Can someone explain in an intuitive way how newtons laws tell you that a particles trajectory is determined for all future time? I.e. that if you specify a position at t=0, you know where that particles going forever?
Second law: F = ma.
The first law is a special case where F = 0, meaning that a = 0. The third law doesn't matter if you are thinking about a single particle.
The second law says that if you know the force on a particle at all times, then you know the acceleration at all times. If you know the acceleration, then you know how much the velocity is changing. If you know the velocity, then you know how much the position is changing.
It's just a cascade of information.
That's it, really. So you need to know the original position and velocity, and the force at all times.
Formulas on /sci/ parse weird, even when they work in the previewer.
Generally, the problem can be solved by just throwing whitespace between all the control characters, parentheses, braces, operators, etc.