Actually, something like that already exists. Take a closer look at most all of the Hybrid cars out there, such as the Toyota Prius. These vehicles are accelerated by two motors, one being the gas engine, while the second being electric. If the electric motor can handle the needs of the vehicle, it takes over and does it. When at rest at a stoplight, the gas engine will completely shut down if not needed (for accessories or what have you).
Here is a more detailed version of how it works from the How Stuff Works website:
The Prius mainly relies on two features to optimize efficiency and reduce emissions:
Its engine only runs at an efficient speed and load - In order to reduce emissions, the Prius can accelerate to a speed of about 15 mph (24 kph) before switching on the gasoline engine. The engine only starts once the vehicle has passed a certain speed. And once the engine starts, it operates in a narrow speed band.
It uses a unique power split device - Gasoline engines can be tuned to run most efficiently in certain speed and load ranges. The power split device on the Prius, which we'll talk about in a minute, allows the engine to stay in its most efficient load and speed range most of the time.
Toyota designed the 1.5-liter engine in the Prius to run at a maximum speed of only 5,000 rpm, where it makes 76 horsepower. Keeping the maximum speed of the engine low allows for the use of lighter components that improve efficiency.
The electric motor on the Prius is rated at 67 horsepower from 1,200 to 1,540 rpm. It produces 295 pound-feet of torque from 0 to 1,200 rpm, which is more than enough to get the car going without the aid of the gasoline engine.
The GM Volt uses a gasoline engine to recharge its batteries, but runs almost entirely off of electric motors. Here is an excerpt from the GM website:
Q: How is the Chevy Volt different than other cars on the road?
A: The car is a plug-in range-extended electric vehicle with an on-board gasoline generator. It has a large battery that stores power from your home electric outlet and which is connected to an electric motor. The electric motor directly propels the car. The battery can power the car for the first 25 to 50 miles. After that, should one continue to need to drive, the on-board gasoline generator provide electricity for the motor and participate in driving the car.
Q: How is the Chevy Volt different than conventional hybrids, like the Prius?
A: Today’s hybrids are called parallel hybrids. They use a small electric motor for low speed driving, but switch to a regular gas engine for acceleration and faster speed driving with the electric motor providing enhancement, hence both engines work side by side or in parallel.
The Volt is a series vehicle meaning only the electric motor powers the car at all times, the gas engine is just a generator for making electricity once the battery is depleted. A little like the Prius, the engine does help spin the wheels after the battery is depleted. GM engineers chose to do this because it improved efficiency by 10 to 15 percent.
There are some interesting articles out there on the subject of anti-idling, where it is suggested that vehicles should not idle more than three (3) minutes at any given time. If I remember correctly, I believe you cannot idle your engine more than one (1) minute after start-up in England (UK) when you get ready to go in the morning, so no "warming up" your vehicle before you are on your way. I read an article from a very green website which stated idling your vehicle more than 10 seconds will put out more emissions than if you stopped and restarted your engine. I don't know the validity of the statement, but here is the source if you want to read what they have to say.
There is also plenty of thought out there which suggests, if you know you're going to be sitting at one place for an extended period of time, you should just shut your vehicle off, or at least stop the engine. This can save you gas and emissions if the period of time is extended you may be sitting there. I'm talking about such as in traffic jams or at a very long traffic light. There is a lot of conjecture on the subject, so do not want to add to it here.
The bottom line is, these engines already exist. I don't think I'd call them a "variable engine", but the term is fine. I believe as manufacturers move towards greater efficiencies, they will be doing more like what you are asking about and have exactly that, engines which turn off at stop lights, while stopped for more than a few minutes, etc.