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 P: 2 First time poster so I apologize any problems with location or content. By the way, you guys are awesome. Just some context: last semester I had a theoretical mechanics class and the teacher said, while teaching us about the Lagrangian formulation, that particles cannot, ever, reach a potential's extremum and stay there (zero velocity). I was very curious at the time but never actually thought about tackling the problem. So today it popped in my head while cooking lunch. I went at the problem in a uni-dimensional way, starting by defining a general potential with an extremum, approximating it, at that point, by a Taylor series to terms of second order and seeing what it meant in terms of equations of motion via the Lagrangian formulation: ${\frac{dV}{dx}|_a=0}$ ${V(x)\approx{V(a)+\frac{dV}{dx}|_a (x-a)+\frac{1}{2}\frac{d^2 V}{dx^2}|_ a (x-a)^2}}$ ${L=\frac{m\dot{x}^2}{2}-\frac{1}{2}\frac{d^2 V}{dx^2}|_ a (x-a)^2}$ (the first and second terms on the right hand side of the potential are, respectively, null and a constant that does not influence the equation of motion) This leads to, using the Euler-Lagrange equations, to the following equation of motion: ${m\ddot{x}+\xi (x-a)=0}$ This leads to: ${x=Ae^{-i\alpha t}+Be^{i\alpha t}+a}$ Applying the conditions of zero velocity at point "a" at a certain time one gets: ${x=a}$ This shows that the only possible way for a particle to exist in a potential extremum with zero velocity is if it is "put" there with zero velocity. I now want to tackle the problem of how much time it takes for the particle to reach such an extrema. Any ideas? Thanks in advance!