On the geometry of Hamiltonian vector fields

. It is known that vector fields play an important role in many areas of mathematics and technology, in particular, in the theory of dynamical systems. Hamiltonian dynamical systems are generated by Hamiltonian vector fields. The paper studies the geometry of a singular foliation of a four-dimensional Euclidean space generated by the orbits of two Hamiltonian vector fields. It is proved that the regular leaf of this foliation is a surface of positive normal curvature and nonzero normal torsion .


Introduction
Let M be a Riemannian (smooth) manifold of dimension n .Definition 1.A partition F of a Riemannian manifold M by path-connected immersed submanifolds L  is called a singular foliation of M if it verifies condition: for each leaf L  and each vector p v T L   at the point p there is X XF  such that () X p v  , where p TL  is the tangent space of the leaf L  at the point p , XF is the module of smooth vector fields on M tangent to leaves ( XF acts transitively on each leaf) [1][2][3][4].
If the dimension of L is maximal, it is called regular, otherwise L is called singular.It is known that orbits of vector fields generate singular foliation.Let us consider a set of vector fields () D V M  of the Lie algebra of all smooth (class C  ) vector fields ()  VMand the smallest Lie sub algebra containing D by ( ).AD Let () be an integral curve of the vector field X with the initial point x for 0, t  which is defined in some region () Ixof real line.Definition 2. The orbit () Lxof a system D of vector fields through a point x is the set of points y in M such that there exist 12 , ,..., k t t t R  and vector fields 12 , ,..., where k is an arbitrary positive integer.
The fundamental result in study of orbits is Sussmann theorem [4], which asserts that every orbit is an immersed submanifold of .M we obtain an involutive distribution : ( ) x then the distribution : ( ) is completely integrable by the Frobenius theorem.If the dimension dim ( ) x AD depends on x , then we can use Hermann theorem for distributions of variable dimension, which gives condition for the complete integrability of distributions which is finitely generated [5,6].
Jacobi Identity: {{ , }, } {{ , }, } {{ , }, } 0 (Here  denotes the ordinary multiplication of real-valued functions.)A manifold M with a Poisson bracket is called a Poisson manifold, the bracket defining a Poisson structure on .M The notion of a Poisson manifold is slightly more general than that of a simplistic manifold, or manifold with Hamiltonian structure; in particular, the underlying manifold M need not be even-dimensional.This is borne out by the standard examples from classical mechanics.
Let M be the Euclidean space m R , with coordinates F p q z and ( , , ) H p q z are smooth functions, we define their Poisson bracket to be the function: This bracket is clearly bilinear and skew-symmetric; the verifications of the Jacobi identity and the Leibniz rule are simple exercises.
in which i and j run from 1 to n , when t and The corresponding flow is obtained by integrating the system of ordinary differential equations , , 0 There is a fundamental connection between the Poisson bracket of two functions and the Lie bracket of their associated Hamiltonian vector fields, which forms the basis of much of the theory of Hamiltonian systems.It is well known following theorem:

On the geometry of Hamiltonian vector fields
Let us consider a family of , D X X  vector fields on four-dimensional Euclidean space 4 E with Cartesian coordinates 1 2 1 2 , , , p p q q where 1 1 The family of orbits of the vector fields ( 1) is a singular foliation regular leaf of which is a surface with a positive normal curvature and with nonzero normal torsion.
Proof.First, we note that for Lie bracket of the vector field 12 , XXit holds   12 , 0. XX  follows from Hermann theorem this system generates completely integrable distribution The vector field generates following one parametrical group of transformations   ( , , , ) , , , , p p q q p cht q sht p cht q sht p sht q cht p sht q cht      generates following one parametrical group of transformations: , , )  , ,  , ,  p p q q p chs q shs p p shs q chs q s R     We to find the invariant functions of the groups generated by vector fields.It is easy to check the following equalities ( ) 0, ( ) 0, ( ) 0, ( ) 0 for the functions: ( , , , ) , ( , , , ) I p p q q p q I p p q q p q    It follows the functions 12 , IIare invariant functions [6].
This invariant functions give us a family of two dimensional surfaces Let L is a regular connected surface defined by equations (2).For a point pL  the orbit () Lp of the family of vector fields contained in .L According to Sussmann theorem [4,7] the orbit () Lp is a two-dimensional manifold and therefore it is an open subset of L This implies that the surface L is a union of orbits.Due to the connectedness of L , we get ( ) .L p L  Now we can check metric characteristics of the surface .L Let us recall some notions on the geometry of two dimensional surface in a fourdimensional Euclidean space 4 .E Consider on a two-dimensional surface F at the point x some direction given by non- zero vector .
t The vector t and the normal plane N of the surface at the point x define a hyper plane E that intersects over the validity of F along some curve  .The curve  is called the normal section at the point x in the direction .t According to its construction the curve  is a three-dimensional curve.Curvature  


of the curve  at the point x is called, respectively, the normal curvature and normal torsion of the surface at the point x in the direction t .Geometry of two dimensional surfaces in the four dimensional Euclidean space is studied by many authors [8][9][10].
We will parameterize the surface by the following equations where ,   is the inner product.
We need two normal vectors to find coefficients of second quadratic forms From this formula we have that normal curvature at the any point x(x1,x2) in any direction t(t1,t2) is a positive and equal to:  Now we calculate normal torsion at the point x(x1,x2) in the direction t(t1,t2) by the formula from and we have following: H p p q q p q  In fact the vector field X1 generates Hamiltonian system which is completely integrable in the sense of Liouville.
Invariant functions are functionally independent first integrals of this Hamiltonian system and following equations

Conclusion
The paper studies the foliation generated by the family of orbits of vector fields.It is shown that regular leaves are two-dimensional manifolds of positive normal curvature and nonzero normal torsion.

Definition 4. A Poisson bracket on a smooth manifold M is an operation that assigns
.., and 0 otherwise.
Hamiltonian vector field, 2) they are functionally independent on M , that is, almost everywhere on M their E3S Web of Conferences 458, 09013 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/2023458090134) the vector fields sgradfi are complete, that is natural parameter on their integral trajectories is defined on the whole number line.into connected components of joint level surfaces of the integrals 1 ,..., n ff is called The Liouville foliation corresponding to the completely integrated system [6] Since 1 ,..., n ff is preserved by sgradH, each leaf of the Liouville foliation is invariant surface.Liouville foliation is consists of regular leaves (which fill almost all M) and special leaves (a subset of zero measure).
1 ,..., n ff are first integrals of sgradH ij ff  for any i and j , 2n M Now we can find two-second quadratic forms of the regular surface.Coefficients of the first of them is calculated by the formula [5,8,9]nown that Gauss torsion of two-dimensional surface in E 4 is calculate by the formula[5,8,9]: