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conjugate momenta For a Lagrangian conservation law A conservation law or
L(q , ˙q ) the conjugate momenta p are conserved quantity of a vector field X is any func-
i
i
i
tion F such that F ◦ φ = F where φ is the flow
t
t
∂L
p = i of X. See constant of motion.
i
∂ ˙q
conservative vector field A vector field F
conjugated system (conjugation) In the n
on a region in R is called conservative if it is
original meaning a conjugated system is a
the gradient of a function f , i.e., F =∇f .
molecular entity whose structure may be
represented as a system of alternating single and
conserved current For a time-independent
multiple bonds. For example, CH = CH − Lagrangian system a conserved current is a first
2
CH = CH , CH 2 = CH − C ≡ N.In integral.
2
such systems, conjugation is the interaction of
In field theory, it is a (m − 1)-form E over
one p-orbital with another across an intervening h
some jet-prolongation J B such that it is closed
σ-bond in such structures. (In appropriate molec-
once evaluated on a critical section σ, i.e., such
ular entities d-orbitals may be involved.) The
that
term is also extended to the analogous interac- h
∗
d[(j σ) E] = 0
tion involving a p-orbital containing an unshared
electron pair, e.g., Cl − CH = CH .
2 See Lagrangian system.
connected Describing a topological space constant of motion A function f : M → R
(X, τ(X)) in which the empty set, ∅, and the on a manifold M is called a constant of motion
whole space, X, are the only subsets which of a vector field X on M if f ◦ F = f , where F t
t
are both closed and open. The open interval is the flow of X.If X is a Hamiltonian vector
H
I = (0, 1) ⊂ R is connected in the standard field, then f is a constant of motion of X if the
H
topology. The union of I and the open interval Poisson bracket {f, H}= 0. See conservation
(−1, 0) is not connected. In fact, both I and law.
(−1, 0) are at the same time open and closed,
since they complement each other.
constitutional unit An atom or group of
See pathwise connected.
atoms (with pendant atoms or groups, if any)
comprising a part of the essential structure of a
connected graph A graph G(V, E) is con-
macromolecule,an oligomer molecule,a block,
nected if there is a path between any two vertices,
or a chain.
{v ,v }∈ V, for all pairs of vertices.
i j
Comment: Visually, connected graphs are k
contact form A 1-form over J B which
thosewhich are in“onepiece.” Note thatif a con- k
identically vanishes on holonomic sections j σ.
nected graph contains a cycle, breaking the cycle
They are locally expressed as
once is guaranteed to preserve the connectedness
of the graph. The effect of subsequent breaks will  i i i ν
ω = dy − y dx
ν
depend on the topology of the graph.  i i ν

 i
ω = dy − y dx
µ
µν
connection of a bundle A distribution (of ...


 i i i ν
rank m = dim(M)) of planes ⊂ T B over ω = dy − y dx .
b b µ 1 ...µ k−1 µ 1 ...µ k−1 ν
the total space B of a bundle (B,M,π; F). The
spaces ⊂ T B are required to be nonvertical One defines three different ideals of contact
b b
so that ⊕ V - T B, where V denotes the forms:
b b b b k
subspace of vertical vectors. ; (J B) is the bilateral ideal gen-
c
erated algebraically by contact 1-forms
i
i
connectivity In a chemical context, the (ω ,ω ,...,ω i µ 1 ...µ k−1 ).
µ
k
information content of a line formula, but omit- ; (J B) is the differential bilateral ideal
∗
c
ting any indication of bond multiplicity. generated by contact 1-forms (i.e., it includes
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC