DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

Here we mean that F is a function of all of the
q i , all of the Pl, and of time.
The equations connecting the old and new
descriptions are then
pl = ∂F
∂q l
Qk = ∂F
∂Pk
K = H+ ∂F
∂t
.
Similar sets of equations can be derived by introducing
different functions F. For instance,
F(pi,Q l ,t)gives
q k = −∂F/∂pk
Pm = − ∂F
∂Q m
K = H+ ∂F
∂t
For the remaining two possibilities, one has the
following rules: K=H+∂F/∂t always holds;
derivatives with respect to the variables always
come with fixed signs. That is, the other two sets
ofequationsinvolveoneof∂F/∂q l or −∂F/∂pk
and one of ∂F/∂Pm and −∂F/∂Q l (note the
signs).
Solution of one of these sets or its equivalent
provides the coordinate transformation. The
solution proceeds as follows: If, for instance
F=F(q k ,Pj,t), then ∂F/∂q l is generally a
function of q k ,Pj , and t. Hence Pl =∂F/∂q l
can be inverted to solve for the new momenta
Pl in terms of the old coordinates and momenta
q k ,pj , andt. Similarly,∂F/∂Pk is a function of
q k ,Pl, andt. Substituting in the expression for
Pl in terms of q k ,pl ,t already found gives an
explicit expression forQ p in terms ofq l ,pk,t.
Finally, the new Hamiltonian K must be expressed
entirely in terms of the new variables
Q k ,Pl,tusing the relationships just found. See
Hamilton–Jacobi Theory.
Canopus -0.2 magnitude star, of spectral
type FO at RA06 h 23 m 57.1 s dec −52 ◦ 41’44”.
cap cloud An orographic stratocumulus
cloud mass that forms closely over the top of
a mountain, arising from the condensation of
© 2001 by CRC Press LLC
.
carbon-14 dating
water in air as it rises to the peak; a particular
case of pileus.
CAPE (Convective Available Potential Energy)
Provides a measure of the maximum
possible kinetic energy that a statically unstable
parcel can acquire, assuming that the parcel ascends
without mixing with the environment and
instantaneously adjusts to the local environmental
pressure. See available potential energy.
Capella 0.08 magnitude binary star of spectral
types G8+ FO at RA05 h 16 m 41.3 s , dec
+45 ◦ 51 ′ 53 ′′ .
capillarity correction Due to the fact that
mercury does not wet glass, in glass/mercury
barometers the top surface of the mercury is convex.
A correction for the curvature of the surface
must be made; the mercury height is slightly less
than the expected height for a given air pressure.
The corrections are slight, on the order of 2 mm
or less.
capillary fringe The zone directly above the
water table that is saturated (soil wetness = 1)
and under tension (pressure < 0). The height
of the capillary fringe (hcr) is inversely proportional
to the radius of the soil grains (r) and
directly proportional to the surface tension (σ )
and the cosine of the contact angle (θc):
hcr =
2σ cos θc
ρgr
where ρg is the fluid weight density.
capillary wave A water wave in which the
primary restoring force is surface tension; waves
with wavelength < ∼1.7 cm are considered capillary
waves.
carbon-14 dating A method to date a carbonaceous
object by measuring the radioactivity
of its carbon-14 content; this will determine how
long ago the specimen was separated from equilibrium
with the atmosphere-plant-animal cycle
(“died”). Carbon-14 is continuously produced
in the atmosphere by cosmic-ray bombardment
and decays with a half-life typically described
as 5568 years; dating is accomplished by comparing
the carbon-14 activity per unit mass of