A hydraulic system for operating a hydraulic fall-back support of a work machine including at least one hydraulic fall-back support cylinder for tracking and limiting movements of a boom of the work machine and a hydraulic pump by means of which the fall-back support cylinder and at least one further hydraulic consumer can be supplied with hydraulic fluid. A stop valve having a blocking position and a passage position is connected between the hydraulic pump and the fall-back support cylinder, by means of which the stop valve and a load-bearing cylinder space of the fall-back support cylinder is blockable and a hydraulic store connected to the load-bearing cylinder space is provided between the stop valve and the fall-back support cylinder.