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.     

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.     

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.     

Über die instabilität konservativer systeme mit gyroskopischen kräften

Summary Routh's theorem states that a steady motion of a discrete, conservative mechanical system is stable if the dynamic potential W(q)=U(q)–T₀(q) assumes a minimum. This is a generalized version of the theorem on the stability of equilibrium at a minimum of the potential energy, which is due to Dirichlet. It is well known that a steady motion may also be stable if W(q) assumes a maximum instead of a minimum. The stability is then due to the gyroscopic terms in the equations of motion, without which the steady motion would be unstable. Here it is shown that the steady motion is always unstable if not only W(q) but also H ₀(q) assumes a maximum, H ₀(q) being the part of the Hamiltonian that does not depend on the momenta. It is astonishing that this unexpectedly simple criterion was not found before now. In the proof, a variational formulation is used for the problem, and the instability is shown directly from the existence of certain motions which diverge from the trivial solution.

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Vorgelegt von C. Truesdell     

Necessary and sufficient conditions for complex balancing in chemical kinetics

Summary In a recent publication (Horn & Jackson [1]) it was shown that complex balancing together with mass action type rate laws ensures certain stability properties of a kinetic system, thereby precluding sustained oscillations, bistability and other types of irregular dynamics. In this paper a necessary condition for complex balancing in general kinetics and necessary and sufficient conditions for complex balancing in mass action systems are derived. A theorem is stated which excludes the occurence of equilibria in certain composition regions of general kinetic systems. For mass action systems it is shown that it is sometimes true that the algebraic structure of the reactions suffices to ensure complex balancing, while in other cases complex balancing occurs only if certain relations between the rate constants are satisfied. The number of these relations, called the deficiency of the mass action system is determined by the algebraic structure of the set of reactions underlying that system.     

Fourier Law,Phase Transitions and the Stationary Stefan Problem

We study the one-dimensional stationary solutions of the integro-differential equation which, as proved in Giacomin and Lebowitz (J Stat Phys 87:37–61, 1997; SIAM J Appl Math 58:1707–1729, 1998), describes the limit behavior of the Kawasaki dynamics in Ising systems with Kac potentials. We construct stationary solutions with non-zero current and prove the validity of the Fourier law in the thermodynamic limit showing that below the critical temperature the limit equilibrium profile has a discontinuity (which defines the position of the interface) and satisfies a stationary free boundary Stefan problem. Under-cooling and over-heating effects are also studied: we show that if metastable values are imposed at the boundaries then the mesoscopic stationary profile is no longer monotone and therefore the Fourier law is not satisfied. It regains its validity however in the thermodynamic limit where the limit profile is again monotone away from the interface.     

Empirical foundation and axiomatic treatment of non-equilibrium temperature

Summary On the basis of well known empirems of classical thermodynamics, such as the First and Zero^th Laws, which lead to the definition of calorimeter systems by which heat exchanges can be measured, thermal interaction between thermally homogeneous systems is investigated by methods of set theory. In connexion with Carnap's axiom for a classical measuring quantity a thermodynamic analogue of the thermostatic temperature can be developed by generalized empirems which govern the thermal interaction between non-equilibrium and thermally homogeneous systems. An empirical identification of the contact temperature and its field formulation are given.I thank Assistant Prof. G. Brunk for stimulating discussions. This paper was prepared in unit FPS 9/1 at the Technische Universität Berlin.     

Propagating phase boundaries: Formulation of the problem and existence via the Glimm method

This paper treats the hyperbolic-elliptic system of two conservation laws which describes the dynamics of an elastic material having a non-monotone strain-stress function. FollowingAbeyaratne &Knowles, we propose a notion of admissible weak solution for this system in the class of functions of bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary) and an initiation criterion (for the appearance of new phase boundaries). We prove theL ¹-continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proofs are based onGlimm's scheme and in particular on the techniques ofGlimm andLax. In order to deal with the kinetic relation, we prove a result of pointwise convergence of the phase boundary.     

Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact

In this paper, we will give conditions under which the equilibrium set of multi-degree-of-freedom non-linear mechanical systems with an arbitrary number of frictional unilateral constraints is attractive. The theorems for attractivity are proved by using the framework of measure differential inclusions together with a Lyapunov-type stability analysis and a generalisation of LaSalle’s invariance principle for non-smooth systems. The special structure of mechanical multi-body systems allows for a natural Lyapunov function and an elegant derivation of the proof. Moreover, an instability theorem for assessing the instability of equilibrium sets of non-linear mechanical systems with frictional bilateral constraints is formulated. These results are illustrated by means of examples with both unilateral and bilateral frictional constraints.     

