In moving object databases, object locations in some multi-dimensional space depend on time. Previous work focuses mainly on moving object modeling (e.g., using ADTs, temporal logics) and ad hoc query optimization. In this paper we investigate logical properties of moving objects in connection with queries over such objects using tools from differential geometry. In an abstract model, object locations can be described as vectors of continuous functions of time. Using this conceptual model, we examine the logical relationships between moving objects, and between moving objects and (stationary) spatial objects in the database. We characterize these relationships in terms of position, velocity, and acceleration. We show that these fundamental relationships can be used to describe natural queries involving time instants and intervals. Based on this foundation, we develop a concrete data model for moving objects which is an extension of linear constraint databases. We also present a preliminary version of a logical query language for moving object databases.