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This work studies the relationships between the generalized Hamilton and fractional gradient systems. We provide a new condition for generalized Hamilton systems to be any order (integer or non-integer) fractional gradient systems, which differs from previous studies that only gave a second-order fractional gradient presentation for generalized Hamilton systems. Based on its definition, we also discuss the condition for fractional gradient systems to be generalized Hamilton systems, which deduces a family of fractional dynamical models for generalized Hamilton systems. The reported results are a special case of our results. Our results are showcased in two examples to illustrate its applicability.