---
abstract: |
    In this study, Lagrangian and Hamiltonian systems, which are mathematical models of mechanical systems, were introduced on the horizontal and the vertical distributions of tangent and cotangent bundles. Finally, some geometrical and physical results related to Lagrangian and Hamiltonian dynamical systems were deduced.\
    [**Keywords:**]{} Tangent-Cotangent Bundles, Lagrangian-Hamiltonian Systems.\
    [**PACS:**]{} 02.04; 03.40.
author:
- |
    Mehmet Tekkoyun [^1]\
    [Department of Mathematics, Pamukkale University,]{}\
    [20070 Denizli, Turkey]{}
title: '**Complex Dynamics Effect on Distributions**'
---

Introduction
============

In modern differential geometry, the tangent-cotangent bundles of any differentiable manifold $M$ are seen to be phase-spaces of velocity and momentum of a given configuration space. Therefore Lagrangian and Hamiltonian systems in Classical Mechanics are explained by means of basic structures on the bundles so that this structures can be given by Liouville vector field $C$, Liouville form $\lambda $, tangent structure $J$, complex structure $F$, vertical distribution $V$, horizontal distribution $H$ and semispray $X,$ . The following symbolical equation expresses the dynamical equations for both Lagrangian and Hamiltonian systems: $$i_{X}\Phi =\digamma  \label{1.1}$$ If anybody studies the Lagrangian systems then equation (\[1.1\]) is the intrinsical form of Euler-Lagrange equations, where $\Phi =\Phi
_{L}=-dd_{F}L $ and $\digamma =dE_{L}$ such that $L$ defined by $L:TM\rightarrow \mathbf{R} $ is Lagrangian function, $E_{L}\mathbf{\ }$given by $E_{L}=C(L)-L$ is energy function associated to $L.$

If one studies the Hamiltonian theory then equation (\[1.1\]) is the intrinsical form of Hamiltonian equations, where $\Phi =\phi _{\mathbf{H}}=-d\lambda $ , $\digamma =d\mathbf{H}$ and $\mathbf{H}$ is a $C^{\infty }-$ function such that $\mathbf{H}:T^{*}M\rightarrow \mathbf{R.}$ Mathematical expressions of mechanical systems are generally given by the Hamiltonian and Lagrangian systems. These expressions, in particular geometric expressions in mechanics and dynamics, are given in some studies [@mcrampin; @nutku; @deleon; @deleon1]. It is known that Lagrangian distribution on symplectic manifolds is used in geometric quantization and a connection on a symplectic manifold is an important structure to obtain a deformation quantization [@etayo]. Paracomplex geometry and the framework of para-Kählerian manifolds being geometric model of generalized Lagrange spaces were introduced [@cruceanu]. Para-complex analogues of the Lagrangians and Hamiltonians were obtained in the framework of para-Kählerian manifold and the geometric conclusions on a para-complex dynamical systems were obtained [@tekkoyun2005]. As determined references and there in, real-complex-paracomplex geometry and mechanical-dynamical systems were analyzed successfully, but they have not dealt with complex dynamical systems on horizontal distribution and vertical distribution of tangent-cotangent bundles of any manifold $M$.

Therefore, here, Euler-Lagrange and Hamiltonian equations related to complex dynamical systems on the distributions used in obtaining geometric quantization have been obtained.

Preliminaries
=============

In this study all mappings are considered of the class $C^{\infty },$expressed by the words ”differentiate” or ”smooth”.  The indices $i$,$j$,...run over set $\{1,..,n\}$ and Einstein convention of summarizing is adopted over all this paper. $\mathbf{R}$, $\mathcal{F}(TM)$, $\chi (TM)$ and $\chi (T^{*}M)$ denote the set of real numbers, the set of real functions on $TM$, the set of vector fields on $TM$ and the set of 1-forms on $T^{*}M,$ respectively.

