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We describe an accurate, one-dimensional, spherically symmetric, Lagrangianhydrodynamics/gravity code, designed to study the effects of radiative coolingand photo-ionization on the formation of protogalaxies. The code can treat anarbitrary number of fluid shells (representing baryons) and collisionlessshells (representing cold dark matter). As a test of the code, we reproduceanalytic solutions for the pulsation behavior of a polytrope and for theself-similar collapse of a spherically symmetric, cosmological perturbation. Inthis paper, we concentrate on the effects of radiative cooling, examining theability of collapsing perturbations to cool within the age of the universe. Incontrast to some studies based on order-of- magnitude estimates, we find thatcooling arguments alone cannot explain the sharp upper cutoff observed in thegalaxy luminosity function.Comment: 33 pages, uuencoded compressed postscript with figures, Ap.J. (in  press), corrections to axes in Fig

Hydrodynamic simulations of galaxy formation. 1: Dissipation and the maximum mass of galaxies