Game Engines

... are frameworks offering services such as:

Real-time rendering and animation, which includes scene management, typically through scene graphs
Interaction
Real-time audio
Game-AI such as pathfinding and state machines
Networking
Resource Management
Physical simulations
Encapsulation of the APIs for graphics, networking, audio, etc., so that the code gets platform-neutral

There is a zillion of half-baked game engines. Please don't add your own to that list. We take a closer look at one quite promising one for .NET: PurpleSharp.

Game (and Animation) Physics

Why simulate physics for film production and for games?

Less work, especially when many objects are to be animated
Plausibly-looking result
Real-time simulation allows interactive animation

Simulation of physics is one important of the many jobs that may be taken over by a game engine or by a specialized engine such as Newton. Note that there are .NET bindings for Newton.
Physical effects appear in different forms:

Particles (point-like objects subject to force fields; typically no collisions; cheap to compute)
Rigid body (for applications such as bowling and ragdolls = bodies without stiffness; medium cost of computation)
Soft body (for material such as rubber, cloth, skin; very expensive to compute; typically not real-time; fraught with numerical instabilities)
Liquids (see Maya Unlimited)

Examples in Cinema 4D and Newton
The three main ingredients of rigid-body physics:

Physics of motion of extended bodies subjected to force fields, possibly with friction or buoyancy
Collision detection
Collision handling

Collision detection is needed heavily in both rigid-body and soft-body physics.

n objects can collide in n(n-1)/2 pairs: The cost of a naive collision test grows quadratically with the number of objects. Note that soft-body physics even have to handle complex self-collisions.
Reduce the work needed:

Use spatial coherency: Start with rough tests in a "broad phase", then apply precise tests in a "narrow phase" to the pairs of objects that are still unresolved. Several levels of hierarchy may be used. The broad phase is typically on approximations such as bounding boxes (axis-aligned = AABB or oriented = OBB), bounding spheres, BSP trees, and quadtrees/octress.
Use temporal coherency: Objects that are far apart in one frame remain so in the next frames, depending on their velocity.