An alternative approach to the n-dimensional small oscillations problem is presented. This method is based on the finding of n new independent constants of motion to get the n eigenfrequencies and the n normal coordinates of the problem. These constants of motion exist and may be explicitly constructed for any small oscillations problem. Three examples are presented. One of them involves solving a five-dimensional small oscillations problem whose solution is usually obtained by finding the roots of a quintic algebraic equation. The approach constructed here is especially suited to deal with high-dimensional problems. Applications to small oscillations as well as to high-degree algebraic equation solutions are discussed.