Confusing variable change calculation

[AxiomOfChoice]

Suppose you start with a system of N particles identified by position vectors [itex]r_1, r_2, \ldots, r_N[/itex] and masses [itex]\mu_1, \mu_2, \ldots, \mu_N[/itex]. Then (quantum mechanically) the kinetic energy operator for this system is given by (assuming [itex]\hbar = 1[/itex])

[tex]
T = \sum_{i = 1}^N -\frac{\Delta_i}{2\mu_i},
[/tex]

where [itex]\Delta_i = \Delta_{r_i}[/itex] is the Laplacian in the variable [itex]r_i[/itex]. Now, let's make the change of variables [itex]\eta_i = r_i - r_N[/itex]. According to the book I have, the kinetic energy in the new variables looks like

[tex]
T' = \sum_{i = 1}^{N-1} -\frac{\Delta_i}{2\mu_i'} + \sum_{i < j} \nabla_i \cdot \nabla_j,
[/tex]

where [itex]\Delta_i = \Delta_{\eta_i}[/itex], [itex]\nabla_i = \nabla_{\eta_i}[/itex], and

[tex]
\frac{1}{\mu'_i} = \frac{1}{\mu_i} + \frac{1}{\mu_N}.
[/tex]

I simply do not see how this is true. It seems that, if one applied the chain rule, one obtains [itex]\Delta_{\eta_i} = \Delta_{r_i}[/itex]; is that not true?

 

[fresh_42]

No. It depends on the connection we use, in this case probably Levi-Civita, because the coordinate transformation changes the evaluation point of the gradient field.