How to construct a set of variables that automatically satisfies certain constraints

Question

I ran into this problem when trying to do a project on one-electron reduced matrices of fermions. The math can be formulated as following:

Let $\{a_i\}_{i=1...m}$ be a set of variables with additional constraints

1. $\sum_{i=1}^ma_i = N$,
2. $a_i \in [0, 2]$.

Obviously, we also have $\text{min}(m) > N/2$.

So my question is that is there any way we could construct another set of variables, i.e. $\{b_i\}$, such that they could satisfy these constraints automatically?

For examples for constraint 2, we could let $b_i = 2 (1- \text{cos}(\theta_i))$, so that $\{b_i\}$ satisfy constraint 2 by construction.

Is there any way to do that for constraint 1 too?