The three body problem |
Integrals
The N-body problem in n dimension
has
(n^{2}+3n+2)/2 classical integrals of motion:
n from the total momentum, n from the position of the center of mass,
and n (n-1)/2 from the total angular momentum
(because
is zero using
)
and finally,
1 from the energy H(x,y).
Because 2nN-1 integrals of motion were necessary to integrate the
N-body problem, there are in general not enough classical integrals
if N>2.
Already the Newtonian 3-body problem in n=2 contains all the
complexity a dynamical system can have.
Rough history
Before Newton, the dynamics of celestial objects was described empirically
which means that laws were drived from observations. Examples are Kepler's
laws. Newton found the laws for classical dynamics and developed differential
and integral calculus simultaneously with Leibnitz. This allowed him to prove
Kepler's laws from more fundamental principles.
Laplace, Lagrange and Poisson belong to the ancestors of celestial mechanics.
With Poincaré at the end of the last century and Birkhoff in the
twenties of this century, the subject was studied with new
geometric and topological methods. Geometrical methods are involved for
example when studying singularities, energy surfaces, equilibrium configurations,
invariant manifolds. Topological methods are invoked for example when proving
the existence of solutions through fixed point theorems.
In the middle of the 20'th century, probabilitistic tools from
ergodic theory and statistical mechanics have been invoked. Ergodic theory
deals with the structure of invariant measures and the
measure-preserving flows, when an invariant probability
measure is present. In statistical mechanics, one is interested in
the evolution of measures on the Fock space of classical mechanics or in
limits, when the particle number goes to infinity. In the second half of
this century, progress on stability questions has been achieved with
perturbative methods. Newer renormalisation proofs of stability also suggest
universal behavior can be relevant in n-body problems, when parameters like
masses are changed. The study of periodic orbits using variational and
topological index methods became a subject in itself. Subquestions include
the study of the stability of orbits or bifurcations depending on parameters
and other questions related with universality.
Other minimal systems coming from subshifts or quasiperiodic motion on
Cantor set were found by variational, perturbative or geometric arguments.
Partial differential equations enter when the formalism is extended to
Vlasov dynamics, a more general dynamical system which allows to evolve more
general measures like smooth densities.
The corresponding evolution equation, the Vlasov equation is fundamental
in stellar dynamics and contains as a special case of discrete measures
the n-body problem. Stability of equilibria solutions or qualitative analysis
of the evolution are quite modern topics of research. Noncollision
singularities of the Newtonian n-body problem in three dimensions have been
found by Xia (n=5) and Gerber (large n).
While it is known that collision singularities have Lebesgue
measure zero and are Baire meager it is now known whether the Newtonian
n-body problem has global solutions for almost all initial conditions. This
is one of the important open problems in celestial mechanics.
Research topics Research on n-body problems can include
Literature
Good starting points in the literature of n-body problems are
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© 2000, Oliver Knill , dynamical-systems.org |