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Today's CfA Theory Seminar

*** CfA Theory group lunch seminar ***

Tue, Sep 30, 12:30pm

Michael Efroimsky (US Naval Observatory)

"Implicit Gauge Invariance in the N-body Problem of Celestial Mechanics:
Practical Applications"

The seminar will be held in Pratt Conference Room (G04) at CfA,
60 Garden St.

Following is the abstract of the talk.


We revisit the Lagrange and Delaunay systems of equations of
celestial mechanics, and point out  a previously neglected aspect
of these equations: in both cases  the orbit resides  on a certain
9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space
spanned by the orbital elements and their time derivatives. We
demonstrate that there exists a vast freedom in choosing  this
submanifold. This freedom of choice (=freedom of gauge fixing)
reveals a symmetry hiding behind Lagrange's and Delaunay's
systems, which is, mathematically, analogous to the gauge
invariance in electrodynamics.

Historically, Lagrange removed this freedom by imposing, "by hand",
a convenient constraint. This was the condition of the instantaneous
ellipse (or hyperbola) being always tangential to the physical velocity.
It turns out that the Lagrange constraint is also implicitly instilled
into the Hamilton-Jacobi treatment of the N-body problem.

Imposure of any supplementary condition different from the Lagrange
constraint (but compatible with the equations of motion) is legitimate
and will not alter the physical trajectory or velocity (though will
much alter the mathematical form of the planetary equations).

Existence of this internal freedom has consequences for the stability
of numerical integrators. Another important aspect of this symmetry is
that any gauge different from that of Lagrange makes the Delaunay system
non-canonical. In a more general setting, when the disturbance depends
not only upon positions but also upon velocities, there exists a
"generalised Lagrange gauge" wherein the Delaunay system is symplectic.
This special
gauge renders orbital elements that are osculating in the phase space. It
coincides with the regular Lagrange gauge when the perturbation is

We provide a practical example illustrating how the gauge formalism
considerably simplifies the calculation of satellite motion about an
oblate precessing planet.