Mathematical relationship between N1 and N2
 

Assuming a "standard" LP/HP jet engine (say that of a B737 NG), what is the relationship between N1 and N2? As there is no direct linkage between the two shafts the ratio between N1/N2 won't be constant. Is there a way to calculate N1 from N2 (if the parameters such as air density, temperature etc... are known)?

Thanks!

You answered your own question. There is indeed no fixed relationship.
Yes, if you have access to either the engine model (or "deck" as its often called, after the "decks" of cards used to calculate it in days of yore) or engine "running line" data (basically, the steady state outputs from the deck). That would be for the specific engine type/variant (and, to be very precise, data for the actual engine in question)

That would be valid for a stable engine condition. Under acceleration the relationship would be subtly different, but could be modelled if one knew the time history of the other parameters, rather than a single set of conditions.

.. engine decks ...

.. think we both must be getting to the venerable stage now ?

They are still called decks - I issued one only the other day!

Engine Decks
 

And given the data from the deck, it is possible to non-dimensionalise (is that a word?) and cross-plot the parameters to give you a steady state engine model that will work across the flight envelope. Indeed I spent a happy couple of weeks doing just that, much number crunching through Excel, of all things!

Tuning the accels and decels is a whole different ball game though.