You are thinking of gravity as a force and nothing in L1 is at risk of 'falling into the sun'. Radial forces cause orbits to be eccentric. If you were on a sp

You are thinking of gravity as a force and nothing in L1 is at risk of 'falling into the sun'. Radial forces cause orbits to be eccentric. If you were on a space walk on the IIS and wanted to deorbit your hammer and tossed it down, it would be in an orbit with the same period as you and hit the space station at some point in the future. To deorbit it you would have to throw it backwards faster enough that its closest approach entered the atmosphere. L1 can be thought of as a saddle shape in gravitational potential with your horse oriented in the direction of travel. With the back of the saddle giving you just enough push or pull to have the same sidereal speed. Having a periodic oscillation to stay centered on that saddle to make sure you don't roll off the side is the hard part. That is still a flawed analogy, but the important part is, ignoring gravity assist. The most energy efficient way to 'fall into the sun' is actually to enter an orbit way out past Jupiter and then reduce sidereal speed to hit the sun. From earth you need accelerate a rocket to 30 km/s to hit the sun or about 100% of the earths velocity, while you an escape the solar system by just going 40% faster.