= Solution
For <conservative forces> and ideal constraints that do no <work>, <mechanical energy> $E=T+U$ is constant. Work-energy balance remains useful with nonconservative <forces>: their power gives $dE/dt$. In one dimension,
$$
\tfrac12m\dot q^2+U(q)=E
$$
locates accessible regions and turning points, and reduces the trajectory to a quadrature $dt=dq/\sqrt{2(E-U)/m}$. A local minimum of potential is a stable equilibrium; a local maximum is unstable. Near a nondegenerate minimum, $U=U_0+K(q-q_0)^2/2+\cdots$ gives $\omega^2=K/m$.
For a <central force>, conserved <angular momentum> reduces radial motion to an <effective potential> $U_{\rm eff}=U(r)+L^2/(2mr^2)$. Its extrema determine circular orbits and their radial stability. In several coordinates, <small oscillations> follow from the kinetic-energy <mass matrix> $M$ and potential Hessian $K$ through $\det(K-\omega^2M)=0$.
<Energy> is also central to the <Lagrangian> $\mathcal L=T-U$ and the <Euler-Lagrange equations>. A time-independent natural mechanical system has conserved <energy>; <cyclic coordinates> yield additional momenta. <Energy> alone does not normally determine every coordinate of a multi-degree-of-freedom motion. \b[Its value is to expose allowed motion, stability and reductions before solving the full differential equations.]
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