Third order
Planck (1858-1947) stated that a photon has an energy equal to a constant times the frequency. This is equation 1 (eq.1). De Broglie (1892-1987) stated that all mass moves with a certain wave length. This wave length is dependent on the impulse p. This is equation 2. Assume that there are n wave in one orbit. This is equation 3. Rearrange equation 3 to equation 4. Impulse p is electron mass times its velocity. (eq.5) Using this leads to equation 7.
Now suppose that the Coulomb(1736-1806) attraction (eq.8) is equal to the Coriolis (1792-1843) centrifugal force (eq.9). Combining leads to equation 12. Replace the velocity v by using equation 7. This leads to equation 13. (eq.13). Restructure this equation to equation 14. Out of this equation the Bohr (1885-1962) radius is defined. This is the equation 18. Follow the green arrow. The energy of this system is the kinetic energy plus the potential energy. This is equation 15. Use equation 12 to produce equation 16. And then use equation 14 to produce equation 17. The energy in the system is equation 17. N is the number of wavelength in the orbit and Z is the number of protons in the nucleus of the atom. Changing the number of wavelength in the orbit of the electron result is emitting a certain amount of energy and that energy has a typical frequency because of equation 1. The frequencies of the hydrogen atom can be explained in this way.
Lorentz (1858-1947) and Le Verrier (1811-1877) introduce equations used for transformation to other equations. This is equation 19 and 20. Differentiating the time t out of these equations lead to a differential equation of a higher order.
If we now look at the red arrows be see a problem. The equality of centrifugal force and the Coulomb force is based upon the second order space by time relation. This is equation 10. The extra time t dependency leads to 1 order higher differential equations. So the equation 11 should be used. One did not expect that this reasoning could go wrong because one did not know equation 11. It is reasonable to construct a differential equation that has a result on the computer that is equal to a waving orbit around the nucleus for an electron as stated by Bohr(1885-1962). It is reasonable to produce results on the computer that are assumed to be.
If this is so we have a task to do.
Let's make a differential equation that has a waving second order ellipse orbit around a positive nucleus as the result as Bohr stated.
Stefan Boersen October 2017