t)$, $y(t)$ themselves.

Suppose, a new parameter $\tau$ is chosen such that $t=t(\tau)$, where $\frac{d t}{d\tau}>0$, and the interval $\left[{{t_0}\,.\,.\,{t_1}}\right]$ is mapped onto $[\tau_0, \tau_1]$.

Since $\Phi$ is positive-homogeneous of degree 1 in $\dot{x}$ and $\dot{y}$, it follows that