Does Anyone Use Newtonian Physics w/ Arbitrary Time Coordinates?

[pervect]

It's fairly common to use Lagrangian mechanics to handle arbitrary space coordinates. But is it ever actually used to handle arbitrary time coordinates, particularly in Newtonian mechanics?

For example, does anyone consider the transformation

x = X
t = T + uX

where (x,t) are normally synchronized coordinates, and (X,T) are abnormally synchronized coordinates.

I fiddled with this, and I'm currently getting the following. Consider a free particle, and let X' = dX / dT

Then I get:

$$L(T,X,X') = \left(\frac{m}{2}\right) \, \frac{X'^2 }{(1 + u\,X')}$$

$$p = \frac{\partial L}{\partial X'} = m\,X'\,\frac{(1+\frac{1}{2}u\,X') }{ (1+u\,X')^2}$$
which gives an energy function, h of
$$h = pX' - L = \frac{m}{2} \left(\frac{X'}{1+u\,X'}\right)^2$$

It seems to work, because the energy function h is what one expects, i.e h = (1/2) m v^2, where v = dx/dt, the velocity in normally synchronized coordinates.

as
dx = dX, and dt = dT + udX,

so dx/dt = (dX/dT) / (1 + u (dX/dT)) = X' / (1+u X')

however, I had to try a couple of times to get the Lagrangian right - while L has the same numerical value independent of changes in the spatial coordinates, it's NOT independent of the changes in the time coordinates , rather L dt is the same, and dt != dT.

Anyway, if this is legitimate, is there any paper or textbook that considers the problem?

 

[atyy]

I don't think this is exactly it, but perhaps something like this?

http://arxiv.org/abs/hep-th/9205092