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

Theorem
Let $ x = x \left ( { t } \right ) $ and $ y = y \left ( { t } \right ) $ be real functions.

Let $ J \left [ { x, y } \right ] $ be a functional of the form

$ \displaystyle J \left [ { x, y } \right ] = \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 \left [ { x, y } \right ] $ depends only on the curve in the xy-plane defined by the parametric equations $ x = x \left ( { t } \right ) $, $ y = y \left ( { t } \right ) $

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 \left [ { y } \right ] = \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 \left ( { t } \right ), x \left ( { t } \right ) } \right ) $ rather than $ \left ( { y \left ( { x } \right ), 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 \left ( { t } \right ) $ and $ \dot y \left ( { t } \right ) $:

$ \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 \left ( { t } \right ) $ and $ y = y \left ( { t } \right ) $,

and not on the functions $ x \left ( { t } \right ) $, $ y \left ( { t } \right ) $ themselves.

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

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