Invariant Characterization of Liouville’s System
with Two Degrees of Freedom
ALEXANDER S. SUMBATOV
Federal Research Center “Computer Science and Control”
of the Russian Academy of Sciences
Vavilov str., 44 bl. 2, 119333 Moscow
RUSSIA
Abstract: - The local problem of finding separated generalized coordinates in a holonomic natural system with
two degrees of freedom (if the solution exists) is reduced to the problem of integrability of the Pfaffian
systems of differential equations.
Key-Words: - Separation of variables, Liouville’s system, differential invariants and parameters,
overdetermined PDE system, Pfaffian equations
Received: July 15, 2021. Revised: October 16, 2022. Accepted: November 7, 2022. Published: December 2, 2022.
1 Problem Statement
Let
(1)
be the Lagrangian of a holonomic system with
two degrees of freedom (the Ricci
summation convention is applied throughout the
paper). The system refers to local coordinates
chosen on its configuration 2-D
manifold . All occurring functions of
coordinates are supposed to be smooth locally
up to desired order. Dot denotes the derivative
with respect to time.
As known, [1], the Lagrange equations are
integrable in certain generalized coordinates
(2)
with the help of the method of separation of
variables iff the Lagrangian takes the Liouville form
(3)
+
after transformation to these coordinates
(are arbitrary functions).
In the case the problem of recognizing
the existence of the Liouville form for a metric
(4)
was stated in the Theory of Surfaces and has some
history, [2], [3], [4], [5], [6], [7]. Nevertheless, it
was solved in the explicit form quite recently, [7].
The obtained conditions are very cumbersome but it
is the consequence of nature of this problem not of
the method used in [7].
The way to specify the separated variables (2) for
the metric (4) is not discussed in literature except
for [3]. Below we do this for (1) provided that
()
2 Reducing to the Pfaffian Systems
2.1 Basic Differential Parameters
For any functions of let us introduce
Θ
where
, =
These formulae give the basic differential
parameters of functions and , [2]. If by any
reversible point transformation (2) the old metric (4)
becomes the new one, then each of these
parameters , where is in the accented
variables.
Any functions of and their derivatives
satisfying this condition is called a differential
invariant of the form (4). The classic example is
the Gaussian curvature of the configuration
manifold endowed with a metric (4).
2.2 One Hidden Cyclic Coordinate Existence
When in (3) the coordinate is cyclic.
The problem to recognize, if a metric (4) given in
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2022.4.9
Alexander S. Sumbatov
E-ISSN: 2766-9823
74
Volume 4, 2022
arbitrary generalized coordinates has a hidden (in
Hertz’s sense) generalized coordinate, was first
solved in [2]. Namely, when the hidden
cyclic coordinate exists iff the differential
invariants and are functions of the
Gaussian curvature . When , the
geodesic flow for (4) has 3 linear first integrals.
A hidden cyclic coordinate in (1) exists iff the
differential parameters and depend on the
force function only, [8]. Hence, the particular
case of the considered problem is exhausted.
2.3 Generic Case
Let the Lagrangian (1) have the Liouville form (3).
Then the metric (4) is
and the unknown function (or
satisfies the equations , (5)
written in this metric, and vice versa, [3].
But according to (3) the conformal metric
also takes the Liouville form in the same
coordinates Hence, we have 2 supplemental
equations such as (5) and (6) but referred to this
conformal metric.
It can be easily proved, [9], that after
transforming these supplemental equations to the
original metric (4), we obtain (5) once more and (6)
in the form
(7)
Thus, if the invariant equations (5), (6) and (7)
have a non-trivial solution the
Lagrangian (1) can be transformed to the Liouville
form, and vice versa.
The system (5-7) of PDE is overdetermined,
there are three PDE for one function. Two equations
are of the second order and one of the third order.
