• theEric
This is for mechanics II. We're working with a particle of mass $$m$$ in one-dimensional motion. The potential energy is given by $$V(x)=-\frac12kx^2$$ and the force is anti-restoring at $$F(x)=kx$$ It is also given that we see unstable equilibrium at $$x=0$$. We want to consider the initial conditions $$\quad$$$$t=0\\x=x_0\\\dot x=0$$ And we want to show that the motion is an "exponential 'runaway'" $$x(t)=\frac12x_0(e^{\alpha t}+e^{-\alpha t})$$ where $$\alpha=\sqrt{\frac km~~}$$
Physics

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