A physics engine is an approximate simulation of certain simple physical systems. These includes objects collisions and fluid dynamics.
GTA III — GTA Vice City Stories
From GTA III up to GTA Vice City Stories (excluding GTA Advance), their physics engines are most likely similar with very little improvements over the year. The engine is probably built in house since Renderware was mostly used for models and textures in these GTA games.
Object collisions were extremely crude. Dynamic objects and vehicles will obey the perceived laws of physics under normal conditions. They can bounce of other objects and slide against other objects. Dynamic objects will not experience gravity unless affected by the environment. If a dynamic object is placed incorrectly above ground, the object can appear to hover above the ground. The collision models were made not to match the model of the object exactly. This caused many anomalies, including the appearance of going through objects, invisible walls, and the appearance of standing on air. Dynamic objects also have problems when colliding at high speeds. This can cause objects to get stuck within other objects or can cause objects to go through other objects entirely.
The physics of the vehicles has improved with each iteration of the games. In GTA III, the engine fully supports the basic physics of four-wheeled land vehicles and a crude support of airborne vehicles like the Dodo and the police helicopter. GTA Vice City introduced support for two-wheeled motor vehicles and a more proper way to support airborne vehicles.
Grand Theft Auto IV uses the Rockstar Advanced Game Engine, which includes a proprietary physics engine, and runs along with the Euphoria animation engine. Improvements to the physics engine has been vast. The engine still suffers from some anomalies experienced in previous games but they have been dramatically reduced.
Most dynamic objects will experience gravity, even if not acted by the environment. Dynamic objects placed above other dynamic objects will travel in tandem. The water has been dramatically improved. When objects land in water, the water will ripple and create a wave in response.
The introduction of Euphoria lets characters be able to interact with the environment in a physically realistic way. Dead characters will conform with the environment instead of laying flat. Characters bailing out of vehicles or falling off a building will bounce off objects.