**Exercise 5.32.** **Hierarchical Jacobi coordinates**

A Hamiltonian for the *n*-body problem is

where *x*_{i} is the tuple of rectangular coordinates for body *i* and
*p*_{i} is the tuple of conjugate linear momenta for body *i*.

The potential energy of the system depends only on the relative positions of the bodies, so the relative motion decouples from the center of mass motion. In this problem we explore canonical transformations that achieve this decoupling.

**a**. Canonical heliocentric coordinates. The coordinates
transform as follows:

where *X* is the center of mass of the system,
and

for *i*>0, the differences of the position of body *i* and the body
with index 0 (which might be the Sun). Find the associated
canonical momenta using an *F*_{2}-type generating function. Show that
the potential energy can be written solely in terms of the coordinates
for *i*>0. Show that the kinetic energy is not in the form of a sum
of squares of momenta divided by mass constants.

**b**. Jacobi coordinates. The Jacobi coordinates isolate the
center of mass motion, without spoiling the usual diagonal quadratic
form of the kinetic energy. Define *X*_{i} to be the center of mass of the bodies with
indices less than or equal to *i*:

The Jacobi coordinates are defined by

The coordinates *x*'_{i} for 0 < *i*< *n* are the difference of the
position of body *i* and the center of mass of bodies with lower
indices; the coordinate *x*'_{0} is the center of mass of the system.
Complete the canonical transformation by finding the conjugate momenta
using an *F*_{2}-type generating function. Show that the kinetic energy
can still be written in the form

for some constants *m*'_{i},
and that the potential *V* can be written solely in terms of the
Jacobi coordinates *x*'_{i} with indices *i* > 0.

**c**. Hierarchical Jacobi coordinates. Define a ``body'' as a
tuple of a mass and a rectangular position tuple. An *n*-body
``system'' is a tuple of *n* bodies: (*b*_{0}, *b*_{1}, `...` , *b*_{n-1}).
Define a ``linking'' transformation _{jk} for bodies *j* and
*k* that takes an *n*-body system and returns a new linked system:

The bodies in the new system are the same as the bodies in the old system
*b*_{i}' = *b*_{i} except for bodies *j* and *k*:

This is a transformation to relative coordinates and center of mass
for bodies *j* and *k*. Extend this transformation to phase space and
show that it preserves the form of the kinetic energy

Show that the transformation to Jacobi coordinates of part **b** is
generated by a composition of linking transformations:

Interpret the coordinate transformation produced by such a succession of linking transformations; why do we call this a ``linking'' transformation? What requirement has to be satisfied for a composition of linking transformations to isolate the center of mass of the system (make it one of the coordinates)? Taking this constraint into account, find hierarchical Jacobi coordinates for a system with six bodies, arranged as two triple systems, each of which is a binary plus a third body. Verify that one of the coordinates is the center of mass of the system, and that the kinetic energy remains a sum of squares of the momenta divided by an appropriate mass constant.

**Exercise 5.33.** **Symplectic Integration**

Consider a system for which the Hamiltonian *H* can be split into two
parts, *H*_{0} and *H*_{1}, each of which describes a system that can be
efficiently evolved:

Symplectic integrators construct approximate solutions for the
Hamiltonian *H* from those of *H*_{0} and *H*_{1}.

We construct a map of the phase space onto itself in the following way
(see Wisdom [47]).
Define _{2}(*t*) to be an infinite sum of Dirac delta
functions, with interval 2,

Recall that a function has the property that _{-a}^{a} *f*
= *f*(0) for any positive *a* and continuous real-valued function *f*.
It is fruitful to think of the delta function as a limit of a
function _{h} that has the value _{h}(*t*) = 1/*h* in the interval `-` *h*/2
< *t* < *h*/2 and zero otherwise.
Now consider the mapping Hamiltonian

The evolution of the system between the delta functions is governed
solely by *H*_{0}. To understand how the system evolves across the
delta functions think of the delta functions in terms of _{h} as
*h* goes to zero. Hamilton's equations contain terms from *H*_{1} with
the factor 1/*h*, which is large, and terms from *H*_{0} that are
independent of *h*. So as *h* goes to zero, *H*_{0} makes a negligible
contribution to the evolution. The evolution across the delta
functions is governed solely by *H*_{1}. The evolution of *H*_{m}
is obtained by alternately evolving the system according to the
Hamiltonian *H*_{0} for an interval *t* = 2 / and then
evolving the system according to the Hamiltonian *H*_{1} for the same time
interval. The longer-term evolution of *H*_{m} is obtained by iterating
this map of the phase space onto itself. Fill in the details to show this.

In terms of Lie series, the evolution of *H*_{m} for one delta function
cycle *t* is generated by

The evolution of *H*_{m} approximates the evolution of *H*.
Identify the noncommuting operator *A* with *L*_{H0} and *B* with
*L*_{H1}. Use the Baker-Campbell-Hausdorff identity
(equation 5.475) to deduce that the local truncation
error (the error in the state after one step *t*) is
proportional to ( *t*)^{2}. The mapping is a first-order
integrator.

By merely changing the phase of the delta functions, we can reduce the truncation
error of the map, and the map becomes a second-order
integrator. Instead of making a map by alternating a full step
*t* governed by *H*_{0} with a full step *t*
governed by *H*_{1}, we can make a map by evolving the system for a half
step *t* / 2 governed by *H*_{0}, then for a full step *t* governed
by *H*_{1}, and then for another half step *t* / 2 governed by
*H*_{0}. In terms of Lie series the second-order map is generated by

Confirm that the Hamiltonian governing the evolution of this map is
the same as the one above but with the phase of the delta functions shifted.
Show that the truncation error of one step of this second-order map is
indeed proportional to ( *t*)^{3}.

Consider the Hénon-Heiles system. We can split the Hamiltonian into two solvable Hamiltonians in the following way (Wisdom [48]):

Hamiltonian *H*_{0} is the Hamiltonian of two uncoupled linear
oscillators; Hamiltonian *H*_{1} is a nonlinear coupling. The
trajectories of the systems
described by each of these Hamiltonians can be expressed in closed form.
Write programs that implement
first-order and second-order maps for the Hénon-Heiles problem. Examine
the evolution of the energy for both chaotic and quasiperiodic initial
conditions. How does the magnitude of the energy error scale with the
step size; is this consistent with the order of the integrator deduced
above? How does the energy error grow with time? Generate surfaces
of section using the second-order map. Does the map preserve the
chaotic or quasiperiodic character of trajectories?