The minimum theoretical refrigerator power for cooling mechanical supports for a cryogenic apparatus is determined with the calculus of variations. A mathematical approach is developed that can be used to evaluate the thermal performance of any mechanical support which has fixed geometry. Four practical, non-optimum cooling arrangements are compared with the minimum power case. The comparison shows that a smaller refrigerator power is required when continuous cooling is provided at intermediate temperatures along the conducting support. A combination of two gas streams in series closely approximates the thermodynamically optimum cooling arrangement. © 1974.