while taking the vector calculus course, I had trouble solving the problem which is as follows:
Suppose a particle with mass $m$ moves in the space under the force field $- \nabla V $ where $V: \mathbb{R}^3 \rightarrow \mathbb{R}$ is a function of class $C^2$. If the origin is the point where $V$ attains the strict local minimum, prove that the origin is the stable equilibrium point. (i.e. given $\rho , \epsilon >0$, $\exists \rho_0 , \epsilon_0 >0$ such that every solution satisfying $\parallel c(0) \parallel < \rho_0 ,m\parallel c'(0) \parallel^2 < \epsilon_0 $ satisfies the condition $\parallel c(t) \parallel < \rho ,m\parallel c'(t) \parallel^2 < \epsilon $ for all $t>0$ where $c(t)$ denotes the position)
I've found some interesting facts while solving these questions.
Since we can yield the equation $-\nabla V = m c''(t)$, we can find that $(\dfrac{1}{2}m\parallel c(t) \parallel^2+V(c(t)))' = m c(t) \cdot c'(t) + \nabla V \cdot c'(t) = 0$ so $\dfrac{1}{2}m\parallel c(t) \parallel^2+V(c(t))$ is constant.
Also, I found that 'Lyapunov's second method for stability' is related to the conditions given in the problem so I tried to apply it. However, it was hard to prove the initial setup for the Lyapunov's method, which is the equation $\dfrac{d}{dt} V <0$ (the other 2 conditions will be guaranteed when we restrict to our region to some neighborhood containing origin, since the origin is the strict local minimum)
I tried these kinds of methods, but I failed to unify all these facts to solve this question.
Please help us if you have any comments that could help to solve these questions.
Thanks in advance!