Basic Structures
----------------

In this subsection, some definitions were derived taking from [@radu]. Let $TM$ be tangent bundle of a real $n$-dimensional differentiable manifold $M$. Then it will be denoted a point of $M$ by $x$ and its local coordinate system by $(U,\varphi )$ such that $\varphi (x)=(x^{i}).$ Such that the projection $\pi :TM\rightarrow M$, $\pi (u)=x,$ a point $u\in TM$ will be denoted by $(x,y)$, its local coordinates being $(x^{i},y^{i})$. There are the natural basis $(\frac{\partial }{\partial x^{i}},\frac{\partial }{\partial y^{i}})$ and dual basis $(dx^{i},dy^{i})$ of the tangent space $T_{u}TM$ and the cotangent space $T_{u}^{*}(TM)$ at the point $u\in TM$**,** respectively$.$ Consider the $\mathcal{F}(TM)-$ and $\mathcal{F}(T^{*}M)-$ linear mappings $J:\chi (TM)\rightarrow \chi (TM)$and $J^{*}:\chi (T^{*}M)\rightarrow \chi (T^{*}M)$ given by $$\begin{array}{l}
J(\frac{\partial }{\partial x^{i}})=\frac{\partial }{\partial y^{i}},\,J(\frac{\partial }{\partial y^{i}})=0,\,
\end{array}$$ and $$\begin{array}{l}
J^{*}(dx^{i})=dy^{i},J^{*}(dy^{i})=0.
\end{array}$$ The tangent space $V_{u}$ to the fibre $\pi ^{-1}(x)$ in the point $u\in TM$ is locally spanned by $\{\frac{\partial }{\partial y^{1}},..,\frac{\partial }{\partial y^{n}}\}$. Therefore, the mapping $V$: $u\in
TM\rightarrow V_{u}\subset T_{u}TM$ provides a regular distribution generated by the adapted basis $\{\frac{\partial }{\partial y^{i}}\}.$Consequently, $V$ is an integrable distribution on $TM$. $V$ is called the vertical distribution on $TM$. Let $N$ be a nonlinear connection on $TM$. $N $ is characterized by $v$, $h$ vertical and horizontal projectors. We consider the vertical projector $v:\chi (TM)\rightarrow \chi (TM)$ defined by $v(X)=X,\,\forall \,X\in \chi (VTM);$ $v(X)=0,\forall \,X\in \chi (HTM).$ Similarly, the mapping $H$: $u\in TM\rightarrow H_{u}\subset T_{u}TM$provides a regular distribution determined by the adapted basis $\{\frac{\delta }{\delta x^{i}}\}.$ Consequently, $H$ is an integrable distribution on $TM$. $H$ is called the horizontal distribution on $TM$. There is a $\mathcal{F}(TM)-$linear mapping $h:\chi (TM)\rightarrow \chi (TM),$ for which $h^{2}=h,$ $Ker$ $h=\chi (VTM).$ Any vector field $X\in \chi (TM)$ can be uniquely written as follows $X=hX+vX=X^{H}+X^{V}$. Therefore $X^{H}$ and $X^{V}$ are horizontal and vertical components of vector field $X$. Therefore, any vector field $X$ can be uniquely written in the form $$\begin{array}{l}
X=X^{H}+X^{V}
\end{array}$$ such that $$\begin{array}{l}
X^{H}=X^{i}(\frac{\partial }{\partial x^{i}}-N_{j}^{i}(x,y)\frac{\partial }{\partial y^{j}}),\,\,\,X^{V}=X^{i}N_{j}^{i}(x,y)\frac{\partial }{\partial
y^{j}}
\end{array}$$ where $N_{j}^{i}$ is local coefficient of nonlinear connection $N$ on $TM.$

