We recall that Isaac Newton invented calculus for the purpose of his mechanics. Likewise we need to develop some calculus ideas for the relational mechanics.

Concretely we have some pseudo partial derivatives which we have not yet evaluated. Actually i think we've been confused because we should not be speaking of "partial derivatives" at all because partial derivatives assume the existence of independant Cartesian coordinates. Instead of partial derivatives $\partial r_{ij} /\partial r_{kl}$ we should be strictly defining $d r_{ij} / d r_{kl}$. But there are too many partial derivative symbols to replace at this stage.

The symbol $r_{ij}$ represents the radial distance between the $i$th and $j$th particles. The symbol $\nu_{ij}$ repressents the relative radial velocities $\nu_{ij}:={\hat_{ij}}} \cdot v_{ij}$.

[Notation] Here $r^2_{ij}:=|x_i - x_j|^2$, and $v_{ij}$ is the relative velocity of the particles $i,j$ and ${\hat{r}}_{ij}$ is the relative unit vector from particles $i,j$.

If we have an $N$ body system with particles $x_i$ and velocities $v_i$, then we define the relational Weber Hamiltonian by the expression $$H=H(r_{ij}, \nu_{ij}) = \sum_{i,j} \frac{q_i q_j}{r_{ij}} (1-\frac{\nu_{ij}^2}{2c^2}) + \frac{\mu^*_{ij}}{2} v_{ij}^2.$$ The sum is over all pairwise interactions.

[Notation] Here $\mu_{ij}^*$ is the reduced kinetic energy mass of the restricted two-body system with masses $\mu_i, \mu_j$.

The point is that constructing and integrating the Weber-Hamilton equations of motion requires computing some pseudo-partial derivatives. Or we might call them "relational derivatives" which are essential in the calculus of relational force laws. For example, we naturally need to evaluate the expressions $dH / d r_{ij}$ and $d H / d \nu_{ij}.$

Ultimately using chain rule everything reduces to the evaluation of $$\partial r_{ij} / \partial r_{kl}, ~~~\partial \nu_{ij} / \partial \nu_{kl}.$$

But the relational functions $\{r_{ij}\}$ and $\{\nu_{ij}\}$ are not independant, so all these derivativse do not reduce to Kronecker deltas. They have different evaluations, and these evaluations are the computational basis for integration of the relational Hamilton-Weber equations.

The formalism of chain rule would suggest that the above pseudo partial derivatives can be expanded in Cartesian coordinates $x_i=(x_i^1, x_i^2, \dots)$ as follows:

$$\frac{\partial r_{ij}}{\partial r_{kl}} = \sum_{m, p} \frac{\partial r_{ij}}{\partial x_{m}^p} \frac{\partial x_m^p}{\partial r_{kl}} =\sum_{m,p} \frac{\partial r_{ij}}{\partial x_m^p} (\frac{\partial r_{kl}}{\partial x_m^p})^{-1}.$$

Caution needs be exercised in this last equality, because it's not actually clear when we have equality between $\frac{\partial x_m^p}{\partial r_{kl}}$ and $(\frac{\partial r_{kl}}{\partial x_m^p})^{-1}$. Technically most of these derivatives are equal to zero especially when the indices $m, k, l$ are all disjoint.

So what can we say about evaluating $\partial r_{ij}/ \partial x_{m}^p$?

We find $$\frac{\partial r_{ij}}{\partial x_{m}^p} = \frac{1}{r_{ij}} \frac{\partial (r^2_{ij}~/2)}{\partial x_{m}^p} = \frac{1}{r_{ij}} \frac{\partial}{\partial x_{m}^p} (x^p_i - x^p_j)^2/2= r_{ij}^{-1} \cdot x_{ij}^p \cdot (\delta_{mi}-\delta_{mj}).$$

Thus $ \partial r_{ij} / \partial r_{kl} = 0$ if $\{i,j\}$ and $\{k,l\}$ are disjoint.
[Incomplete] If $\{i,j\} = \{i, k\}$, then we have $$ \partial r_{ij} / \partial r_{ik} = \sum_p \frac{r_{ik}}{r_{ij}}\frac{x_{ij}^p}{x_{ik}^p}$$
[Error?] If $\{i,j\} = \{k,l\}$, then we have $ \partial r_{ij} / \partial r_{kl} = \partial r_{ij} / \partial r_{ij} = 2\times 3 = 6.$

What can we say about $\frac{\partial \nu_{ij}}{\partial x_{m}^p}$ and $ \frac{\partial \nu_{ij}}{\partial v_{m}^p}$ ?

This gets tedious.

[Insert evaluation. Incomplete].