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is the kinetic and V the potential energy. The called the configuration bundle, and L is a hori-
k
variational principle states zontal m-form over the k-jet prolongation J B
[m being the dimension of the base manifold
b
i
i
δ L(q , ˙q , t)dt = 0 m = dim(M)]. The dynamical system is induced
a by the action functional on the sections σ of the
configuration bundle:
which leads to the Euler-Lagrange equations of

motion k
∗
A (σ) = (j σ) L
D
d ∂L ∂L D
− = 0 ,i = 1,...,n.
dt ∂ ˙q i ∂q i where (j σ) L denotes evaluation of the
k
∗
Lagrangian L along the (jet prolongation) of the
See also Lagrangian system. configuration σ. See also action functional.
Lagrangian (sub)manifold (in a symplectic lamp A source of incoherent radiation.
manifold [P, ω]) A submanifold M ⊂ P
of a symplectic manifold (P, ω) which is both language A language L = (K, N,c,X)
isotropic and coisotropic, i.e., such that can be described either as a tuple of the lan-
guage’s alphabet (K), the set of nonterminals
o
T M = (T M) . (N), the symbol denoting a construct in the lan-
p
p
guage (c), and the set of grammar rules (X); or
The dimension of a Lagrangian manifold M is equally, the nonempty countable set of allowed
exactly half of the dimension of P. constructs C, C ∈ K , where K is the set of all
∗
∗
possible constructs formed over the alphabet K.
Lagrangian symmetry A transformation Comment: Programs (for example, databases
leaving a Lagrangian system invariant. and queries upon them) are necessarily expressed
For example, let (M, L) be a in some arbitrary language.
time-independent Lagrangian system; a
n
transformation ) : M → M, locally given Laplace equation Let U ⊂ R open and

by )(x) = x , can be lifted to the tangent u : U → R. The Laplace equation for u is
space T) : TM → TM, locally given by
n

T )(x, v) = (x ,v ). Itisa Lagrangian

u = u = 0.
symmetry if the following identity is satisfied x i x i
i=1

L(x ,v ) = L(x, v).
Lax-Milgram lemma This lemma asserts
the existence and uniqueness of solutions of vari-
In field theory, a bundle morphism ) : B →
k
B, which lifts to the jet prolongation j ) : ational problems with elliptic sesquilinear forms.
It is a particular case of more general results for
k
k
J B → J B, such that
sesquilinear forms satisfying inf-sup conditions.
k
∗
(j )) L = L.
left action (of a group on a space X ) A map
λ : G × X → X such that:
Lagrangian system A dynamical system
which is defined by the action functional induced (i.) λ(e, x) = x;
by a Lagrangian.
(ii.) λ(g · g ,x) = λ(g , λ(g ,x));
1
2
2
1
A time-independent Lagrangian system is a
pair (Q, L) where Q is a manifold, called the where G is a group, e its neutral element, · the
configuration space, and L : TQ → R is a func- product operation in G, and X a topological
tion on the tangent space of Q. The function L space. The maps λ g : X → X defined by
is called the Lagrangian of the system. λ (x) = λ(g, x) are required to be homeomor-
g
A Lagrangian field theory is a pair (B,L) phisms. A map ˆ λ : G → Hom(X) defined
where B = (B,M,p; F) is a fiber bundle, by ˆ λ(g) = λ is thence associated to a left
g
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