Game Engines

... are frameworks offering services such as:

Real-time rendering, which includes scene management, typically through scene graphs
Animation, in particular through predefined bone motion (skinning) and simulated physics
Collision detection (also used for physics)
Interaction
Real-time audio
Scripting
Game-AI such as pathfinding and state machines
Networking
Resource Management
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 look at some promising ones for .NET: 3dState 3D Engine, Brume, PurpleSharp plus the helper libraries Artifical Engines.

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. You can even get an inexpensive physics processor in hardware AGEIA PhysX
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 pins and ragdolls [bodies without stiffness]; medium cost of computation)
Soft body (for material such as rubber, cloth, skin; very expensive to compute; currently one rarely 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.