The first of the three conservation laws of the Two-Body Problem.
Conservation of Energy
- Gravity is a conservative force: per unit mass it is the gradient of a potential,
- Dot the equation of motion with the velocity :
Corrected from the notes
- The handwriting writes (the is missing) and
- Dotting with gives a sum equal to zero, as written above; the next line of the notes () uses the sum
- Both terms are total time derivatives:
- The quantity in parentheses is the (specific) energy ; the gravitational potential energy is
Specific Mechanical Energy
- Kinetic plus potential energy per unit mass is constant along the orbit
Energy and Semimajor Axis
- From the relation of physical and geometric quantities:
- and inversely
- So the energy fixes the size of the orbit; its sign classifies the conic (see Energy of the Conic Section)
- Energy is not an independent constant once and are known (see Conserved Quantities)