# [FOM] Answer to Richard -- Continued

José Félix Costa fgc at math.ist.utl.pt
Tue Dec 13 06:36:08 EST 2005

In fact we can have a logical geometrical system around Newtonian gravity.
The following (partial) axiomatization is due to Elie Cartan (I toke it from
Misner et al., «Gravitation»):

Axiom 1
There exists a function t (universal time) and a symmetric covariant
derivative V.

Axiom 2
The 1-form dt is covariantly constant.

Axiom 3
Spatial vectors are unchanged by parallel transport around infinitesimal
closed curves.

Axiom 4
All vectors are unchanged by parallel transport around infinitesimal,
spatial, closed curves.

Axiom 5
The Ricci curvature tensor has the form 4 \pi \ro dt times dt, where \ro is
the density of mass.

Axiom 6
There exists a metric « . » defined on spacial vectors only, which is
compatible with the covariant derivative in this sense: for any spatial w
and v, and for any u, V_u  (w.v) = (V_u w).v + w. (V_u v).

Axiom 7
The Jacobi curvature operator J is self-adjoint.

Axiom 8
Ideal rods ...  Ideal clocks measure ...

I don't know about gaps in logical deduction from this axiomatization, if we
confine the physical realm to a limited scope. It is really amazing
formulation as beautiful as Landau's formulation of Lagrangian's mechanics
out of axioms of isotropy and homogenity of space and time plus Galilean
relativity principle.

J  Felix

P.S.: Logical deduction of Kepler's laws from Newtonian gravity is not a
standard subject in the sense that derivations in Analytical Mechanics have
almost nothing to do with Newton's derivation. For the Newton's geometric
derivation we can look inside Chandraseckar big book on Newton's Principia
made (more) simple or in Feynman's Last Lecture.

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J. Felix Costa
Departamento de Matematica
Instituto Superior Tecnico
Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL
tel:      351 - 21 - 841 71 45
fax:     351 - 21 - 841 75 98
e-mail:   fgc at math.ist.utl.pt
www:    http://fgc.math.ist.utl.pt/jfc.htm
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