# Background

A sub-problem of some velocity planning problems is to determine the velocity curve where for some k: $$\frac{d^kv}{dt^k}=\text{constant}\tag{1}$$ However, the velocity should be solved in the form $v=v(s)$, rather than $v=v(t)$. Where $s$ is the arc length, so that $$\frac{ds}{dt}=v\tag{2}$$

Therefore we have a change-of-variable problem for an ODE.

# The Problem

Given ODE (1) and (2), I want to transform equation (1) from an equation of v(t) to an equation of v(s).

Here is what I have tried:
\begin{align} \frac{d^kv}{dt^k}&=(\frac{d}{dt})^kv\\ &=(\frac{ds}{dt}\frac{d}{ds})^kv \end{align} Therefore, it is natural to consider a general formula for $(\frac{ds}{dt}\frac{d}{ds})^k$.

I've calculated formulas for the first few values of k: $$(\frac{ds}{dt}\frac{d}{ds})^2=\frac{d^2s}{dt^2}\frac{d}{ds}+(\frac{ds}{dt})^2\frac{d^2}{ds^2}$$ $$(\frac{ds}{dt}\frac{d}{ds})^3=\frac{d^3s}{dt^3}\frac{d}{ds}+3\frac{ds}{dt}\frac{d^s}{dt^2}\frac{d^2}{ds^2}+(\frac{ds}{dt})^3\frac{d^3}{ds^3}$$

These are similar problems, but they have different answers. And one answer seems not giving me an insight for another.

Is there a general formula for my problem? Is there a general method for these problems? BTW, is there a software which can calculate these formulas automatically?

Thank you!

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