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Hamilton principle The principle accord-
ing to which the action functional determines
configurationsofmotion(alsocalledcriticalcon-
H figurations). Critical configurations are those
for which the action functional is stationary with
respect to all compactly supported deformations.
For this reason, the Hamilton principle is also
h-version of finite elements A strategy that
called the principle of stationary action.
seeks to achieve a sufficiently small discretiza-
In Lagrangian systems the Hamilton principle
tion error of a finite element scheme for a bound-
ary value problem by using fine meshes. This can is equivalent to the Euler-Lagrange equations,
which, depending on the number of independent
mean a small global meshwidth or local refine-
variables (n = 1or n> 1), are called equations
ment. The latter is employed in the context of
of motion or field equations, respectively.
adaptive refinement.
See also Lagrangian system and action func-
half-life The time t required for one half
1/2 tional.
of a population to change randomly from some
state ∫ to another state ∫ . At least one of these Hamiltonian system Let (M, ω) be a sym-

two states must be observable. plectic manifold and H : M → R a smooth
Comment: The classic example of half-life is
function. The corresponding Hamiltonian vec-
radioactivity. In any sample of matter, some of
tor field X H is determined by the condition
the nuclei will be radioactive isotopes and the
ω(X ,Y) = dH · Y. The flow of X in canon-
H
H
remainder not. The decline in the radioactivity ical coordinates satisfies Hamilton’s equations
of the sample is governed by a first-order Poisson
and is called a Hamiltonian system.
process More generally, an evolution equation is
N = N e −λt
0
called a Hamiltonian system if it can be written in
1
where N and N are the current and original the form F ={F, H} with respect to some Pois-
˙
0
number of radioactive atoms in the sample and son bracket { , }. H is called the Hamiltonian of
λ the characteristic decay constant for an iso- the system.
tope. Thus t = ln 2/λ. Judged by the metric
1/2
of radioactivity, nonradioactivity is a nonobserv-
Hamiltonian vector field Let (P, { , }) be
able state (though it can of course be observed
Poisson manifold and H ∈ C (P ). The Hamil-
∞
by other assays).
tonian vector field X of H is defined by
H
n
1
Hamilton equations Let (q ,...,q ,p ,
1 ∞
...,p ) be canonical coordinates and H a X (G) ={H, G}, for all G ∈ C (P ).
H
n
smooth function. Hamilton’s equations for H
are Hammettequation(Hammettrelation) The
∂H ∂H equation in the form:
i
˙ q = , ˙p =− i ,i = 1,...,n.
i
∂ ˙p i ∂ ˙q
lg(k/k ) = ρσ
0
n
Hamilton-Jacobi equation Let U ⊂ R be
open and u : U ×R → R. The Hamilton-Jacobi or
equation for u is lg(K/K ) = ρσ
0
u + H(Du, x) = 0,
t applied to the influence of meta-or para-
substituents X on the reactivity of the func-
where Du = D u = (u ,...,u ) denotes the
x
x 1
x n
gradient of u with respect to the spatial variable tional group Y in the benzene derivative m-or
p-XC H Y. k or K is the rate or equilibrium
x = (x ,...,x ). 6 4
n
1
constant, respectively, for the given reaction of
1 This notation is restricted to this discussion and char- m-or p-XC H Y; k or K refers to the reac-
4
acteristic of that seen in other texts on this subject; do not 6 0 0
confuse it with notation on reactions elsewhere, especially tion of C H Y, i.e., X = H; σ is the substituent
5
6
the use of N and λ. constant characteristic of m-or p-X: ρ is the
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