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then [c] and [a] are disjoint. The set A/ ∼ of Euler Leonhard Euler (1707–1783) Swiss
all equivalence classes is a partition of the set A, mathematician. Somesaythemostprolificmath-
and it is called the quotient of A with respect to ematician in history and the first modern math-
the relation ∼. ematician universalist. He worked at the St.
Example: Let ≡ be the relation in Z defined Petersburg Academy and the Berlin Academy of
n
by a ≡ b if and only if n divides a − b, Science.
n
i.e., if there exists an integer k ∈ Z such that
a − b = nk. It is an equivalence relation. If
Euler equations The motion of a perfect
a ≡ b we say that a is congruent to b modulo n.
n
The equivalence classes are [a] = {a + kn : fluidinadomainM (asmoothRiemannianmani-
n
k ∈ Z}. The quotient space is denoted by fold with boundary) is governed by the Euler
equations
Z = {[0] , [1] ,..., [n − 1] } and it is finite.
n
n
n
n
The structure of additive group of Z induces a
∂u +∇ u =−∇p
structure of additive group on Z with respect to ∂t u
n
the operation [a] +[b] = [a+b] which is well divu = 0
n
n
n
defined; i.e., it does not depend on the representa-
where u is the velocity field of the fluid and p is
tives chosen for the equivalence classes. In fact,
the pressure.

n
if a ∈ [a] and b ∈ [b] , then [a ] + [b ] = The motion of a rigid body with angular
n
n
n

[a + b ] = [a + b] = [a] + [b] .

n
n
n
n
2
3
1
Specializing to the case n = 2, the quotient momentum X = (X ,X ,X ) and principal
3
space Z is made of two elements [0] and [1] . moments of inertia I = (I ,I ,I ) is governed
1
2
2 2 2
The class [0] is the neutral element with respect by the Euler equations
2
to the additive structure, while one has [1] +
2 I 2 −I 3
[1] = [0] . By using the function exp(iπn), ˙ X = X X 3
2
1
2 2 I 2 I 3
Z with the additive structure is mapped into the ˙ X = I 3 −I 1 X X 3
1
2
2 I 1 I 3
group of signs with the obvious multiplicative ˙ X = I 1 −I 2 X X .
3 I 1 I 2 1 2
group structure.
Euler-Lagrange equations The Lagrang-
equivalent norms Two norms and 2
1
on a normed vector space X such that there is a ian formulation of a classical mechanical sys-
1
n
−1 tem described by a Lagrangian L(q ,...q ,
positive number c such that c x ≤ x ≤ 1 n
2
1
c x forallx ∈ X. Onfinitedimensionalvector ˙ q ,... ˙q )isgivenbytheprincipleofleastaction,
1
spaces all norms are equivalent. which leads to the Euler-Lagrange equations
d ∂L ∂L
equivariant See principal bundle. − = 0 ,i = 1,...,n.
dt ∂ ˙q i ∂q i
Erlanger program A plan initiated by Felix
These are equivalent to Hamilton’s equations.
Klein in 1872 to describe geometric structures in
In field theory where the Lagrangian L
terms of their groups of automorphisms.
depends on fields ϕ (x) and their derivatives
i
∂ ϕ (x), the Euler-Lagrange equations become
essential singularity A singularity of an µ i
analytic function that is not a pole.
∂L(x) ∂L(x)
essential spectrum For a linear operator − ∂ µ = 0 ,i = 1,...,n.
∂ϕ (x) ∂[∂ ϕ (x)]
µ i
i
A : X → X on a normed vector space X the
set of scalars λ such that either ker(A − λI) is
infinite dimensional or the image of A − λI fails
to be a closed subspace of finite codimension. For a mechanical Lagrangian system (Q, L)
over the configuration space Q with local
i
essentially self-adjoint operator See self- coordinates q the Lagrangian is a function
i
i
i
i
adjoint operator. L(t, q ,u )with (q ,u ) local coordinates over
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC