Existence Of First Integrals Research Articles

Generalized nonlinear superposition principles for planar polynomial vector fields

We study the possible existence of first integrals of the form I ( x , y ) = ( y - g 1 ( x ) ) α 1 ( y - g 2 ( x ) ) α 2 ⋯ ( y - g ℓ ( x ) ) α ℓ h ( x ) , where g 1 ( x ) , … , g ℓ ( x ) are unknown particular solutions of d y / d x = Q ( x , y ) / P ( x , y ) , α i are unknown constants and h ( x ) is an unknown function. For certain systems, some of the particular solutions remain arbitrary and the other ones are explicitly determined or are functionally related to the arbitrary particular solutions. We obtain in this way a nonlinear superposition principle that generalize the classical nonlinear superposition principle of the Lie theory. In general, the first integral contains some arbitrary solutions of the system but also quadratures of these solutions and an explicit dependence on the independent variable, see García et al. (J. Lie theory, to appear).

On the symmetry approach to polynomial conservation laws of one-dimensional Lagrangian systems

In this paper we analyze polynomial conservation laws of one-dimensional non-autonomous Lagrangian dynamical systems x ̈ =−∂ Π(t,x)/∂x . The analysis is based upon application of Noether's theorem which relates the existence of conservation laws to the symmetries of Hamilton's action integral. It is shown that the existence of first integrals depends on the solution of the system of first-order partial differential equations — generalized Killing's equations. General solution of the problem is formally determined. It is demonstrated that the final form of dynamical system and corresponding conservation law depends on the solution of the so-called potential equation. However, the structure of symmetry transformations, which generate particular class of conservation laws, could be prescribed independent of the solution of potential equation. This fact is used to underline phenomenological aspect of symmetry approach. Its pragmatic value is confirmed through several concrete examples.

The problem of first integral of nonholonomic systems

In this paper, the problem of first integrals of a nonholonomic system is discussed. The aim of this work is concentra.ted on finding the condition for existence of first integrals. The obtained results are applied for the construction of linear and quadratic integrals of nonholonomic systems. It obtains two important affirmations; they are: Any first integral could be treated as a particular nonholonomic constraint and contrarily, any nonholonomic constraint could be regarded as a first integral of the nonholonomic system.

The conditions of existence of first integrals and hamiltonian structures of Lotka-Volterra and Volterra systems

A study of a certain subset of Volterra equations has revealed that some statements about time-independent constants of motion, Hamiltonian functions, and Poisson structure matrices appearing in the Lotka-Volterra equations, either regarded as proven or of the sort that could be proven, are not valid, in fact. Particular cases are given as examples to explain the reasons for the occurring phenomena.

Darboux integrability for 3D Lotka-Volterra systems

We describe the improved Darboux theory of integrability for polynomial ordinary differential equations in three dimensions. Using this theory and computer algebra, we study the existence of first integrals for the three-dimensional Lotka-Volterra systems. Only working up to degree two with the invariant algebraic surfaces and the exponential factors, we find the major part of the known first integrals for such systems, and in addition we find three new classes of integrability. The method used is of general interest and can be applied to any polynomial ordinary differential equations in arbitrary dimension.

Darboux First Integral Conditions and Integrability of the 3D Lotka-Volterra System

We apply the Darboux theory of integrability to polynomial ODE’s of dimension 3. Using this theory and computer algebra, we study the existence of first integrals for the 3–dimensional Lotka–Volterra systems with polynomial invariant algebraic solutions linear and quadratic and determine numerous cases of integrability.

Multivector field formulation of Hamiltonian field theories: equations and symmetries

We state the intrinsic form of the Hamiltonian equations of first-order Classical Field theories in three equivalent geometrical ways: using multivector fields, jet fields and connections. Thus, these equations are given in a form similar to that in which the Hamiltonian equations of mechanics are usually given. Then, using multivector fields, we study several aspects of these equations, such as the existence and non-uniqueness of solutions, and the integrability problem. In particular, these problems are analyzed for the case of Hamiltonian systems defined in a submanifold of the multimomentum bundle. Furthermore, the existence of first integrals of these Hamiltonian equations is considered, and the relation between {\sl Cartan-Noether symmetries} and {\sl general symmetries} of the system is discussed. Noether's theorem is also stated in this context, both the ``classical'' version and its generalization to include higher-order Cartan-Noether symmetries. Finally, the equivalence between the Lagrangian and Hamiltonian formalisms is also discussed.

Connection between the existence of first integrals and the painlevé property in two-dimensional lotka-volterra and quadratic systems

Taking advantage of the considerable amount of work done in the search for first integrals (invariants) for the two-dimensional Lotka-Volterra system and the quadratic system (lvs and qs), we compare the relations needed to exhibit invariants (one for the lvs, at least three for the qs) to the two conditions of the Painleve test (index and compatibility). We find that, eventually restricting the invariants to those which are analytic (all exponents integers) and thereby adding new constraints, these constraints always coalesce with the two Painleve conditions. We conclude that straightforward application of the Painleve test picks up only these simple analytic invariants and that possession of the Painleve property is too strong a condition for the existence of the invariants.

