We study the equilibrium behavior and dynamics of a polymer collapse transition for a system composed of a short Lennard-Jones (LJ) chain immersed in a LJ solvent for solvent densities in the range of rho=0.6-0.9 (in LJ reduced units). The monomer hydrophobicity is quantified by a parameter lambdais an element of[0,1] which gives a measure of the strength of attraction between the monomers and solvent particles, and which is given by lambda=0 for a purely repulsive interaction and lambda=1 for a standard LJ interaction. A transition from the Flory coil to a molten globule is induced by increasing lambda. Generally, the polymer size decreases with increasing solvent density for all lambda. Polymer collapse is induced by changing the hydrophobicity parameter from lambda=0 to lambdagreater than or equal to0.5, where the polymer is in a molten globule state. The collapse rate increases monotonically with increasing hydrophobicity and decreases monotonically with increasing solvent density. Doubling the length of the chain from N=20 to N=40 monomers increases the collapse time roughly by a factor of 2, more or less independent of the hydrophobicity and solvent density. We also study the effect of conformational restrictions on polymer collapse using a chain model in which the bond angles are held near 109.5degrees using a stiff angular harmonic potential, but where free internal rotation is allowed, and find that the collapse times increase considerably with respect to the fully flexible polymer, roughly by a factor of 1.6-3.5. This increase is most pronounced for high solvent densities.