On the Lagrange and Hill stability of motion in the three-body problem

In the three-body problem, we consider the Lagrange and Hill stability including the Lagrange stability for the manifold of symmetric motions that exists in the case where two of three bodies have equal masses. To analyze the stability, in addition to integrals of energy and angular momentum we use the Lagrange–Jacobi equality. We prove theorems on the Lagrange and Hill stability. The theorem on the Hill stability has effective application in the case where the mass of a body is much less than masses of two other bodies. In this case, as it is known, the model of the restricted three-body problem is usually applied.