Direct Method of Finding First Integrals of Finite Dimensional Systems and Construction of Nondegenerate Poisson Structures

We present a novel method of finding first integrals and nondegenerate Poisson structures for a given system. We consider the given system as a system of differential 1-forms. After multiplying this system by a set of multiplicative functions, we demand the existence of first integrals. More interesingly these multipliers play a crucial role in constructing the required Poisson structures, if it exists. We illustrate this procedure with a class of physically interesting systems.

Lagrangian formulation and first integrals of piecewise linear dissipative systems

The Lagrangian description of the motion of a damped oscillator is used as a starting point for deducing the canonical formulation and studying the existence of first integrals according to Noether's generalized theorem. It is shown that first integrals exist only if the restoring force is linear. Finally, results are applied to the study of some important linear or piecewise-linear cases such as the Reid oscillator and the oscillator with bilinear hysteresis.

Geodesic first integrals with explicit path-parameter dependence in Riemannian space–times

In a Riemannian space Vn general formulas are obtained for geodesic first integrals which are mth order polynomials in the tangent vector and which are assumed to depend explicitly on the path parameter s. It is found that such first integrals must also be polynomials in s. Necessary and sufficient conditions are found for the existence of these first integrals. The existence of many well-known symmetries such as homothetic motions (scale change), affine collineations, conformal motions, projective collineations, conformal collineations, or special curvature collineations are shown to be sufficient for the existence of such first integrals with explicit path-parameter dependence. To illustrate the theory, geodesic first integrals of this type have been calculated for four Riemannian space–times of general relativity.

On a corollary of the noether theorem for the two-dimensional problem of the Mayer type

The group-theoretical approach to the construction of differential laws of conservation is considered for two-dimensional Mayer type problems. Approach is based on the use of the Lie-Ovsiannikov theory and the Noether theorem /4,5/ which is the fundamental tool for deriving the laws of conservation for uncontrolled physical processes (see, e.g., /6–9/). Certain classes of optimally controlled processes were earlier considered in /10–12/ from the point of view of the Noether theory. Sufficient conditions of existence of first integrals of two-dimensional variational problems of the Mayer type are obtained. As an example, the problem of heat transfer optimization in the boundary layer of a compressible gas is considered.

Noether's theory in classical nonconservative mechanics

Noether's theorem and Noether's inverse theorem for mechanical systems with nonconservative forces are established. The existence of first integrals depends on the existence of solutions of the generalized Noether-Bessel-Hagen equation or, which is the same, on the existence of solutions of the Killing system of partial differential equations. The theory is based on the idea that the transformations of time and generalized coordinates together with dissipative forces determine the transformations of generalized velocities, as it is the case with variations in a variational principle of Hamilton's type for purely nonconservative mechanics [17], [18]. Using the theory a few new first integrals for nonconservative problems are obtained.

Criteria for Pervasive Damping of Mechanical Systems with First Integrals of Motion

AbstractThis paper determines necessary and sufficient criteria for pervasive (or complete) damping of mechanical systems with first integrals of motion. The pervasiveness of the damping is essential when the stability of a system is investigated through its linearized set of equations of motion by Liapunov's method: the sufficient conditions for asymptotic stability then become necessary conditions. The existence of first integrals of motion restricts the asymptotic stability to a certain class of initial conditions. The pervasiveness of the damping will be considered in the same sense.

Conservation laws in classical mechanics for quasi-coordinates

Noether's theorem and Noether's inverse theorem for mechanical systems with gauge-variant Lagrangians under symmetric infinitesimal transformations and whose motion is described by quasi-coordinates are established. The existence of first integrals depends on the existence of solutions of the system of partial differential equations — the so-called Killing equations. Non-holonomic mechanical systems are analysed separately and their special properties are pointed out. By use of this theory, the transformation which corresponds to Ko Valevskaya first integral in rigid-body dynamics is found. Also, the nature of the energy integral in non-holonomic mechanics is shown and a few new first integrals for non-conservative problems are obtained. Finally, these integrals are used in constructing Lyapunov's function and in the stability analyses of nonautonomous systems. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting a Lagrangian function.

A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians

In this paper a procedure is given for finding first integrals of non-conservative mechanical systems in which the Lagrangian is gauge-variant under infinitesimal transformations as functions of time, position and generalized velocities. The procedure is founded on the generalized Noetherian theorem. The existence of first integrals depends on the existence of solutions of the system of partial differential equations. Two examples are analyzed in detail using this theory. The considerations are based on mechanical systems but the results obtained can be used on all problems in physics, engineering and mathematics for which one can construct an action integral in Hamilton's sense.

Noether's theorem for optimum control systems

Abstract In this paper Noether's theorem of classical mechanics and the variational calculus is developed for optimum control systems. Also, it is shown that the existence of first integrals of Pontryagin's maximum principle equations depend on the existence of solutions of a system of partial differential equations—generalized Killing's equations —for optimum control systems.

A group-variational procedure for finding first integrals of dynamical systems

In this work a group-variational procedure is given for finding first integrals of non-conservative dynamical systems. The existence of first integrals depends on the existence of solutions of the system of partial differential equations—generalized Killing's equations. Two examples are analyzed in detail.