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.     

Conservation laws in linear elastodynamics

Noether's theorem on variational principles invariant under a group of infinitesimal transformations is used to obtain a class of conservation laws associated with linear elastodynamics. These laws represent dynamical generalizations of certain path-independent integrals in elastostatics which have been of considerable recent interest. It is shown that the conservation laws obtained here are the only ones obtainable by Noether's theorem from invariance under a reasonably general group of infinitesimal transformations.     

Poincaré and the shortcomings of the Hamilton-Jacobi method for classical or quantized mechanics

A great theorem was proven by H. Poincaré in celestial mechanics. It states that, in the most general problems of mechanics, the total energy of the system is the only well behaved first integral of the system, while other so-called integrals cannot be represented by uniform and convergent series. This very important result can be explained and visualized by comparison with standard methods of discussion, as, for example, the Hamilton-Jacobi procedure. The discussion shows that there are serious limitations to the use of this procedure, which collapses in the most general problems (Poincaré theorem) and can be used only for “almost separated” variables. The Poincaré theorem appears to provide the distinction between determinism in mechanics and statistical mechanics according to Boltzmann. The research presented here done under Contract Nonr 266(56) and was first described in a Quarterly Report dated July 31, 1959.     

Hagedorn's theorem in some special cases of rheonomic systems

Hagedorn's theorem on instability [Arch. Rational Mech. Anal. 58 (1976) 1], deduced from Jacobi's form of Hamilton's principle, refers to scleronomic mechanical systems. In this paper we shall prove that Hagedorn's methodology can be generalized to a class of rheonomic mechanical systems with differential equations of motion which allow the existence of Painlevé's integral of energy. The application of this methodology to the case of rheonomic systems which allow, together with Painlevé's integral, cyclic integrals, as well as to the mechanical systems having resultant motion, with prescribed transport motion, and, finally, to the systems having Mayer's rheonomic potential, are also considered. Obtained results are illustrated by an example.     

Existence of solutions for nonlinear impulsive Hammerstein integral equations in Banach spaces

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One type of integrals for the equations of motion of higher-order nonholonomic systems

This paper presents one type of integrals and its condition of existence for theequations Of motion of higher-order nonholonomic systems,including I-order integral(generalized energy integral),2-order integral and p-order integral(P>2).All of theseintegrals can be constructed by the Lagrangianfunction of the system.Two examples aregiven to illustrate the application of the suggested method.     

FIRST INTEGRALS AND INTEGRAL INVARIANTS FOR VARIABLE MASS NONHOLONOMIC SYSTEM IN NONINERTIAL REFERENCE FRAMES

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Die umkehrung der stabilitätssätze von Lagrange-Dirichlet und Routh

Summary The Lagrange-Dirichlet theorem states that the equilibrium position of a discrete, conservative mechanical system is stable if the potential energy U(q) assumes a minimum in this position. Although everything seems to indicate that the equilibrium is always unstable in case of a maximum of the potential energy, this has yet to be proven. In all existing instability theorems the function U(q) has to satisfy additional requirements which are very restrictive.In this paper instability is proved in the case of a maximum of U(q)C ², without further restrictions. The instability follows directly from the existence of certain types of motions which are not found as solutions of differential equations, but as the solutions of a variational problem. Existence theorems are given for the variational problem, based on a result found by Carathéodory.In similar way an inversion of Routh's theorem on the stability of steady motions in systems with cyclic coordinates is also given. The result obtained here is not as general as the inversion of the Lagrange-Dirichlet theorem because the equations of motion are of a more complex type.

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Vorgelegt von C. Truesdell

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Von der Fakultät für Mathematik der Universität Karlsruhe (TH) angenommene Habilitationsschrift.     

