If $\nabla\times\mathbf F=0$ on a simply connected region containing the bounded trajectory, then the <Poincare lemma> gives a scalar potential $U$ with $\mathbf F=-\nabla U$. The work is
$$
W=\int\mathbf F\mathbin\cdot d\mathbf x
=U(\mathbf x_1)-U(\mathbf x_2),
$$
which remains bounded when $\mathbf x$ and $\mathbf F$ remain bounded. Since $H$ is also bounded, $\Delta Q=W-\Delta H$ cannot grow linearly with the observation time.

Thus sustained linear heat dissipation in this setting requires a <nonconservative force>, and, under the stated simply connectedness assumption,
$$
\boxed{\nabla\times\mathbf F\ne0.}
$$
Such a force can perform nonzero work on repeated bounded cycles. On a multiply connected domain, a curl-free force can have nonzero circulation, so the topological assumption is essential.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 3 / f / Solution

A nonconservative force has path-dependent work. On a simply connected domain, a continuously differentiable force is nonconservative exactly when its curl does not vanish identically.


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