$(\frac{\delta }{\delta x^{i}},\frac{\partial }{\partial y^{i}})$ is a local basis adapted to the horizontal distribution $HTM$ and the vertical distribution $VTM$. Then $(dx^{i},\delta y^{i})$ is dual basis of $(\frac{\delta }{\delta x^{i}},\frac{\partial }{\partial y^{i}})$ basis. We have $$\begin{array}{l}
\frac{\delta }{\delta x^{i}}=\frac{\partial }{\partial x^{i}}-N_{j}^{i}(x,y)\frac{\partial }{\partial y^{j}}.
\end{array}$$ and $$\begin{array}{l}
\delta y^{i}=dy^{i}+N_{j}^{i}(x,y)dx^{j}.
\end{array}$$ $F$ is an almost complex structure on $TM.$ $F^{*}$ is the dual structure of $F.$ For the operators $h,v,F,F^{*}$ we get $$\begin{array}{l}
h+v=I;\,\,F^{2}=-I;\,F^{*2}=-I \\
h(\frac{\delta }{\delta x^{i}})=\frac{\delta }{\delta x^{i}};\,h(\frac{\partial }{\partial y^{i}})=0;\,v(\frac{\delta }{\delta x^{i}})=0;v(\frac{\partial }{\partial y^{i}})=\frac{\partial }{\partial y^{i}},\, \\
F(\frac{\delta }{\delta x^{i}})=-\frac{\partial }{\partial y^{i}};\,\,F(\frac{\partial }{\partial y^{i}})=\frac{\delta }{\delta x^{i}}, \\
F^{*}(dx^{i})=-\delta y^{i};\,F^{*}(\delta y^{i})=dx^{i}.
\end{array}$$ **Lagrangian Dynamical Systems**

