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i mean given a system with a conserved Energy in one dimension

$$ E= \frac{p^{2}}{2m}+V(x) $$

then the 'solution' to this problem is implicitly given by

$$ t(x)= \frac{1}{2m} \int_{0}^{x}\frac{du}{\sqrt{E-V(u)}} $$

so apparently from this equation we could know all the quantities $ p(x) $ and $ x(t) $ so for a one dimension all the mechanical problems are solvable isn't it ??

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1 Answer 1

up vote 2 down vote accepted

Yes, and generally it is solvable numerically just like a differential equation of motion is generally solvable numerically.

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a question .. what would happen if when we invert $ t(x) $ to obtain $ x(t) $ we get that $ x(t) $ is a multivalued function with many branches ? :D –  Jose Javier Garcia Jan 21 '13 at 22:35
@JoseJavierGarcia: I did not think of it, but I guess we can chose one of branches that "starts" from the right initial conditions by continuity. A differential (Newton) equation has also two-parametric family of solutions, and only initial conditions fix the parameters. –  Vladimir Kalitvianski Jan 22 '13 at 8:24

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