Quasi‐One‐Dimensional Riemann Problems and Their Role in Self‐Similar Two‐Dimensional Problems

We study two‐dimensional Riemann problems with piecewise constant data. We identify a class of two‐dimensional systems, including many standard equations of compressible flow, which are simplified by a transformation to similarity variables. For equations in this class, a two‐dimensional Riemann problem with sectorially constant data becomes a boundary‐value problem in the finite plane. For data leading to shock interactions, this problem separates into two parts: a quasi‐one‐dimensional problem in supersonic regions, and an equation of mixed type in subsonic regions. We prove a theorem on local existence of solutions of quasi‐one‐dimensional Riemann problems. For 2 × 2 systems, we generalize a theorem of Courant & Friedrichs, that any hyperbolic state adjacent to a constant state must be a simple wave. In the subsonic regions, where the governing equation is of mixed hyperbolic‐elliptic type, we show that the elliptic part is degenerate at the boundary, with a nonlinear variant of a degeneracy first described by Keldysh. (Accepted December 4, 1997)     

Existence of entropy as a consequence of asymptotic stability

For a class of physical systems whose temporal evolution is governed by ordinary differential equations, the consequences of an assumption of asymptotic stability for equilibrium states in isolation remarkably resemble various forms of the second law of thermodynamics. Here we apply a known converse to Lyapunov's stability theorem to motivate both Gibbs' theory of thermostatics and the use of the Clausius-Duhem inequality for systems which are out of equilibrium and exchanging heat with their surroundings. We also discuss conditions under which the entropy of a system can be expressed as a sum of the entropies of its material points.     

Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations

This paper presents mechanical quadrature methods (MQMs) for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O(h ³) and low computation complexity. Moreover, the mechanical quadrature methods are simple without computing any singular integration. A nonlinear system is constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system are proved based on an asymptotical compact theory and the Stepleman theorem. Using the h ³-Richardson extrapolation algorithms (EAs), the accuracy to the order of O(h ⁵) is improved. To slove the nonlinear system, the Newton iteration is discussed extensively by using the Ostrowski fixed point theorem. The efficiency of the algorithms is illustrated by numerical examples.     

One-dimensional statistical models

The properties of one-dimensional statistical systems are studied. A consistent comparison of the values of the binary correlation function obtained from the configuration integral and from Bogolyubov chain equations in various approximations is presented. The results obtained here are discussed briefly.The existing methods for studying the behavior of real statistical systems are usually based on perturbation theory, and the presence of small parameters characterizing the proximity of the system to an ideal system is assumed. Strongly interacting systems do not permit the isolation of small parameters; therefore, there are no effective methods for studying them at present. In this connection, it appears to be of interest to turn to one-dimensional systems which enable the investigation to be advanced much further and, in particular, permit a consideration of the case of strong interaction. Comparison of the exact results with approximate results obtained by methods for decomposing chains of recurrence equations for correlation functions [1] may be regarded as a qualitative criterion of the accuracy of the latter.Configuration integrals for one-dimensional statistical systems were first obtained in reference [2]. Papers have recently appeared in which one-dimensional models are studied by methods of the theory of stochastic processes [3–6].     

On the equations of age-dependent population dynamics

The aim of this work is to give a direct and constructive proof of existence and uniqueness of a global solution to the equations of age-dependent population dynamics introduced and considered by M. E. Gurtin & R. C. MacCamy in [3]. The linear theory was developed by F. R. Sharpe & A. J. Lotka [10] and A. G. McKendrick [8] (see also [1], [9]) and extended to the nonlinear case by M. E. Gurtin & R. C. MacCamy in [3] (see also [4] [5] [6]). In [3], the key of the proof of existence and uniqueness was to reduce the problem to a pair of integral equations. In fact, as we shall see, the problem can also be solved by a simple fixed point argument. To outline more clearly the ideas of the proof, we will first discuss the setting and the resolution of the linear case, and then we will generalize the results of [3].     

NOTES ON A STUDY OF VECTOR BUNDLE DYNAMICAL SYSTEMS (Ⅱ)─PART 2

In the part 2, theorem 3.1 studied in part 1^[15] is proved first. The proof is obtained via a way of changing variables to reduce the original system of differential equations to a form concerning standard systems of equations in the theory of differentiable dynamical systems. Then by using theorem 3.1 together with the preliminary theorem 2.1, the main theorem of this paper announced in part 1 is proved. The definition of admissible perturbation is contained in the appendix of part 2. The meanings of the main theorem is described in the introduction of part 1.     

A Cauchy integral approach to Hele-Shaw problems with a free boundary: The case of zero surface tension

In this paper, we study a nonlinear and nonlocal free-boundary dynamics — the Hele-Shaw problem without surface tension when the fluid domain is either bounded or unbounded. The key idea is to use a global quantity, the Cauchy integral of the free boundary, to capture the motion of the boundary. This Cauchy integral is shown to be linear in time. The free boundary at a fixed time is then recovered from its Cauchy integral at that time. The main tool in our analysis isCherednichenko's theorem concerning the inverse properties of the Cauchy integrals.As products of our approach, we establish the short-time existence and uniqueness of classical solutions for analytic initial boundaries. We also show the non-existence of classical solutions for all smooth but non-analytic initial boundaries when there is a sink at either a finite point or at infinity. When the fluid domain is bounded, all solutions except the circular one break down before all the fluid is sucked out from the sink. Regularity results are also obtained when there is a source at a finite point or at infinity.     

Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

This paper is on the so called inverse problem of ordinary differential equations, i.e. the problem of determining the differential system satisfying a set of given properties. More precisely we characterize under very general assumptions the ordinary differential equations in \(\mathbb {R}^N\) which have a given set of either \(M\) partial integrals, or \(M first integral, or \(M partial and first integrals. Moreover, for such systems we determine the necessary and sufficient conditions for the existence of \(N-1\) independent first integrals. We give two relevant applications of the solutions of these inverse problem to constrained Lagrangian and Hamiltonian systems respectively. Additionally we provide the general solution of the inverse problem in dynamics.     

Existence of homoenergetic affine flows for the Boltzmann equation

An existence theorem is proved for homoenergetic affine flows described by the Boltzmann equation. The result complements the analysis of Truesdell and of Galkin on the moment equations for a gas of Maxwellian molecules. Existence of the distribution function is established here for a large class of molecular models (hard sphere and angular cut-off interactions). Some of the data lead to an implosion and infinite density in a finite time, in agreement with the physical picture of the associated flows; for the remaining set of data, global existence is shown to hold.     

NEW THEORY FOR EQUATIONS OF NON-FUGHSIAN TYPE REPRESENTATION THEOREM OF TREE SERIES SOLUTION (Ⅰ)

In the analytic theory of differential equations the exact explicit analytic solution has not been obtained for equations of the non-Fuchsian type (Poincare's problem). The new theory proposed in this paper for the first time affords a general method of finding exact analytic expres-sion for irregular integrals.By discarding the assumption of formal solution of classical theory,our method consists in deriving a cor-respondence relation from the equation itself and providing the analytic structure of irregular integrals naturally by the residue theorem. Irregular integrals are made up of three parts: noncontracted part,represented by ordinary recursion series,all-and semi-contracted part by the so-called tree series. Tree series solutions belong to analytic function of the new kind with recursion series as the special case only.     

Existence and Stability of Viscoelastic Shock Profiles

We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic–parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in our investigations of numerical Evans function computations determining stability of large-amplitude and nonclassical type shock profiles.     

A direct construction of first integrals for certain non-linear dynamical systems

A direct, constructive approach to the problem of finding first integrals of certain non-linear, second order ordinary differential equations is presented. The idea is motivated by the construction of the energy integral for the equations of motion of the corresponding conservative systems. Although the method developed for the class of equations studied herein is elementary, it yields the same results as the more advanced group-theoretical methods, such as the use of symmetries] in the context of Noether's theorem. The approach reveals some interesting features when it is specialized to the case of linear equations. Finally, a two-dimensional example is considered by extending the methodology developed for scalar equations to their vector counterparts. It is shown that, as a consequence, a first integral which is independent of the energy integral exists for a particular Hamiltonian of the Contopoulos type.