N-body problem of first-order differential equations, with initial conditions as initial position coordinates and initial momentum values. Symmetries in the -body problem yield global integrals of motion that simplify the problem. Translational symmetry of the problem results in the center of mass moving with constant velocity, so that , where is the linear velocity and is the initial position. The constants of motion and represent six integrals of the motion. Rotational symmetry results in the total angular momentum being constant where × is the cross product. The three components of the total angular momentum yield three more constants of the motion. The
Escape velocity kg or 1,000 kg; what differs is the amount of energy required. For an object of mass formula_9 the energy required to escape the Earth's gravitational field is "GMm / r", a function of the object's mass (where "r" is the radius of the Earth, "G" is the gravitational constant, and "M" is the mass of the Earth, ). A related quantity is the specific orbital energy which is essentially the sum of the kinetic and potential energy divided by the mass. An object has reached escape velocity when the specific orbital energy is greater or equal to zero. An