It is shown straightforwardly in this note that the orbits of a mass m around a mass M in the frame (1, 2, 3) of the Euler angles can be found by solving simultaneously six Euler Lagrange equations in six Lagrange variables. The problem is far too complicated to be solved by hand but can be solved numerically using Maxima to give twenty one new types of orbit described on page 4. There are nutations and precessions in the Euler angles phi, theta and chi. The usual assumption made in orbital theory is that the orbit is planar. Note carefully that the inverse square law is still being used for the force of attraction between m and M, but the Eulerian orbits contain far more information than the theory of planar orbits. The inverse square law does not give precession of a planar orbit as is well known. However the same inverse square law gives rise to many types of nutation and precession in the twenty one types of orbit described by the Euler angles. This is a completely new discovery, and it should be noted carefully that it is based on classical rotational dynamics found in any good textbook. It could have been defined in the eighteenth century, but orbital theory became ossified in planar orbits. This theory is part of ECE2 generally covariant unified field theory because all of rotational dynamics are described by a spin connection of Cartan geometry. Eq. (10) of this note is a special case of the definition of the Cartan covariant derivative. This is another major discovery. An art gallery full of graphics can be prepared based on these results, of greatest interest are plots of the twenty one types of orbit, looking for precessions of a point in the orbit such as the perihelion.

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