What are the effects of higher derivatives in the slow-motion approach?

[eljose]

Hello i didn,t understand the slow motion approach of course i know that:

$$\frac{\partial f }{\partial x^{0}}\sim \epsilon \frac{\partial f }{\partial x^{a}}$$ according to this approach with epsilon<<<1 small parameter for every smooth function my doubts are.

a)what would happen to higher derivatives of f with respect to time and spatial coordinates?..x,y,z ?

b)If we only want effect upto first order in epsilon parameter then..what would happen to:

$$\nabla ^{2}f$$ (Laplacian)

$$(1,1,1)*Gra(f)$$ (scalar product involving the gradient)

or if we had $$\epsilon div(f)$$ would it mean that the only term that should be kept is df/dt if we consider effects only to first order?..

thanks.

 

[pervect]

I've been trying to fill in the context of this post, with little success. Regarding what problem are you taking a "slow motion" approach? Are you doing low-velocity, weak-field approximations to GR?

 

[Entropia]

The slow-motion approach is a mathematical technique used to simplify the analysis of systems with small parameters. In this approach, the small parameter (represented by epsilon) is multiplied by derivatives of a function (represented by f) with respect to time and spatial coordinates.

To answer your questions, the effects of higher derivatives in the slow-motion approach depend on the specific system being analyzed. Generally, higher derivatives will have a smaller impact on the overall behavior of the system compared to lower derivatives. This is because the small parameter (epsilon) is multiplied by the derivative, making higher derivatives even smaller.

If we only want to consider effects up to first order in the epsilon parameter, then higher derivatives will have even less of an impact. In this case, the Laplacian (second derivative with respect to spatial coordinates) and scalar product involving the gradient (first derivative with respect to spatial coordinates) would be negligible compared to the first derivative with respect to time.

For example, if we had an equation with the term \epsilon \nabla^{2}f, and we are only considering effects up to first order, then we can neglect this term and only consider the term \epsilon \frac{\partial f }{\partial t}. This would simplify the analysis and make it easier to solve.

In summary, the effects of higher derivatives in the slow-motion approach are smaller compared to lower derivatives, and can be neglected if we are only considering effects up to a certain order. However, the specific impact of higher derivatives will depend on the system and the specific equations being analyzed.