In this section, Euler-Lagrange equations for Classical Mechanics are made by means of almost complex structure $F$ under the consideration of the basis $\{\frac{\delta }{\delta x^{i}},\,\frac{\partial }{\partial y^{i}}\}$ on distributions $HTM$ and $VTM$ of tangent bundle $TM$ of manifold $M.$ Let $(x^{i},y^{i})$ be its local coordinates. Let semispray be the vector field $X$ given by $$\begin{array}{l}
X=X^{i}\frac{\delta }{\delta x^{i}}+\stackrel{.}{X}^{i}\frac{\partial }{\partial y^{i}},\stackrel{.}{\,\,X}^{i}=X^{i}N_{j}^{i}
\end{array}
\label{3.1}$$ where the dot indicates the derivative with respect to time $t$. The vector field denoted by $C=F(X)$ and expressed by $$\begin{array}{l}
C=-X^{i}\frac{\partial }{\partial y^{i}}\stackrel{.}{+X}^{i}\frac{\delta }{\delta x^{i}}
\end{array}
\label{3.2}$$ is called *Liouville vector field* on the bundle $TM$. The maps given by $\mathbf{T,P}:TM\rightarrow \mathbf{R}$ such that $\mathbf{T}=\frac{1}{2}m_{i}(x^{i})^{2},\mathbf{P}=m_{i}gh$ are called *the kinetic energy* and *the potential energy of the mechanical system,* respectively.* *Here* *$m_{i},g$ and $h$ stand for mass of a mechanical system having $m$ particles, the gravity acceleration and distance to the origin of a mechanical system on the tangent bundle $TM$, respectively. Then $L:TM\rightarrow \mathbf{R}$ is a map that satisfies the conditions; i) $L=\mathbf{T-P}$ is a *Lagrangian function, ii)* the function given by $E_{L}=C(L)-L$ is *a Lagrangian energy*. The operator $i_{F}$ induced by $F$ and shown by $$\begin{array}{l}
i_{F}\omega (X_{1},X_{2},...,X_{r})=\sum_{i=1}^{r}\omega
(X_{1},...,F(X_{i}),...,X_{r})
\end{array}
\label{3.3}$$ is said to be *vertical derivation,* where $\omega \in \wedge
^{r}{}TM,$ $X_{i}\in \chi (TM).$ The *vertical differentiation* $d_{F}
$ is defined by $$\begin{array}{l}
d_{F}=[i_{F},d]=i_{F}d-di_{F}
\end{array}
\label{3.4}$$ where $d$ is the usual exterior derivation. For an almost complex structure $F$, the closed fundamental form is the closed 2-form given by $\Phi
_{L}=-dd_{F}L$ such that $$\begin{array}{l}
d_{F}:\mathcal{F}(TM)\rightarrow {}T^{*}M
\end{array}
\label{3.5}$$ Then we have $$\begin{array}{l}
\Phi _{L}=-(\frac{\delta }{\delta x^{j}}dx^{j}+\frac{\partial }{\partial
y^{j}}\delta y^{j})(-\frac{\partial L}{\partial y^{i}}dx^{i}+\frac{\delta L}{\delta x^{i}}\delta y^{i}) \\
\,\,\,\,\,\,\,\,\,\,=\frac{\delta (\partial L)}{\delta x^{j}\partial y^{i}}dx^{j}\wedge dx^{i}-\frac{\delta ^{2}L}{\delta x^{j}\delta x^{i}}dx^{j}\wedge \delta y^{i}+\frac{\partial ^{2}L}{\partial y^{j}\partial y^{i}}\delta y^{j}\wedge dx^{i}-\frac{\partial (\delta L)}{\partial y^{j}\delta
x^{i}}\delta y^{j}\wedge \delta y^{i}.
\end{array}
\label{3.6}$$ Let $X$ be the second order differential equation (semispray) determined by **Eq.** (\[1.1\]). Then we get $$\begin{array}{l}
i_{X}\Phi _{L}=\Phi _{L}(X)=X^{i}\frac{\delta (\partial L)}{\delta
x^{j}\partial y^{i}}\delta _{i}^{j}dx^{i}-X^{i}\frac{\delta (\partial L)}{\delta x^{j}\partial y^{i}}dx^{j}-X^{i}\frac{\delta ^{2}L}{\delta
x^{j}\delta x^{i}}\delta _{i}^{j}\delta y^{i}+\stackrel{.}{X}^{i}\frac{\delta ^{2}L}{\delta x^{j}\delta x^{i}}dx^{j} \\
+\stackrel{.}{X}^{i}\frac{\partial ^{2}L}{\partial y^{j}\partial y^{i}}\delta _{i}^{j}dx^{i}-X^{i}\frac{\partial ^{2}L}{\partial y^{j}\partial y^{i}}\delta y^{j}-\stackrel{.}{X}^{i}\frac{\partial (\delta L)}{\partial
