1. Higher order resonances in the second fundamental mode of resonance B. Érdi Astronomy Department Eötvös University 5th Austrian-Hungarian Workshop on Celestial Mechanics Wien, April 8-10, 2010

2. First model of resonance: the pendulum

3. The second fundamental mode of resonance: Henrard, Lemaitre (1983) Murray, Dermott (1995) capture into resonance passage through resonance Action, angle variables:

4. Transformation of the variables: Transformed Hamiltonian:

5. pericentric libration apocentric libration is possible First order resonances:

6. Second order resonances: pericentric and apocentric libration

7. Third order resonances:

8. Evolution of level curves depending on δ

9. Local extrema of the energy function:

10. Phase plane trajectories:

11. Restricted three-body problem:

12. 7:4 resonance librational trajectories a=0.688612 e=0.225657

13. Series of librational trajectories: the variation of e is small amplitude of libration: not much change

14. 4:1 resonance: a=0.396851 e=0.899320

15. amplitude of libration: case of exact resonance: libration exists for all e leaving the resonance, libration exist above certain values of e

16. Fourth order resonances: to be studied yet