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the tangent bundle T Q. Euler-Lagrange equa- evolutionequation Theequationsofmotion
tions are given by of a dynamical system. They describe the time
evolutionofasystemandaregivenbydifferential

i
u = ˙q i
equations of the form
∂L d ∂L i = 0
∂q i − dt ∂u
dx(t)
where dot denotes time derivative. = F(x(t)).
dt
For a Lagrangian system (B, L) of a field
theory over the configuration bundle B =
evolution operator As time passes, the state
(B, M, π, F) let us choose local fibered coord-
µ
i
i
i
inates (x , y , y ,..., y µ 1 ...µ k ) over the k-jet ψ of a physical system evolves. If the state is ψ 0
µ
0
k
prolongation J B and the Lagrangian of the at time t = 0 and it changes to ψ at a later
i
i
µ
t
0
form L = L(x , y , y ,..., y i ) ds, where time t, one sets F (ψ ) = ψ. The operator F t
µ µ 1 ...µ k is called the evolution operator. Determinism
ds is the standard local volume element of the
base manifold M. Euler-Lagrange equations are is expressed by the group property F ◦ F =
s
t
F ,F = identity.
given by t+s 0

i
i
y = d y ,... , y i = d y i
 µ µ µ 1 ...µ k µ 1 ...µ k k
 exact form An exterior k-form ω ∈ ; (M)
∂L i − d ∂L i + ·· · + (−1) d
k
∂y µ ∂y µ µ 1 ...µ k which is the exterior differential of a (k − 1)-

 ∂L
 = 0 form ω, i.e., ω = dθ. Of course, exact forms are
i
closed, i.e., dω = 0. See also closed form.
∂y µ 1 ...µ k
2
where d denotes the total derivative with respect Example: The form ω = xdx + ydy in R is
µ
µ

i
i
2
2
ν
to x (i.e., d = ∂ + y ∂ + y ∂ + ··· ). an exact form, since ω = d
1 2 (x + y ) .
µν i
µ
µ i
µ
See Hamilton principle and Lagrangian sys-
tem.
excitability The concept of excitability first
Euler’s formula e iθ = cos θ + i sin θ appeared in the literature on neural cells
reveals a profound relationship between complex (neurons). It was shown experimentally that a
numbers and the trigonometric functions. small trigger in the membrane current can lead
to large, transient response in membrane elec-
Euler’s integral The representation for the trical potential. This is known as action potential,
gamma function and it is responsible for the rapid communica-
tions between neurons. The mathematical model
∞
z−1 −t
N(z) = t e dt. for this phenomenon was Huxley and it exhibits,
0
among many other features, the threshold phe-
Euler’s method The simplest numerical nomenon (see also threshold phenomenon).
integration scheme for a differential equation
˙ x = f (x). The update rule is x n+1 = x +
n
f (x ) + t. excluded volume A polymer is made of
n
a chain of molecules. These molecules cannot
evaluation map Let F(M) denote a set of occupy the same position in three-dimensional
functions on a manifold M. The evaluation map space. In the simple theory for polymers (see
Gaussian chain), one neglects this effect. A more
is given by ev : F(M) × M → R : ev(f, x) =
f (x) , f ∈ F(M), x ∈ M. realistic models for a polymer must to consider
this effect.
even function A function f (x) such that
f (−x) = f (x), for all x. If g(−x) = −g(x),
for all x, then g is an odd function. expansion In a network (or graph), the
replacement of a smaller subnetwork by a larger
event In probability theory, a measurable one using a sequence of operations on nodes,
set. In relativity, a point in space-time. edges, parameters, and labels.
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC