Could the suns energy be used to move our solar system near a younger solar system over about 4 billion years?

To move the entire solar system one would need to move the Sun and every planet and even every last one of the countless objects orbiting the Sun otherwise so many objects will get their orbits messed up that all life in the solar system would be at risk of getting annihilated by collisions.

There is simply no reason to even attempt moving the entire solar system, or even a single planet for that matter. It takes a lot of energy to move heavy objects. Planets may have a lot of resources to take advantage of and live off of for a long trip but most of it will be dead weight. A better option would be to simply move a large resource rich asteroid with a colony built within it. It would be much easier to accelerate, it could contain plenty of metal, water, and fuel (in the form of helium 3 and deuterium for fusion most likely) and on top of that its thick exterior would protect the colonists within from radiation and interstellar matter the "ship" would collide with during the journey.

The Europeans didn't sail the entire continent to the Americas, they just sent ships. When you go to the grocery store you don't move your entire house. There is absolutely no reason to move the entire solar system.

If our civilization was a Type 3 or 4 Civilization, we would be able to move our solar system in a matter of days without the lift of a finger. A Type 3 or 4 Civilization would actually have already done this thousands of times. A Type 3 or 4 Civ can harness the power of the Sun and dry it up in weeks. They also could harness the power of rougly 98 % of the stars in our galaxy. So basically we would have no chance unless; again we were a Type 3 or 4. We expect to reach this advancement in less than 1 million years, more than enough time to get ready and move , so to answer your question, yes!

F=ma

K=(1/2)mv^2

E=mc^2

t=4 billion years.

1.Add up the mass of the whole system.

2.Now figure out how far you need to go, and how fast you need to go.

3.Now calculate how much energy is needed.

4.Now completely convert the sun's mass to energy.

Case in point, the answer from 3 will be greater than the answer from 4.

It can't be done. Unless you find a way to passively harness fundamental string energy. Which we don't know exists yet.

ok, think about this for a second. If we find a young solar system and take 4 billion years to get there... it won't be young anymore.

If you were able to control the gravitational pull of the center of the Milky Way, then you can direct the Sun.

4 billion years is a short time in cosmology but a long long time in Mankind...

Provied that our petty differences doe not wipe ourselves out,will Mankind still be around then?

Well, we have 3 billion years to answer this question, assuming you and we will still be around.

if we are even around at that point, we should have long before mastered the technology that would make this a moot point.