Isentropic Approximation¶of the Compressible Euler System¶in One Space Dimension

Using the stability results of Bressan & Colombo [BC] for strictly hyperbolic 2 × 2 systems in one space dimension, we prove that the solutions of isentropic and non-isentropic Euler equations in one space dimension with the respective initial data (ρ₀, u ₀) and (ρ₀, u ₀, &\theta;₀=ρ₀ ^{γ− 1}) remain close as soon as the total variation of (ρ₀, u ₀) is sufficiently small. Accepted April 25, 2000?Published online November 24, 2000     

Hopf Bifurcation in the presence of symmetry

Using group theoretic techniques, we obtain a generalization of the Hopf Bifurcation Theorem to differential equations with symmetry, analogous to a static bifurcation theorem of Cicogna. We discuss the stability of the bifurcating branches, and show how group theory can often simplify stability calculations. The general theory is illustrated by three detailed examples: O(2) acting on R ², O(n) on R ⁿ , and O(3) in any irreducible representation on spherical harmonics.The work of second author was also supported by a visiting position in the Department of Mathematics, University of Houston     

Stability of Poisson Equilibria and Hamiltonian Relative Equilibria by Energy Methods

We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum Methods. Using a topological generalisation of Lyapunovs result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the topology of the symplectic leaf space at that point. This criterion is applied to generalise the energy-momentum method to Hamiltonian systems which are invariant under non-compact symmetry groups for which the coadjoint orbit space is not Hausdorff. We also show that a G-stable relative equilibrium satisfies the stronger condition of being A-stable, where A is a specific group-theoretically defined subset of G which contains the momentum isotropy subgroup of the relative equilibrium. The results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid.Acknowledgement This work was partially supported by an EPSRC Visiting Fellowship (GR/L57074) and an NSERC individual research grant for GWP, an EPSRC Research Grant (GR/K99893), a Scheme Four grant from the London Mathematical Society, a European Community Marie Curie Fellowship (HPMF-CT-2000-00542) for CW, and by European Community funding for the Research Training Network MASIE (HPRN-CT-2000-00113). We thank the University of Warwick Mathematics Institute for its hospitality during several visits when parts of the paper were written. We are also grateful to TUDOR RATIU for some very helpful remarks.     

The Erpenbeck High Frequency Instability Theorem for Zeldovitch–von Neumann–D?ring Detonations

The rigorous study of spectral stability for strong detonations was begun by Erpenbeck (Phys. Fluids 5:604–614 1962). Working with the Zeldovitch–von Neumann–D?ring (ZND) model (more precisely, Erpenbeck worked with an extension of ZND to general chemistry and thermodynamics), which assumes a finite reaction rate but ignores effects such as viscosity corresponding to second order derivatives, he used a normal mode analysis to define a stability function V(t,e){V(\tau,\epsilon)} whose zeros in ${\mathfrak{R}\tau > 0}${\mathfrak{R}\tau > 0} correspond to multidimensional perturbations of a steady detonation profile that grow exponentially in time. Later in a remarkable paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966; Stability of detonations for disturbances of small transverse wavelength, 1965) he provided strong evidence, by a combination of formal and rigorous arguments, that for certain classes of steady ZND profiles, unstable zeros of V exist for perturbations of sufficiently large transverse wavenumber e{\epsilon} , even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense defined (nearly 20 years later) by Majda. In spite of a great deal of later numerical work devoted to computing the zeros of V(t,e){V(\tau,\epsilon)} , the paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966) remains one of the few works we know of [another is Erpenbeck (Phys. Fluids 7:684–696, 1964), which considers perturbations for which the ratio of longitudinal over transverse components approaches ∞] that presents a detailed and convincing theoretical argument for detecting them. The analysis in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) points the way toward, but does not constitute, a mathematical proof that such unstable zeros exist. In this paper we identify the mathematical issues left unresolved in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) and provide proofs, together with certain simplifications and extensions, of the main conclusions about stability and instability of detonations contained in that paper. The main mathematical problem, and our principal focus here, is to determine the precise asymptotic behavior as e?￥{\epsilon\to\infty} of solutions to a linear system of ODEs in x, depending on e{\epsilon} and a complex frequency τ as parameters, with turning points x _* on the half-line [0,∞).     

Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data

The concept of very weak solution introduced by Giga (Math Z 178:287–329, 1981) for the Stokes equations has hardly been studied in recent years for either the Navier–Stokes equations or the Navier–Stokes type equations. We treat the stationary Stokes, Oseen and Navier–Stokes systems in the case of a bounded open set, connected of class C^1,1{\mathcal{C}^{1,1}} of \mathbbR³{\mathbb{R}^3}. Taking up once again the duality method introduced by Lions and Magenes (Problèmes aus limites non-homogènes et applications, vols. 1 & 2, Dunod, Paris, 1968) and Giga (Math Z 178:287–329, 1981) for open sets of class C^￥{\mathcal{C}^{\infty}} [see also chapter 4 of Necas (Les méthodes directes en théorie des équations elliptiques. (French) Masson et Cie, éd., Paris; Academia, éditeurs, Prague, 1967), which considers the Hilbertian case p = 2 for general elliptic operators], we give a simpler proof of the existence of a very weak solution for stationary Oseen and Navier–Stokes equations when data are not regular enough, based on density arguments and a functional framework adequate for defining more rigourously the traces of non-regular vector fields. In the stationary Navier–Stokes case, the results will be valid for external forces not necessarily small, which lets us extend the uniqueness class of solutions for these equations. Considering more regular data, regularity results in fractional Sobolev spaces will also be discussed for the three systems. All these results can be extended to other dimensions.     

New methods to find solutions and analyze stability of equilibrium of nonholonomic mechanical systems

A large proportion of constrained mechanical systems result in nonlinear ordinary differential equations, for which it is quite difficult to find analytical solutions. The initial motions method proposed by Whittaker is effective to deal with such problems for various constrained mechanical systems, including the nonholonomic systems discussed in the first part of this paper, where in addition to differential equations of motion, nonholonomic constraints apply. The final equations of motion for these systems are obtained in the form of corresponding power series. Also, an alternative, direct method to determine the initial values of higher-order derivatives \({\ddot{q}}_0 ,{{\dddot{q}{} }}_{\!0} ,\ldots \) is proposed, being different from that of Whittaker. The second part of this work analyzes the stability of equilibrium of less complex, nonholonomic mechanical systems represented by gradient systems. We discuss the stability of equilibrium of such systems based on the properties of the gradient system. The advantage of this novel method is its avoidance of the difficulty of directly establishing Lyapunov functions aimed at such unsteady nonlinear systems. Finally, these theoretical considerations are illustrated through four examples.     

On the adjacent equilibrium method in the stability theory of conservative elastic continua

The relationship of the adjacent equilibrium method, the regular perturbation method and the energy method for neutral equilibrium is studied. It is shown that unlike the adjacent equilibrium method, the regular perturbation method yields, for the problems under consideration, non-homogeneous perturbation equations and that adjacent states of equilibrium do not exist at the bifurcation point. These results are then compared with the result of the energy criterion for neutral equilibrium V₂[u] = 0. It is found that although the physical arguments are different in the three methods, the resulting stability equations are identical; thus showing why the adjacent equilibrium argument, even for cases when it is incorrect, yields correct critical loads. This is followed by a discussion of an incorrect derivation of a stability condition and a notion about a load type introduced in the stability literature, which are consequences of the assumption of the general existence of adjacent equilibrium states at bifurcation points.     

Measure-theoretic foundations of statistical mechanics

The basic formulas of classical equilibrium statistical mechanics are derived from well-known theorems in measure theory and ergodic theory. The method used is a generalization of the methods of Khinchin and Grad and deals with several, in fact a “complete set”, of “invariants” or “integrals of the motion”. Most of the results are simple corollaries of Birkhoff's ergodic theorem, and since time-averages are used, the whole approach is characterized by an absence of statistical “ensembles” and probability notions. In the course of the development a “generalized temperature” is introduced, and a generalization of the second law of thermodynamics is derived. Formulas for the “microcanonical”, “canonical”, and “grand canonical” distributions appear as special cases of the general theory.     

On the instability of equilibrium when the potential has a non-strict local minimum

Instability is studied for a mechanical system with two degrees of freedom whose potential has a non-strict local minimum at the origin. Under suitable conditions of smoothness it is shown that the corresponding equilibrium is unstable when the potential has a local minimum along a curve passing through the origin. As a particular case, we get instability if the potential is analytic and has a non-isolated minimum at the origin. These results improve those of G. Hamel [1] and Silla [5] on the same subject.

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Résumé On étudie l'instabilité d'un système mécanique à deux degrés de liberté quand le potentiel a un minimum non isolé a l'origine. On montre que l'équilibre correspondant est instable si le potentiel est assez régulier et a un minimum local en chaque point d'une courbe passant par l'origine. Comme cas particulier, on prouve l'instabilité de l'origine si le potentiel est analytique et y présente un minimum non isolé. Ces résultats améliorent ceux de G. Hamel [1] et L. Silla [5] sur le même sujet.