The computation of the forward dynamics plays an important role in simulating the motion of interconnected rigid bodies while considering the physical properties and constraints of each part. The applications in graphics, animation, and robotics usually require fast computation, which leads to the usage of fast recursive algorithms. In this paper, we present a formulation of the recursive forward dynamics of serial kinematic chains that that is rooted in geometry, which allows coordinate-free view and geometrically meaningful interpretations of the involved quantities. The mathematical framework is called conformal geometric algebra (CGA) and it extends classical vector algebra by introducing a unified representation of a large array of geometric operations, transformations and mathematical objects, such as points, lines, and planes, in a rigorous yet intuitive manner. Hence, using CGA for the computation of the forward dynamics provides a unified mathematical framework that seamlessly integrates both the geometric and dynamic aspects of the system. We validate the computation numerically and provide an implementation of the results in an open-source library, making it immediately available in practice.