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AppendixA
Calculus of Variations
The calculus of variations appears in several chapters of this volume as a means to
formally derive the fundamental equations of motion in classical mechanics as well
as in quantum mechanics. Here, the essential elements involved in the calculus of
variations are briefly summarized.1
Consider a functional F depending on a function y of a single variable x (i.e., y =
y(x)) and its first derivative y′ = dy/dx. Moreover, this functional is defined in terms
of the line or path integral
F[I]= xbI(y,y′,x)dx. (A.1)
xa
Accordingly, the value of F will depend on the path chosen to go from x to x .
a b
The central problem of the calculus of variations [1–3] consists of determining the
path y(x) that makes F an extremum (a maximum, a minimum or a saddle point).
In other words, this is equivalent to determining the conditions for which (A.1)
acquires a stationary value or, equivalently, is invariant under first-order variations
(or perturbations) of the path y(x), i.e.,
δF =δxxbIdx=xxbδIdx=0. (A.2)
a a
Let us thus define the quantities δy = Y(x) y(x) and δI = I(Y,Y′,x)
I(y,y′,x), where Y(x) denotes the perturbed path. Variations are taken with respect
to the same x value, so δx = 0. It is straightforward to show that δy′ = d(δy)/dx and
therefore
1 The brief description of the essential elements involved in the calculus of variations presented
here can be complemented with more detailed treatments, which can be found in well-known
textbooks on mathematical physics, e.g., [1–3].
A. S. Sanz and S. Miret-Artés, A Trajectory Description of Quantum Processes. 265
I. Fundamentals, Lecture Notes in Physics 850, DOI: 10.1007/978-3-642-18092-7,
©Springer-Verlag Berlin Heidelberg 2012
266 Appendix A: Calculus of Variations
δI = ∂I + ∂I d δy. (A.3)
∂y ∂y′ dx
Substituting this expression into (A.2) and then integrating by parts yields
xb ∂I d ∂I δydx= 0, (A.4)
xa ∂y dx ∂y′
since, at the boundaries, δy(x ) = δy(x ) = 0. Because δy is an arbitrary, infinites-
a b
imal increment, it can be chosen so that the integrand in (A.4) vanishes. This leads
to the well-known Euler–Lagrange equation,
∂I d ∂I = 0. (A.5)
∂y dx ∂y′
Thefunction y satisfying this equation, if it exists, is said to be an extremal curve or
extremal.
Equation (A.5) can also be recast as
∂I d ′ ∂I
∂x dx I y ∂y′ =0, (A.6)
which arises after taking into account the dependence of I on x, y and y′ as well as
the fact that
d = ∂ +y′ ∂ +y′′ ∂ . (A.7)
dx ∂x ∂y ∂y′
Equation (A.6) is useful whenever I does not depend explicitly on x, for it becomes
I y′ ∂I = constant, (A.8)
∂y′
which is also an extremal.
Consider now that I depends on several functions y ,y ,...,y of x and their
1 2 N
respectivederivatives,y′ ,y′ ,...,y′ .Then,proceedinginasimilarway,afunctional
1 2 N
F[I]= xbI(y ,y ,...,y ,y′,y′,...,y′ ,x)dx (A.9)
1 2 N 1 2 N
xa
can be defined, which becomes an extremum or stationary when the set of Euler–
Lagrange equations
∂I d ∂I =0, i = 1,2,...,N, (A.10)
∂y dx ∂y′
i i
Appendix A: Calculus of Variations 267
is satisfied. However, it could happen that the search for an extremum condition is
subject to a constraint, as in the so-called isoperimetric problems (e.g., determining
the closed plane curve of maximum area and fixed perimeter). In such cases, given
asetJ ,J ,...,J of constraining conditions that depend on x and the y (i =
1 2 M i
1,2,...,N),thesetofNequations(A.10)isreplacedbythesetofN+M equations
⎧ M
∂I d ∂I ∂J
⎨ + λ (x) j =0
∂y dx ∂y′ j ∂y . (A.11)
⎩ i i j=1 i
J (x,y ,y ,...,y ) = 0
j 1 2 N
The λj functions are the so-called Lagrange undetermined multipliers, M unknown
functions of x (or constants) which have to be determined in order to obtain a full
(complete) solution to the problem.
If the constraints in (A.11) are specified by a set of M functional integral
constraints,
F = xbJ(y ,y ,...,y ,y′,y′,...,y′ ,x)dx = c , (A.12)
j j 1 2 N 1 2 N j
xa
where all c are constant and the F are extrema for the y , a function
j j i
M
K =I + λJ (A.13)
j j
j=1
can be defined. Proceeding as before, one finds that these functions have to satisfy
the Euler–Lagrange equation
∂K d ∂K =0, i = 1,2,...,N, (A.14)
∂y dx ∂y′
i i
as well as the integral constraints (A.12).
In the particular case of mechanical systems, when the variational principle is
applied, power series expansions up to the third order in the displacement are often
considered. In these series expansions, the zeroth-order term gives us the action
integral along the reference trajectory; the second-order is called the first variation,
which vanishes for any path due to the stationarity condition; the third-order or
second variation provides us with information about the nature of the stationary
value (maximum, minimum or saddle point) by analyzing the eigenvalues of the
matrix associated with the corresponding quadratic form in the displacements.
Theformalismdescribedaboveisrathergeneral.AsseeninChap.3,forexample,
it is closely related to the formal derivation of Schrödinger’s wave equation. In this
case, instead of several functions y of a single variable x, one considers a function
i
ψofseveralvariables x . These functions are usually called field functions or fields.
i
Furthermore, a subtle conceptual difference can be found in the application of the
268 Appendix A: Calculus of Variations
calculusofvariationsinclassicalandinquantummechanics.Inclassicalmechanics,
it is tightly connected to the concept of energy (Hamiltonian); different solutions
are then obtained from its application, namely the classical trajectories. In quantum
mechanics,though,thisideaisextendedtofunctionalsofasingledependentvariable
(the wave function field) and several independent variables, thus generalizing the
classical case. Thus, rather than keeping constant the energy along a given path,
energy conservation appears in the calculation of the average or expectation value
of such an observable.
References
1. Morse, P.M., Feshbach, H.: Methods of Theoretical Physics. McGraw-Hill, New York (1953)
2. Margenau, H., Murphy, G.M.: The Mathematics of Physics and Chemistry, 2nd edn. Van
Nostrand, New York (1956)
3. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)
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