What is the physical meaning of fictitious time in celestial mechanics?

[Hugo_Tlahuext]

I apologize for my english in advance. To describe aproaches of two bodies in celestial mechanics, it is often used the regularization theory, in order to transform the equations of motion. The kepler problem has a singularity and is removed by introducing the fictitious time (τ) through the relation dt=rdτ, where r represent the relative distance between the bodies. So my questions is:
Does the fictitious time has a physical meaning? Iknow that the use of this parameter (τ) is like watch a slow motion movie of the system when the two bodies approach a collision , i wonder if this new parameter has another property that makes the system well-behaved (the regularized hamiltonian does not have singularities) with respect these close approaches.
Thanks and saludos from Mexico

 

[Valerie Park]

. The fictitious time has a physical meaning in the sense that it is used to represent the motion of the two bodies. It is a mathematical tool used to make the equations of motion more manageable and to remove singularities from the system. The regularized Hamiltonian does not have singularities because the fictitious time eliminates them, giving the system a more consistent behavior. This makes it easier to analyze the dynamics of the system. In essence, the fictitious time is just a way to represent the motion of the two bodies in a more consistent and manageable way.