Now we construct the first continuation of this
PDE system. By differentiation of (5) and (7) with
respect to we obtain four equations which
are algebraically linear relative to the derivatives of
of the third order
(8)
(the indices specify the numbers of the independent
variables with respect to which the
differentiations were fulfilled). Here, as usual,
,
Since , the list (8) of the
independent derivatives is shorter
(9)
The determinant of the linear system mentioned
above with quantities (9) is not zero: when (4) is
written in isothermal (isometric) coordinates, i.e.
,
the determinant equals
Hence, the unknowns (9) can be specified by
solving the linear system as functions
(10)
( ).
The obtained formulae are cumbersome and we
do not give them here. The formulae can be derived
easily with the help of symbolic computations, e.g.
of the system Mathematica. Copy of the notebook
Mathematica with detailed explanations is available
from the Author.
With the help of (10) the third derivatives of
can be eliminated in (6). As a result, we
obtain the quadratic equation with respect to .
When its discriminant is negative the Lagrangian (1)
has not the Liouville form.
Denote by and the real roots as the
explicit functions of the unknowns .
Substituting each of them successively in the right-
hand sides of (10) we can write two Pfaffian
systems
,
, , (11)
,
Each of these systems is closed with 5 unknown
functions
3 Main Result
Theorem. The Lagrangian (1) has the Liouville
form (3) iff one of the Pfaffian systems (11) is
completely integrable, i.e. the exterior derivatives of
its right-hand sides vanish by virtue of (11).
The converse theorem is evident.
The validity of the direct theorem follows from
the structure of the system (11) and from the
invariance property of the function first differential
( is covector, is vector). Thus,
where the brackets denote any
replacement .
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2022.4.9
Alexander S. Sumbatov
E-ISSN: 2766-9823
75
Volume 4, 2022
4 Conclusion
The formulae of this paper were verified for the
simplest cases (Euclidean metric, linear and
quadratic force function, elliptic and parabolic
coordinates, etc.). Symbolic computations have
shown the effectiveness of the proposed approach.
References:
[1] J. Liouville, L’integration des équations
différentielles du mouvement d’un nombre
quelconque de points matériels, Journal de
mathématiques pures at appliquées, T.14,
1849, pp. 257-299.
[2] F.G. Minding, Wie sich entscheiden lässt, ob
zwei gegebene krumme Flächen auf einander
abwickelbar sind oder nicht; nebst
Bemerkungen über die Flächen von
unveränderlichem Krümmungsmaβe, Journal
für die reine und angew. Math., Bd.19, 1839,
SS. 370-387.
[3] J. Edmund Wright, Invariants of Quadratic
Differential Forms, Cambridge: University
Press, 1908.
[4] V. Vagner, On the problem of determining the
invariant characteristics of Liouville surfaces,
Trudy Sem. Vektor. Tenzor. Analizu, Vol.5,
1941, pp. 246–249.
[5] V.I. Shulikovski, An invariant criterion for a
Liouville surface, Dokl. Akad. Nauk SSSR,
Vol.94, 1954, pp. 29–32.
[6] A.I. Chakhtauri, On Invariant Characterization
of Liouville Surfaces, Trudy Tbilisskogo
gosudarstvennogo universiteta, V.56, 1955,
pp. 69-72.
[7] B.S. Kruglikov, Invariant characterization of
Liouville metrics and polynomial integrals,
Journal of Geometry and Physics, Vol.58,
2008, pp. 979-995.
[8] A.S. Sumbatov, On cyclic coordinates of
conservative dynamical systems with two
degrees of freedom, Teoriya ustoychivosti i
yeyo prilozheniya, Novosibirsk: Nauka, 1979,
pp. 214-221.
[9] A.S. Sumbatov, On integration of the
Hamilton-Jacobi equation by the method of
separation of variables, Aktual’nye problemy
klassicheskoy i nebesnoy mekhaniki, Moscow:
TOO “El’f” LTD, 1998, pp. 138-145.
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INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2022.4.9
Alexander S. Sumbatov
E-ISSN: 2766-9823
76
Volume 4, 2022
