# Gravitational well

1. Jul 30, 2008

### rolly_wood

Wikipedia says: "In a uniform gravitational field, the gravitational potential at a point is proportional to the height. Thus if the graph of a gravitational potential Φ(x,y) is constructed as a physical surface and placed in a uniform gravitational field so that the actual field points in the − Φ direction, then each point on the surface will have an actual gravitational potential proportional to the value of Φ at that point. As a result, an object constrained to move on the surface will have roughly the same equation of motion as an object moving in the potential field Φ itself."
What about the contraints? By looking at a marble moving in an hyperbolic funnel it seems that it stays too long near the well, longer than the 1/r2 force would allow. By projecting the weight of the marble on perpendicular and tangential direction (to the funnel) we obtain the force acting on it (taking in to account the constraint). Further projection on horizontal and vertical direction should give (the former) the central force which should be inverse square. Instead I obtained a relation like r^2/(r^4+1) which is similar to inverse square a great distance from the well but it is completely different near to it. It seems that this explain why the marble path is different than one would expect especially near the well.