Tune optimization for maximum dynamic acceptance; 1, formulation

In order to combine the acceptance limitation due to a mechanical obstacle at radius rmech with that due to magnetic imperfections present in the lattice, a quantity eda to be called ``dynamic accepta nce'' is introduced. Using lowest order theory (with transfer matrices and no Hamiltionian) perturbed linear betatron motion is calculated and used to derive the dependence eda(rmech). Being in analyt ic form, this acceptance reduction provides a figure of merit that can be used to optimize the lattice tunes (thereby refining the prescription ``stay away from low order resonances''). Apart from its definition as an acceptance rather than an aperture, what distinguishes eda(rmech )from the commonly employed ``dynamic aperture'' is its dependence on rmech and the importance of this distinction fad es as rmech becomes large. In this Part~I the method is formulated and, to demonstrate the method, optimal fractional tunes are found with only random errors present-the loss of acceptance is dominate d by sextupole errors. But the intended application is for field errors that are systematic over sections of the lattice, but not necessarily over the whole lattice. Such field errors are unavoidable and are especially important in a high tune accelerator like the LHC.