Q:

How to understand "force of interest" in simple interest?

I am struggling to understand "force of interest" in simple interest. I understand that "force of interest" can be interpreted as compound interest with infinitely small time interval. That is, the new interest generated any infinitely short moment is added to the value. 
However, as simple interest means that the interest has a linear continuous relationship with time, I am not sure how to understand "force of interest" in simple interest. For example, how can I interpret the equation $\delta _t = \frac{i}{1+it}$ realistically in simple interest? 

A:

You seem to have taken the force of interest to be $\dfrac{a'(t)}{a(t)}$ where $a(t)$ is the accumulated amount at time $t$
So your expression $\delta _t = \dfrac{i}{1+it}$ seems to be correct for simple interest with an interest rate of $i$, since $a(t)=a(0) (1+it)$ and $a'(t) = a(0)i$
In this case, $\delta_t$ is the momentary compounding rate, and is a decreasing function of $t$: 

when $t=0$ and the starting amount is $a(0)$, it is $\delta_0=i$, 
while for example when $t=\frac1i$ and the accumulated amount is $2a(0)$, it is $\delta_{1/i}=\frac12 i$. 

