On the Existence of the Conditionally Linear Integral in Conservative
Holonomic Systems 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: - For the Lagrange equations of the 2nd kind the problem of the existence of the conditionally linear
in the velocities integral, which possesses the property that its total time derivative is identically proportional to
the integral itself, is considered. The Lagrange function is assumed to be given in arbitrary generalized
coordinates. The conditions for the existence of such integral are reduced to the study of the compatibility of
two equations in partial derivatives of the 2nd order for one unknown function of two independent arguments.
These equations are written in the invariant form in an arbitrary system of generalized coordinates and the
problem is transformed into the investigation of the set of Pfaffian equations.
Key-Words: - linear integral, point transformation, differential invariants and parameters, overdetermined
PDEs system, Pfaffian equations
Received: January 17, 2023. Revised: March 19, 2023. Accepted: April 6, 2023. Published: May 3, 2023.
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 is referred 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.
The dot denotes the derivative with respect to time.
The Lagrange equations of the 2nd kind, resolved
with respect to the generalized accelerations, have
the form
(2)
where
Consider the expression
(3)
If its total time derivative due to the system (2) is
identically
(4)
where is a certain function, then the integral
curves of the equations (2) with initial conditions
such that for some value of belong
to the set given by the equation (3).
In the considered case of two degrees of freedom,
when the constant is arbitrary, the necessary and
sufficient conditions for the existence of the first
linear integral (3) were found in [1] and [2]. These
conditions have to be satisfied with the functions
, and their derivatives. The functions may be
given in any generalized coordinates.
But the identity (4) may be satisfied only for
some exceptional value of . When the
expression (3) is called the linear invariant relation
of the equations (2). For the case of two degrees of
freedom, the criterion for its existence was given by
[3].
And when the expression (3) will be
called the conditionally linear integral in this paper.
Without loss of generality, we assume .
Conditions for the existence of such kinds of
integrals were the subject of the papers, [4], [5],
[6]. The most general result was obtained in
[6], for the case of arbitrary number of
degrees of freedom. It is proved that when a
conditionally linear integral exists, there are
generalized coordinates of the system in which the
Lagrange function is written in the form
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(5)
where do not depend on . The converse
conclusion is also true. The conditionally linear
integral is .
However, this result does not clarify whether
having the Lagrange function (1) referred to certain
given generalized coordinates, it is possible to find
(in the local sense) the point transformation of the
coordinates so that the transformed Lagrange
function takes the form (5) (then the system has the
conditionally linear integral), or such transformation
does not exist (then there is no conditionally linear
integral).
Below we research this problem in the case of a
naturally conservative system with two degrees of
freedom.
2 Reducing to the Pfaffian System
2.1 Kilmister’s Theorem and Its Invariant
Reformulation
The Lagrange functions found in [4], [5], can be
obtained from (5) under the additional assumption
that the products do not depend on .
Let us exhibit that in the case of and of a
natural conservative system (i.e. its Lagrangian does
not contain linear in the generalized velocities
terms) the following theorem is valid. This theorem
has been proved first by [4], in a different way.
Theorem. For the existence of a conditionally
linear integral in a natural conservative holonomic
system with two degrees of freedom, it is necessary
and sufficient that there exists the nondegenerate
point transformation
(6)
such that the transformed Lagrange function
becomes
(7)
with the force function
. The
network is semi-geodesic, .
Indeed, by opening the brackets in the formula
(5), we obtain
To exclude the terms linear in the velocities, one
should require that be independent of
Then, omitting the exact time derivatives, whose
presence in the Lagrange function does not affect
the form of the Lagrange equations, we obtain (7)
where
.
If the desired functions (6) exist then the
following differential equations
(8)
must be compatible. Here is an
integrating factor for
,
,
( denotes the exterior multiplication).
The system (8) has the invariant form because,
when by any reversible point transformation
the kinematic line element (KLE), [2],
becomes
then for each equation in (8) we have ,
where is written exactly as but in -variables.
The first and second equations in (8) follow from
the fact that the first differential parameters, [7], of
the functions (6) are equal and
in coordinates. The third
equation (8) gives the condition that the differential
form is exact. This equation can be rewritten in
the equivalent form
(9)
where, [7],
are correspondingly the second and mixed
differential parameters of functions and
.
Thus, we have the system of 3 PDEs with two
unknown functions and
The problem is to find the
compatibility conditions of these equations. When
the equations are compatible the conditionally linear
integral takes a place.
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2.2 Elimination of One Unknown Function
For a simplification of the formulae following next
let us set
(12)
The Pfaffian system
(10)
will be of our interest in what follows.
The system of equations (8) and (9) takes the
form
(15)
(11)
From the first two equations we find
(12)
and now one can replace the first two equations (11)
by
(13)
Remark. If then the differential form ω is
exact and the fact, that depends on only,
follows from (13). Since we have
and, hence, according to (9).
But
whence
follows.
Thus, and depend on only, due to
which the Lagrangian (1) has the hidden or explicit
cyclic coordinate, [8]. There exists the first integral
linear in the velocities. The case is
exhausted.
