Appendix B
Simple Models
In this appendix, we give the transformation from momentum and position variables,
(p, x), to action-angle variables, (J, ,), for four one-dimensional model systems
that have been widely used to study the onset of chaos in classical mechanical
systems. They include the pendulum, the quartic double well, the infinite square
well, and one-dimensional hydrogen both with and without a constant external field
(Stark field).
B.1 The Pendulum
The most important one-dimensional model for nonlinear conservative physics is
the pendulum because in many cases it very accurately describes the behavior of
nonlinear resonances. The pendulum can be modeled with a Hamiltonian of the
form
H o =
p 2
2m
− g cos(x) = E o .
(B.1)
A plot of the potential V (x) = −g cos(x) is shown in Fig. B.1 The phase space
plots can be obtained from the expression for the momentum,
p = ±
2m(E o + g cos(x))
(B.2)
and are shown in Fig. B.2. As can be seen from Figs. B.1 and B.2, there are two
types of motion, libration and rotation. They must be considered separately. Let us
first consider libration.
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3
405
Simple Models
In this appendix, we give the transformation from momentum and position variables,
(p, x), to action-angle variables, (J, ,), for four one-dimensional model systems
that have been widely used to study the onset of chaos in classical mechanical
systems. They include the pendulum, the quartic double well, the infinite square
well, and one-dimensional hydrogen both with and without a constant external field
(Stark field).
B.1 The Pendulum
The most important one-dimensional model for nonlinear conservative physics is
the pendulum because in many cases it very accurately describes the behavior of
nonlinear resonances. The pendulum can be modeled with a Hamiltonian of the
form
H o =
p 2
2m
− g cos(x) = E o .
(B.1)
A plot of the potential V (x) = −g cos(x) is shown in Fig. B.1 The phase space
plots can be obtained from the expression for the momentum,
p = ±
2m(E o + g cos(x))
(B.2)
and are shown in Fig. B.2. As can be seen from Figs. B.1 and B.2, there are two
types of motion, libration and rotation. They must be considered separately. Let us
first consider libration.
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3
405