finding a particular solution differential equation systems

Integrable Hamiltonian Systems on Lie groups: Kowalewski type. Optimal Control and Geometry: Integrable Systems Cambridge Studies in Advanced Mathematics 154 Cambridge University Press: Cambridge, UK, 2016. Note sur l’intégration des équations différentielles de la Dynamique, présenté au Bureau des Longitudes le 29 juin 1853. Calculus of Variations and Partial Differential Equations of the First Order Second (revised) English Translation Chelsea Publishing Co.: White River Junction, VT, USA, 1982. Mathematical Methods of Classical Mechanics Springer: Berlin/Heidelberg, Germany, 1989. Vorlersungen Über Dynamic Druck und Verlag: Berlin, Germany, 1884. Les Méthodes Nouvelles de la Mécanique Célecte Tome I Gauther-Villars: Paris, France, 1892. Sur les inégalités séculaires des moyens mouvemens des planétes. This leap to infinite dimensional Hamiltonians and related hierarchies of commuting Hamiltonians further illustrates the relevance of Lie algebraic methods in the theory of integrable systems.

Since this function can be also expressed as 1 2 ∫ 0 L κ 2 ( s ) d s where κ ( s ) is the geodesic curvature of the projected curve in the underlying symmetric space, elastic curves appear naturally in this setting (see also for related results). In particular, Heisenberg’s magnetic equation and Schroedinger’s non-linear equation appear as Poisson equation associated with f 0 ( g ( s ) ) = 1 2 ∫ 0 L | | d Λ d s ( s ) | | 2 d s where Λ ( s ) = g − 1 ( s ) d g d s ( s ), | | Λ ( s ) | | = 1. More generally it was shown in that the space of periodic horizontal curves of fixed length L in the isometry group G over a three dimensional space of constant curvature can be given a structure of an infinite dimensional Poisson manifold relative to which some famous equations of mathematical physics appear as Poisson equations associated with geometric invariants of curves on the base space. Curves p ( t ) defined on some interval are called elastic if for each t ∈ ( 0, T ) there exits an interval ⊂ over which the elastic energy of p ( t ) is minimal relative to the boundary conditions p ˙ ( t 0 ) and p ˙ ( t 1 ) .Īpart from the above remarks, there is another spectacular property of elastic curves that makes them special: elastic curves appear as soliton solutions in the non-linear Schroedinger equation. The integral 1 2 ∫ 0 T κ 2 ( s ) d s is known as the elastic energy of the curve p ( t ) . Here κ ( t ) = | | d D p ( t ) d t ( p ˙ ( t ) ) | |, where d D p ( t ) d t denotes the covariant derivative along p ( t ).

It consists of finding a continuously differentiable curve p ( t ) in M in an interval, with its tangent vector p ˙ ( t ) of unit length and its covariant derivative bounded and measurable in that satisfies fixed tangential directions p ˙ ( 0 ) = v 0, v 0 ∈ T p ( 0 ) M and p ˙ ( T ) = v 1, v 1 ∈ T p ( T ) M along which the integral 1 2 ∫ 0 T κ 2 ( s ) d s minimal among all other curves that satisfy the same boundary conditions. We will now introduce another optimal problem intertwined with the rolling problem. We also introduce a rolling geodesic problem that admits a spectral representation on symmetric Riemannian spaces and we then show the relevance of the corresponding integrals on the nature of the curves whose elastic energy is minimal. We then show that the classical integrals of motion associated, with the Kowalewski top, the two-body problem of Kepler, and Jacobi’s geodesic problem on the ellipsoid can be all derived from the aforementioned Hamiltonian systems. As a consequence, the spectral invariants associated with the spectral curve produce a large number of integrals of motion, all in involution with each other, that often meet the Liouville complete integrability criteria. We show that there is a particular sub-class of these Hamiltonians that admit a spectral representation on the Lie algebra of G. We then use the Maximum Principle of optimal control to generate the Hamiltonians whose solutions project onto the extremal curves of these problems. This paper follows Jacobi’s prophetic observations by introducing certain “meta” variational problems on semi-simple reductive groups G having a compact subgroup K. Since there is no general rule for finding the right choice, it is better to introduce special variables first, and then investigate the problems that naturally lend themselves to these variables. Jacobi summarized difficulties with differential equations by saying that the main problem in the integration of differential equations appears in the choice of right variables.

Finding a particular solution differential equation systems