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- Jan 26, 2012

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**Problem**: Consider a constant mass $m$ moving under the influence of a conservative force field $-\mathrm{grad}\,\Phi(x)$ defined by a potential function $\Phi:W_0\rightarrow\Bbb{R}$ on an open set $W_0\subset\Bbb{R}^3$. The corresponding dynamical system on the state space $W=W_0\times\Bbb{R}^3 \subset \Bbb{R}^3\times\Bbb{R}^3$ for $(x,v)\in W_0\times\Bbb{R}^3$ is given by

\[\left\{\begin{aligned}\frac{dx}{dt} &= v\\ \frac{dv}{dt} &= -\mathrm{grad}\,\Phi(x) \end{aligned}\right.\]

Let $(\overline{x},\overline{v})\in W_0\times\Bbb{R}^3$ be an equilibrium point of the above dynamical system. Determine when the dynamical system is stable at $(\overline{x},\overline{v})$.

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