Reduction of Lagrangian mechanics on Lie algebroids

We prove that if a surjective submersion which is a homomorphism of Lie algebroids is given, then there exists another homomorphism between the corresponding prolonged Lie algebroids and a relation between the dynamics on these Lie algebroid prolongations is established. We also propose a geometric reduction method for dynamics on Lie algebroids defined by a Lagrangian and the method is applied to regular Lagrangian systems with nonholonomic constraints.     

Non-Hamiltonian systems separable by Hamilton–Jacobi method

We show that with every separable classical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These systems are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate the conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.     

Elliptic Equation with Singular Terms on Riemannian Manifold

In this paper we deal with the following equation: on a three-dimensional Riemannian manifold . We assume that the volume of Σ, the norm , and are small enough. Using a rather simple argument we show the existence of solution to the problem. Dedicated to Gosia and Basia.     

Goldfish Geodesics and Hamiltonian Reduction of Matrix Dynamics

We describe the Hamiltonian reduction of a time-dependent real-symmetric N×N matrix system to free vector dynamics, and also provide a geodesic interpretation of Ruijsenaars–Schneider systems. The simplest of the latter, the goldfish equation, is found to represent a flat-space geodesic in curvilinear coordinates.      

Is the Hamiltonian geometrical criterion for chaos always reliable?

It is found that the application of a newly developed geometrical criterion, in which negative eigenvalues of the associated matrix determined by the dynamical curvature of a conformal metric for a Hamiltonian system are used to identify the onset of local instability or chaos, is somewhat problematic in some circumstances. In fact, this criterion is neither necessary nor sufficient for the prediction of instability of orbits on a same energy hypersurface because it is not in good agreement with information on unstable or chaotic behavior given by the maximal Lyapunov exponent in general.     

Differential Galois obstructions for non-commutative integrability

We show that if a holomorphic Hamiltonian system is holomorphically integrable in the non-commutative sense in a neighbourhood of a non-equilibrium phase curve which is located at a regular level of the first integrals, then the identity component of the differential Galois group of the variational equations along the phase curve is Abelian. Thus necessary conditions for the commutative and non-commutative integrability given by the differential Galois approach are the same.     

New Jacobian Elliptic Function Solutions to Modified KdV Equation: I

An extended Jacobian elliptic function expansion method presented recently by us is applied to the mKdV equation such that thirteen families of Jacobian elliptic function solutions including both new solutions and Fu's all results are obtained. When the modulus m→1 or 0, we can find the corresponding six solitary wave solutions and six trigonometric function solutions. This shows that our method is more powerful to construct more exact Jacobian elliptic function solutions and can be applied to other nonlinear differential equations.     

New Jacobian Elliptic Function Solutions to Modified KdV Equation: Ⅰ

An extended Jacobian elliptic function expansion method presented recently by us is applied to the mKdVequation such that thirteen families of Jacobian elliptic function solutions including both new solutions and Fu‘s allresults are obtained. When the modulus m → 1 or 0, we can find the corresponding six solitary wave solutions and sixtrigonometric function solutions. This shows that our method is more powerful to construct more exact Jacobian ellipticfunction solutions and can be applied to other nonlinear differential equations.     

A Class of Calogero Type Reductions of Free Motion on a Simple Lie Group

The reductions of the free geodesic motion on a non-compact simple Lie group G based on the G ₊ × G ₊ symmetry given by left- and right-multiplications for a maximal compact subgroup are investigated. At generic values of the momentum map this leads to (new) spin Calogero type models. At some special values the ‘spin’ degrees of freedom are absent and we obtain the standard BC _n Sutherland model with three independent coupling constants from SU(n + 1,n) and from SU(n,n). This generalization of the Olshanetsky-Perelomov derivation of the BC _n model with two independent coupling constants from the geodesics on G/G ₊ with G = SU(n + 1,n) relies on fixing the right-handed momentum to a non-zero character of G ₊. The reductions considered permit further generalizations and work at the quantized level, too, for non-compact as well as for compact G.      

Adiabatic perturbations for compactons under dissipation and numerically-induced dissipation

Compacton propagation under dissipation shows amplitude damping and the generation of tails. The numerical simulation of compactons by means of dissipative schemes also show the same behaviors. The truncation error terms of a numerical method can be considered as a perturbation of the original partial differential equation and perturbation methods can be applied to its analysis. For dissipative schemes, or when artificial dissipation is added, the adiabatic perturbation method yields evolution equations for the amplitude loss in the numerical solution and the amplitude of the numerically-induced tails. In this paper, such methods are applied to the K(2,2)$K(2\text{,}2)$ Rosenau–Hyman equation, showing a very good agreement between perturbative and numerical results.     

Solution of a diffusion equation of smoke molecules due to constant point source in gravitational field

From both the Langevin equation, including a gravitational term, the Fokker-Planck equation based on the dynamical behavior of Brownian particles, and equation of smoke molecular diffusion due to a constant point source is introduced and is solved by applying the Laplace transformation with the convolution theorem.The solution is expressed by the complementary error function with a mean pathway of smoke molecules affected by gravity and is proved to be reduced to conventional forms by a certain restriction neglecting gravity without any forced term.     

An algebraic approach to discrete mechanics

Using basic ideas from algebraic geometry, we extend the methods of Lagrangian and symplectic mechanics to treat a large class of discrete mechanical systems, that is, systems such as cellular automata in which time proceeds in integer steps and the configuration space is discrete. In particular, we derive an analog of the Euler-Lagrange equation from a variational principle, and prove an analog of Noether's theorem. We also construct a symplectic structure on the analog of the phase space, and prove that it is preserved by time evolution.     

Elliptic Solutions to Difference Non-Linear Equations and Related Many-Body Problems

We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for τ-functions. Starting from a given algebraic curve, we express the τ-function and the Baker–Akhiezer function in terms of the Riemann theta function. We show that the elliptic solutions, when the τ-function is an elliptic polynomial, form a subclass of the general algebro-geometric solutions. We construct the algebraic curves of the elliptic solutions. The evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the Ruijsenaars-Schneider many body system. The zeros obey equations which have the form of nested Bethe-ansatz equations, known from integrable quantum field theories. We discuss the Lax representation and the action-angle-type variables for the many body system. We also discuss elliptic solutions to discrete analogues of KdV, sine-Gordon and 1D Toda equations and describe the loci of the zeros. Received: 15 May 1997 / Accepted: 7 September 1997     

One-dimensional approximation to stochastic lattice system with strong coupling

One-dimensional approximation to stochastic lattice system with strong coupling is derived. For a special periodic coupling, the existence of a rotation number is proved, which yields the frequency locking of the system. And by the one-dimensional approximation system, an approximation to the rotation number is also derived.