DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

Hα condensation
Hα condensation The downflow of Hα emitting
material in the chromospheric portions of
solar flares. Typical downflow velocities are of
the order 50 kms −1 and are observed as redshifts
in Hα line profiles.
Hα radiation An absorption line of neutral
hydrogen (Balmer α) which lies in the red part
of the visible spectrum at 6563 Å. At this wavelength,
Hα is an ideal line for observations of
the solar chromosphere. In Hα, active regions
appear as bright plages while filaments appear
as dark ribbons.
Hamiltonian In simple cases, a function of
the coordinates xα , the canonical momenta pσ
conjugate toxσ , and the parameter (time) t.
H = H x α ,pβ,t
dx
= pα
α
dt −L
x δ , dxγ
dt
,t
where
pα = ∂L
∂ dxα ,
dt
andListheLagrangian. Thisrelationisinverted
to express the right side of the equation in terms
ofxα ,pβ, andt. Such a transformation is called
a Legendre transformation. This can be done
only if L is not homogeneous of degree 1 in
dxα dt ; special treatments are needed in that case.
The standard action principle, written in
terms of the Hamiltonian, provides the equations
of motion.
I=
t2
t1
pα˙x α −H x γ ,pδ,t dt
is extremized, subject to x α being fixed at the
endpoints. This yields
−˙pα − ∂H
∂xα= 0
˙x α − ∂H
∂pα
= 0 .
From these can be formed an immediate implication:
dH
dt
∂H
=
∂t
= ∂H
∂t
© 2001 by CRC Press LLC
218
∂H dpσ
+
∂pσ dt
∂H
+
∂x µ
dx µ
dt
since the last two terms on the right side of the
equation cancel, in view of the equations of motion.
Hence H = constant if H is not an explicit
function of t. Also, notice from the equations
of motion that conserved quantities are easily
found. If H is independent of x σ then pσ is a
constant of the motion. See Lagrangian.
Hamiltonian and momentum constraints in
general relativity The Einstein field equations,
as derived by varying the Einstein–Hilbert
action SEH, are a set of 10 partial differential
equations for the metric tensor g. Since the theory
is invariant under general coordinate transformations,
one expects the number of these
equations as well as the number of components
of g to be redundant with respect to the physical
degrees of freedom. Once put in the ADM form,
SEH shows no dependence on the time derivatives
of the lapse (α) and shift (β i ,i = 1, 2, 3)
functions; their conjugated momenta πα and π i β
vanish (primary constraints). This reflects the
independence of true dynamics from rescaling
the time variable t and relabeling space coordinates
on the space-like hypersurfaces t of the
3+1 slicing of space-time.
We denote by π ij the momenta conjugate
to the 3-metric components γij . Once primary
constraints are satisfied, the canonical Hamiltonian
then reads
HG + HM =
t
d 3 x
α (HG + HM) + βi
H i G + Hi
M ,
where the gravitational Hamiltonian density is
HG =
8πGγ −1/2 γikγjl + γilγjk
ij kl 1
−γij γkl π π −
16πG γ 1/2(3) R,
and the gravitational momentum densities are
H i G =
− 2π ij
|j
1
=−
16πG γ il jk
2γjl,k − γjk,l π .
The corresponding quantities for matter have
been denoted by the subscript M in place of G
and their explicit expressions depend on the particular
choice of matter fields that one wishes to
consider.