Frobenius theorem and invariants for Hamiltonian systems
نویسنده
چکیده
We apply Frobenius integrability theorem in the search of invariants for one-dimensional Hamiltonian systems with a time-dependent potential. We obtain several classes of potential functions for which Frobenius theorem assures the existence of a two-dimensional foliation to which the motion is constrained. In particular, we derive a new infinite class of potentials for which the motion is assurately restricted to a two-dimensional foliation. In some cases, Frobenius theorem allows the explicit construction of an associated invariant. It is proven the inverse result that, if an invariant is known, then it always can be furnished by Frobenius theorem.
منابع مشابه
Frobenius kernel and Wedderburn's little theorem
We give a new proof of the well known Wedderburn's little theorem (1905) that a finite division ring is commutative. We apply the concept of Frobenius kernel in Frobenius representation theorem in finite group theory to build a proof.
متن کاملOn the Hamiltonian structure of evolution equations
The theory of evolution equations in Hamiltonian form is developed by use of some differential complexes arising naturally in the formal theory of partial differential equations. The theory of integral invariants is extended to these infinitedimensional systems, providing a natural generalization of the notion of a conservation law. A generalization of Noether's theorem is proved, giving a one-...
متن کاملExtended affine Weyl groups and Frobenius manifolds
We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F (t1, . . . , tn) of WDVV equations of associativity polynomial in t1, . . . , tn−1, exp tn.
متن کاملThe Sign-Real Spectral Radius for Real Tensors
In this paper a new quantity for real tensors, the sign-real spectral radius, is defined and investigated. Various characterizations, bounds and some properties are derived. In certain aspects our quantity shows similar behavior to the spectral radius of a nonnegative tensor. In fact, we generalize the Perron Frobenius theorem for nonnegative tensors to the class of real tensors.
متن کاملTri - Hamiltonian Structures of The Egorov Systems of Hydrodynamic Type ∗
was initiated by Dubrovin and Novikov [2] and continued by Mokhov and Ferapontov [5]. In the present paper, we prove a theorem on the existence of three Hamiltonian structures for a diagonalizable Hamiltonian hydrodynamic type system (1) having two physical symmetries, with respect to the Galilean transformations and to scalings, and possesses some additional properties, namely, the metric of t...
متن کامل