By the substitution of the derivatives (12) in the
third equation (11) and carrying out simplifications
we obtain
(14)
(the factor was canceled and did not enter the
formula).
Thus, the considered problem has been reduced to
researching the consistency of the overdetermined
system of PDEs (13) and (14) with one unknown
function . This system is written in the
invariant form.
See that PDEs (13) and (14) are dependent
linearly on the second partial derivatives and
.
2.3 First Prolongation of the Differential
System
To simplify a little the next formulae let us consider
that, from the outset, the KLE is given in isometric
coordinates
where . As known, such a choice of
coordinates is always possible locally. Of course,
the following generic conclusions do not depend on
a coordinate choice.
Set
In the explicit form, equations (13) and (14) are
correspondingly
(15)
and
(16)
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To obtain the first prolongation of the differential
system (15) and (16) differentiate each equation
relative to the independent variables and
These equations contain independent leading
derivatives
(17)
provided that and .
The matrix of the coefficients at the
quantities (17) is
The rank of this matrix equals if
(18)
We suppose that this condition rejects the trivial
case and is always fulfilled.
The linear combination
leads to the equation
(19)
where
The equations (15), (16), and (19) can be resolved
with respect to and . In virtue of (18), the
rank of the submatrix formed by the
first two rows of the matrix written above, equals .
Hence, according to the linear algebraic system (15)
and (16), the solution for and can be
searched in the form
Here iz unknown parameter and is any
particular solution of the linear inhomogeneous
algebraic system (15-16). Let us pick
,
The substitution of these formulae in (19) leads to
the linear equation with respect to because all the
terms having the second degree of disappear.
Thus, the algebraic system (15), (16), and (19)
specifies the single solution
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After the substitution of the obtained , and
in the right-hand sides of (10) and carrying out the
replacements ,
we derive the set of Pfaffian equations (10) closed
relative to unknowns and .
3 Main Result
Thus, the problem of the existence of the
conditionally linear integral of the Lagrange
equations in the case of two degrees of freedom has
been transformed into a study of the closed set of
Pfaffian equations. When a nontrivial solution
is known one can find from (12)
and then
obtain by quadratures. In -
coordinates the Lagrangian (1) takes the form (7).
If such a solution does not exist there is no
conditionally linear integral of the Lagrange
equations.
4 Conclusion
In the considered problem the analysis of the
overdetermined nonlinear PDEs system of the
second order can be changed by the study of the
nonlinear PDEs system of the first order. Since all
the equations of the latter system are polynomials of
high degrees with respect to and the
problem of finding its integrability conditions is
hard enough, but there are powerful modern relevant
algorithms and computer systems of symbolic
computations which would be useful in concrete
cases. There is a vast set of corresponding
publications. The list of some of them one can find,
e.g., in the bibliography of the book, [9], and in later
sources.
References:
[1] F.G. Minding, Wie sich entscheiden lässt, ob
zwei gegebene krumme Flächen auf einander
abwickelbar sind oder nicht; nebst
Bemerkungen über die Flächen von
unveränderlichen Krümmungsmaβe, Journal
für die reine und angew. Math., Bd.19, 1839,
SS. 370-387.
[2] J. L. Synge, On the Geometry of Dynamics,
Philosophical transactions of the Royal Society
of London, ser. A, vol. 226, 1926, pp. 31-104.
[3] (text in French) G.C. Ricci, T. Levi-Civita,
Méthodes de calcul différentiel absolu et leurs
applications. Mathematische Annalen. Bd. 54.
1901. SS. 125-201.
[4] C. W. Kilmister, The existence of integrals of
dynamical systems linear in the velocities.
Edinburgh Math. Notes, 44, 1961, pp.13-16 in
Proc. Edinburgh Math. Soc., vol. 12 (ser. II),
part 4, 1960-1961.
[5] R. H. Boyer, On Kilmisters’s conditions for the
existence of linear integrals of dynamical
systems. Proc. Edinburgh Math. Soc., vol.14
(ser. II), part 3, June 1965, pp. 243-244.
[6] C. D. Collinson, Integrals of dynamical
systems linear in the velocities // Proc.
Edinburgh Math. Soc., vol.17 (ser. II), part 3,
June 1971, pp. 241-244.
[7] J. Edmund Wright, Invariants of Quadratic
Differential Forms, Cambridge: University
Press, 1908.
[8] A. S. Sumbatov, On cyclic coordinates of
conservative dynamical systems with two
degrees of freedom, Teoriya ustoychivosti i ee
prilozheniya, Novosibirsk: Nauka, 1979, pp.
214-221.
[9] B. Sturmfels, Solving Systems of Polynomial
Equations, CBMS regional conf. ser. in math.
no.97, AMS ed., 2002.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Conducting the research and investigation process,
creation and presentation of the work for publication
have been fulfilled by Alexander S. Sumbatov who
is the only Author of this article.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The Author has no conflict of interest to declare that
is relevant to the content of this article.
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