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
First model of resonance: the pendulum
The second fundamental mode of resonance:
Henrard, Lemaitre (1983)
Murray, Dermott (1995)
capture into resonance
passage through resonance
Action, angle variables:
Transformation of the variables:
Transformed Hamiltonian:
pericentric libration
apocentric libration is possible
First order resonances:
Second order resonances:
pericentric and apocentric libration
Third order resonances:
Evolution of level curves depending on δ
Local extrema of the energy function:
Phase plane trajectories:
Restricted three-body problem:
7:4 resonance
librational trajectoriesa=0.688612
e=0.225657
Series of librational trajectories:
the variation of e is small
amplitude of libration:
not much change
4:1 resonance:
a=0.396851
e=0.899320
amplitude of libration:
case of exact resonance:
libration exists for all e
leaving the resonance,
libration exist above certain
values of e
Fourth order resonances:
to be studied yet