- #1
V0ODO0CH1LD
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Newton's laws says ## F=ma ##. Which, as far as I can see, states that all physical interactions concern the second time derivative of position. And because there is no other way for two bodies to interact in the physical world, the "worst" I can do to a system is change its acceleration, right? My question is: are higher (than three) order derivatives of position with respect to time valid physical concepts? I can clearly see why the third time derivative of displacement, jerk, would be a useful concept. But what I can't see is how defining anything beyond that makes any sense. How can I change the rate at which a body changes its acceleration? I mean, if the only thing I can do is apply more or less force to a body, therefore changing its acceleration. What would I have to do to change anything beyond that?
The other thing is: if I have some equation that relates time and position as ## x=t^3 ##. I have no good reason to stop taking derivatives at the second time around. So the actual "equation of motion" for a particle that behaves in a manner described by that equation would be:
[tex] x(t)=x_0+\dot{x_0}t+\frac{\ddot{x_0}}{2}t^2+\frac{\dddot{x_0}}{6}t^3[/tex]
by the taylor series approximation.
So is it the point that I could never, in reality, do something to a body so that it behaved in such a way that it's trajectory would obey higher (than three) order polynomial approximations, or is it that we just neglect the higher order derivatives by saying; sure.. the equation says ## x=t^n ## but the acceleration is ##(n-1)nt^{n-2}## and that's all we need!
What part of mathematics are Newton's laws constraining?
The other thing is: if I have some equation that relates time and position as ## x=t^3 ##. I have no good reason to stop taking derivatives at the second time around. So the actual "equation of motion" for a particle that behaves in a manner described by that equation would be:
[tex] x(t)=x_0+\dot{x_0}t+\frac{\ddot{x_0}}{2}t^2+\frac{\dddot{x_0}}{6}t^3[/tex]
by the taylor series approximation.
So is it the point that I could never, in reality, do something to a body so that it behaved in such a way that it's trajectory would obey higher (than three) order polynomial approximations, or is it that we just neglect the higher order derivatives by saying; sure.. the equation says ## x=t^n ## but the acceleration is ##(n-1)nt^{n-2}## and that's all we need!
What part of mathematics are Newton's laws constraining?