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action, law of action and reaction (New- action principle (Newton’s second law)
ton’s third law) The basic law of mechanics Any force F acting on a body of mass m induces

asserting that two particles interact so that the an acceleration a of that body, which is pro-
forces exerted by one upon another are equal in portional to the force and in the same direction
magnitude, act along the straight line joining the F = m a.

particles, and are opposite in direction.
action, principle of least The principle
action functional In variational calculus (Maupertius 1698–1759) which states that the
(and, in particular, in mechanics and in field actual motion of a conservative dynamical sys-
theory) is a functional defined on some suitable tem from P to P takes place in such a way that
1
2
space F of functions from a space of independent the action has a stationary value with respect to
variables X to some target space Y; for any all possible paths between P and P correspond-
1
2
regular domain D and any configuration ψ of ing to the same energy (Hamilton principle).
the system it associates a (real) number A [ψ].
D
A regular domain D is a subset of the space X activation energy (Arrhenius activation
energy) An empirical parameter characterizing
(the time t ∈ R in mechanics and the space-time
the exponential temperature dependence of the
point x ∈ M in field theory) such that the action
2
rate coefficient k, E = RT (d ln k/dT ), where
functional is well-defined and finite; e.g., if X is a
R is the gas constant and T the thermodynamic
a manifold, D can be any compact submanifold
temperature. The term is also used for threshold
of X with a boundary ∂D which is also a compact
energies in electronic potential surfaces, in
submanifold.
By the Hamilton principle, the configurations which case the term requires careful definition.
ψ which are critical points of the action func- activity In biochemistry, the catalytic power
tional are called critical configurations (motion of an enzyme. Usually this is the number of sub-
curves in mechanics and field solutions in field strate turnovers per unit time.
theory).
In mechanics one has X = R and the relevant adaptive refinement A strategy that aims to
space is the tangent bundle TQ to the configura- reduce some discretization error of a finite ele-
tion manifold Q of the system. Let ˆγ = (γ, ˙γ) ment scheme by repeated local refinement of the
be a holonomic curve in TQ which projects onto underlaying mesh. The goal is to achieve an
the curve γ in Q and L : TQ → R be the equidistribution of the contribution of individ-
Lagrangian of the system, i.e., a (real) func- ual cells to the total error. To that end one relies
tion on the space TQ. The action is given by on a local a posteriori error estimator that, for

h
A [γ ] = D L(γ (t), ˙γ(t)) dt. D can be any each cell K of the current mesh ; , provides an
D
closed interval. If suitable boundary conditions estimate η of how much of the total error is due
K
are required on γ one can allow also infinite inter- to K.
vals in the parameter space R. Starting with an initial mesh ; , the refine-
h
In field theory X is usually a space-time mani- ment loop comprises the following stages:
fold M and the relevant space is the k-order (i.) Solve the problem discretized by means
k
jet extension J B of the configuration bundle of a finite element space built on ; ;
h
(B,M,π,F) of the system. Let ˆσ be a holo- (ii.) Determine guesses for the total error of
k
nomic section in J B which projects onto the the discrete solution and for the local error contri-
k
section σ in B and L : J B → R be the
butions η . If the total error is below a prescribed
h
Lagrangian of the system, i.e., a (real) func- threshold, then terminate the loop;
k
tion on the space J B. The action is given
(iii.) Mark those cells of ; for refinement
by A [σ] = D L(ˆσ(x)) ds, where L(ˆσ(x)) h
D
denotes the value which the Lagrangian takes whose local error contributions are above the
average error contribution;
over the section; D ⊂ M can be any regular
domain and ds is a volume element. If suitable (iv.) Create a new mesh by refining marked
boundary conditions are required on the sections cells of ; and go to (i.).
h
σ one can allow also infinite regions up to the Algorithms for the local refinement of simplicial
whole parameter space M. and hexaedral meshes are available.
© 2003 by CRC Press LLC