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:
[itex]{\frac{dV}{dx}|_a=0}[/itex]
[itex]{V(x)\approx{V(a)+\frac{dV}{dx}|_a (x-a)+\frac{1}{2}\frac{d^2 V}{dx^2}|_ a (x-a)^2}}[/itex]
[itex]{L=\frac{m\dot{x}^2}{2}-\frac{1}{2}\frac{d^2 V}{dx^2}|_ a (x-a)^2}[/itex]
(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:
[itex]{m\ddot{x}+\xi (x-a)=0}[/itex]
This leads to:
[itex]{x=Ae^{-i\alpha t}+Be^{i\alpha t}+a}[/itex]
Applying the conditions of zero velocity at point "a" at a certain time one gets:
[itex]{x=a}[/itex]
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!
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:
[itex]{\frac{dV}{dx}|_a=0}[/itex]
[itex]{V(x)\approx{V(a)+\frac{dV}{dx}|_a (x-a)+\frac{1}{2}\frac{d^2 V}{dx^2}|_ a (x-a)^2}}[/itex]
[itex]{L=\frac{m\dot{x}^2}{2}-\frac{1}{2}\frac{d^2 V}{dx^2}|_ a (x-a)^2}[/itex]
(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:
[itex]{m\ddot{x}+\xi (x-a)=0}[/itex]
This leads to:
[itex]{x=Ae^{-i\alpha t}+Be^{i\alpha t}+a}[/itex]
Applying the conditions of zero velocity at point "a" at a certain time one gets:
[itex]{x=a}[/itex]
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!