Dynamical simulation, in computational physics, is the simulation of systems of objects that are free to move, usually in three dimensions according to Newton's laws of classical dynamics, or approximations thereof. Dynamical simulation is used in computer animation to assist animators to produce realistic motion, in industrial design (for example to simulate crashes as an early step in crash testing), and in video games. Body movement is calculated using time integration methods.

In computer science, a program called a physics engine is used to model the behaviors of objects in space. These engines allow simulation of the way bodies of many types are affected by a variety of physical stimuli. They are also used to create dynamical simulations without having to know anything about physics. Physics engines are used throughout the video game and movie industry, but not all physics engines are alike. They are generally broken into real-time and the high precision, but these are not the only options. Most real-time physics engines are inaccurate and yield only the barest approximation of the real world, whereas most high-precision engines are far too slow for use in everyday applications.

To understand how these Physics engines are built, a basic understanding of physics is required. Physics engines are based on the actual behaviors of the world as described by classical mechanics. Engines do not typically account for non-classical mechanics (see theory of relativity and quantum mechanics) because most visualization deals with large bodies moving relatively slowly. The models used in dynamical simulations determine how accurate these simulations are.

The first model which may be used in physics engines governs the motion of infinitesimal objects with finite mass called “particles.” This equation, called Newton’s Second law (see Newton's laws) or the definition of force, is the fundamental behavior governing all motion:

This equation allows us to fully model the behavior of particles, but it is not sufficient for most simulations because it does not account for the rotational motion of rigid bodies. This is the simplest model that can be used in a physics engine and was extensively used in early video games.

Bodies in the real world deform as forces are applied to them, so we call them “soft,” but often the deformation is negligibly small compared to the motion, and it is very complicated to model, so most physics engines ignore deformation. A body that is assumed to be non-deformable is called a rigid body. Rigid body dynamics deals with the motion of objects that cannot change shape, size, or mass but can change orientation and position.

To account for rotational energy and momentum, we must describe how force is applied to the object using a moment, and account for the mass distribution of the object using an inertia tensor. We describe these complex interactions with an equation somewhat similar to the definition of force above:

where ${\displaystyle \mathbf {I} }$ is the central inertia tensor, ${\displaystyle {\vec {\omega }}}$ is the angular velocity vector, and ${\displaystyle \tau _{j}}$ is the moment of the jth external force about the mass center.