Derive the Jarzynski equality for the case in which the system of interest is described by classical Hamiltonian \(H_{\lambda}(\boldsymbol{q}, \boldsymbol{p})\) that is coupled to a thermal reservoir with Hamiltonian \(\mathcal{H}\),

with the total Hamiltonian \(\mathscr{H}\) given by \[\mathcal{H}=\mathcal{H}\left(\boldsymbol{q}^{\prime}, \boldsymbol{p}^{\prime}\right)+H_{\lambda}(\boldsymbol{q}, \boldsymbol{p})+H_{\mathrm{int}}\left(\boldsymbol{q}, \boldsymbol{p}, \boldsymbol{q}^{\prime}, \boldsymbol{p}^{\prime}\right).\]
The proof is easiest if you assume that \(H_{\lambda}\) is a small subsystem compared with \(\mathcal{H}\), and that the interaction term \(H_{\text {int }}\) is sufficiently weak, but the Jarzynski equality can be proven without making any assumptions about the relative size of \(\mathcal{H}, H_{\lambda}\), and \(H_{\text {int }}\).