INVESTIGACI
´
ON
REVISTA MEXICANA DE F
´
ISICA
52
(5) 429–432
OCTUBRE 2006
Hamiltonians and Lagrangians of non-autonomous
one-dimensional mechanical systems
G.F. Torres del Castillo
Departamento de F´
ısica Matem´atica, Instituto de Ciencias,
Universidad Aut´onoma de Puebla, 72570 Puebla, Pue., M´exico.
I. Rubalcava Garc´
ıa
Facultad de Ciencias F´
ısico Matem´aticas, Universidad Aut´onoma de Puebla,
Apartado Postal 1152, 72001 Puebla, Pue., M´exico.
Recibido el 25 de mayo de 2006; aceptado el 12 de septiembre de 2006
It is shown that a given non-autonomous system of two ﬁrst-order ordinary differential equations can be expressed in Hamiltonian form. The
derivation presented here allows us to obtain previously known results such as the inﬁnite number of Hamiltonians in the autonomous case
and the Helmholtz condition for the existence of a Lagrangian.
Keywords:
Non-autonomous systems; Hamilton equations; Lagrangians.
Se muestra que un sistema dado, no aut´onomo, de ecuaciones diferenciales ordinarias de primer orden puede expresarse en forma hamil-
toniana. La deducci´on presentada aqu´
ı nos permite obtener resultados previamente conocidos tales como el n´umero inﬁnito de hamiltonianas
en el caso aut´onomo y la condici´on de Helmholtz para la existencia de una lagrangiana.
Descriptores:
Sistemas no aut´onomos; ecuaciones de Hamilton; lagrangianas.
PACS: 45.05.+x; 45.20.-d
1.
Introduction
As is well known, it is very convenient to express a given sys-
tem of ordinary differential equations (not necessarily related
to classical mechanics) as the Euler–Lagrange equations as-
sociated with some Lagrangian,
L
, or as the Hamilton equa-
tions associated with some Hamiltonian,
H
(see,
e.g.
, Ref. 1).
One of the advantages of such identiﬁcations is the possibility
of ﬁnding constants of motion, which are related to symme-
tries of
L
or
H
. Also, the Hamiltonian of a classical system
is usually regarded as an essential element to ﬁnd a quantum
version of the mechanical system.
In the simple case of a mechanical system with forces
derivable from a potential (that may depend on the veloci-
ties), there is a straightforward procedure for ﬁnding a La-
grangian or a Hamiltonian. However, in the case of non-
conservative mechanical systems or of systems not related to
mechanics, the problem of ﬁnding a Lagrangian or a Hamil-
tonian is more involved. A given system of
n
second-order
ordinary differential equations is equivalent to the Euler–
Lagrange equations for some Lagrangian if and only if a set
of conditions (known as the Helmholtz conditions) are ful-
ﬁlled (see,
e.g.
, Refs. 2, 3, and the references cited therein).
The aim of this paper is to give a straightforward pro-
cedure to ﬁnd a Hamiltonian for a given system of two
ﬁrst-order ordinary differential equations (which may not be
equivalent to a second-order ordinary differential equation)
that possibly involves the time in an explicit form. The re-
sults derived here contain the Helmholtz condition for