Let \(V:\mathbb R^n\to\mathbb R\) be \(C^1\) and let \(x''_i=-\partial V/\partial x_i\) on a finite interval \((a,b)\).
Show \(H(t)=\frac12\langle x'(t),x'(t)\rangle+V(x(t))\) is constant.
If \(V\ge M> -\infty\), show \(x,x',x''\) are bounded on \((a,b)\) and prove the limits of all three as \(t\to b\) exist.
Solution
Write \[ x(t)=(x_1(t),\ldots,x_n(t)). \] The system can be written vectorially as \[ x''(t)=-\nabla V(x(t)). \]
1. The energy \(H\) is constant.
Differentiate \[ H(t)=\frac12\|x'(t)\|^2+V(x(t)). \] Using the chain rule and the differential equation, \[ \begin{aligned} H'(t) &=\langle x''(t),x'(t)\rangle +\langle\nabla V(x(t)),x'(t)\rangle\\ &=\langle-\nabla V(x(t)),x'(t)\rangle +\langle\nabla V(x(t)),x'(t)\rangle\\ &=0. \end{aligned} \] Thus \[ \boxed{H(t)=H_0} \] is constant on \((a,b)\).
2. A lower bound on \(V\) gives a uniform bound on \(x'\).
Assume \[ V(x)\ge M \] for all \(x\in\mathbb R^n\). From energy conservation, \[ \frac12\|x'(t)\|^2 =H_0-V(x(t)) \le H_0-M. \] Hence \[ \|x'(t)\|\le C:=\sqrt{2(H_0-M)} \] for all \(t\in(a,b)\).
3. The trajectory \(x(t)\) is bounded and has a limit as \(t\to b\).
Fix \(t_0\in(a,b)\). For \(t\in[t_0,b)\), \[ \begin{aligned} \|x(t)-x(t_0)\| &\le\int_{t_0}^t\|x'(s)\|\,ds\\ &\le C(b-a). \end{aligned} \] The same estimate, with the endpoints of the integral reversed when \(t<t_0\), gives \[ \|x(t)-x(t_0)\|\le C|t-t_0|\le C(b-a) \] for every \(t\in(a,b)\). Thus \(x\) is bounded on the whole interval.
Moreover, if \(s,t\to b\), then \[ \|x(s)-x(t)\| \le C|s-t|\longrightarrow0. \] Thus \(x(t)\) is Cauchy as \(t\to b\). Completeness of \(\mathbb R^n\) gives some \[ c\in\mathbb R^n \] such that \[ \boxed{x(t)\to c.} \]
4. The acceleration is bounded.
The set \[ K=\overline{\{x(t):t\in(a,b)\}} \] is closed and bounded in \(\mathbb R^n\), hence compact. Since \(V\) is \(C^1\), \(\nabla V\) is continuous and therefore bounded on \(K\). Thus there is a constant \(D\) such that \[ \|\nabla V(y)\|\le D \qquad(y\in K). \] Using the equation of motion, \[ \boxed{\|x''(t)\|\le D.} \]
5. The velocity has a limit as \(t\to b\).
For \(s,t\in(a,b)\), \[ \|x'(s)-x'(t)\| \le\int_{\min(s,t)}^{\max(s,t)}\|x''(r)\|\,dr \le D|s-t|. \] Hence \(x'(t)\) is Cauchy as \(t\to b\). Therefore there is some \[ v\in\mathbb R^n \] with \[ \boxed{x'(t)\to v.} \] The uniform bound from step 2 already shows that \(x'\) is bounded.
6. The acceleration also has a limit.
By step 3 and continuity of \(\nabla V\), \[ \nabla V(x(t))\longrightarrow\nabla V(c). \] Since \[ x''(t)=-\nabla V(x(t)), \] we obtain \[ \boxed{x''(t)\longrightarrow-\nabla V(c).} \] Thus \(x,x',x''\) are all bounded on \((a,b)\) and all three have limits as \(t\to b\).