# all the 1-dimensional problems in newtonian mechanics are solvable?

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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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