y^{j}\delta x^{i}}\delta _{i}^{j}\delta y^{i}+\stackrel{.}{X}^{i}\frac{\partial (\delta L)}{\partial y^{j}\delta x^{i}}\delta y^{j}.
\end{array}
\label{3.7}$$ Since the closed 2-form $\Phi _{L}$ on $TM$ is the symplectic structure, it is found $$\begin{array}{l}
E_{L}=C(L)-L=-X^{i}\frac{\partial L}{\partial y^{i}}+\stackrel{.}{X}^{i}\frac{\delta L}{\delta x^{i}}-L
\end{array}
\label{3.8}$$ and hence $$\begin{array}{l}
dE_{L}=-X^{i}\frac{\delta (\partial L)}{\delta x^{j}\partial y^{i}}dx^{j}+\stackrel{.}{X}^{i}\frac{\delta ^{2}L}{\delta x^{j}\delta x^{i}}dx^{j}-\frac{\delta L}{\delta x^{j}}dx^{j}-X^{i}\frac{\partial ^{2}L}{\partial
y^{j}\partial y^{i}}\delta y^{j}+\stackrel{.}{X}^{i}\frac{\partial (\delta L)}{\partial y^{j}\delta x^{i}}\delta y^{j}-\frac{\partial L}{\partial y^{j}}\delta y^{j}
\end{array}
\label{3.9}$$ With the use of **Eq.** (\[1.1\]), considering **Eqs.**(\[3.7\]) and (\[3.9\]) we get $$\begin{array}{l}
X^{i}\frac{\delta (\partial L)}{\delta x^{j}\partial y^{i}}\delta
_{i}^{j}dx^{i}-X^{i}\frac{\delta (\partial L)}{\delta x^{j}\partial y^{i}}dx^{j}-X^{i}\frac{\delta ^{2}L}{\delta x^{j}\delta x^{i}}\delta
_{i}^{j}\delta y^{i}+\stackrel{.}{X}^{i}\frac{\delta ^{2}L}{\delta
x^{j}\delta x^{i}}dx^{j} \\
+\stackrel{.}{X}^{i}\frac{\partial ^{2}L}{\partial y^{j}\partial y^{i}}\delta _{i}^{j}dx^{i}-X^{i}\frac{\partial ^{2}L}{\partial y^{j}\partial y^{i}}\delta y^{j}-\stackrel{.}{X}^{i}\frac{\partial (\delta L)}{\partial
y^{j}\delta x^{i}}\delta _{i}^{j}\delta y^{i}+\stackrel{.}{X}^{i}\frac{\partial (\delta L)}{\partial y^{j}\delta x^{i}}\delta y^{j} \\
=-X^{i}\frac{\delta (\partial L)}{\delta x^{j}\partial y^{i}}dx^{j}+\stackrel{.}{X}^{i}\frac{\delta ^{2}L}{\delta x^{j}\delta x^{i}}dx^{j}-\frac{\delta L}{\delta x^{j}}dx^{j}-X^{i}\frac{\partial ^{2}L}{\partial
y^{j}\partial y^{i}}\delta y^{j}+\stackrel{.}{X}^{i}\frac{\partial (\delta L)}{\partial y^{j}\delta x^{i}}\delta y^{j}-\frac{\partial L}{\partial y^{j}}\delta y^{j}
\end{array}
\label{3.10}$$ or $$\begin{array}{l}
X^{i}\frac{\delta (\partial L)}{\delta x^{j}\partial y^{i}}dx^{j}-X^{i}\frac{\delta ^{2}L}{\delta x^{j}\delta x^{i}}\delta y^{j}+\stackrel{.}{X}^{i}\frac{\partial ^{2}L}{\partial y^{j}\partial y^{i}}dx^{j}-\stackrel{.}{X}^{i}\frac{\partial (\delta L)}{\partial y^{j}\delta x^{i}}\delta y^{j}+\frac{\delta L}{\delta x^{j}}dx^{j}+\frac{\partial L}{\partial y^{j}}\delta y^{j}=0
\end{array}
\label{3.11}$$ If a curve denoted by $\alpha :\mathbf{R}\rightarrow TM$ is considered to be an integral curve of $X,$ i.e$.$ $X(\alpha (t))=\frac{d\alpha (t)}{dt}$ then we have the equations $$\begin{array}{l}
\frac{d}{dt}(\frac{\partial L}{\partial y^{i}})+\frac{\delta L}{\delta x^{i}}=0,\,\,\frac{d}{dt}(\frac{\delta L}{\delta x^{i}})-\frac{\partial L}{\partial y^{i}}=0,
\end{array}
\label{3.12}$$ which are named to be a *Euler-Lagrange equations* which are deduced by means of an almost complex structure $F$ on $HTM$ horizontal and $VTM$ vertical distributions.

Thus the triple $(TM,\Phi _{L},X)$ is a *mechanical system* which is* *structured by means of an almost complex structure $F$ and taking into account the basis $\{\frac{\delta }{\delta x^{i}},\frac{\partial }{\partial y^{i}}\}$ on the distributions $HTM$ and $VTM$.

Hamiltonian Dynamical Systems
=============================

In this section, Hamiltonian equations for Classical Mechanics are obtained on the distributions ${}HT^{*}M$ and $VT^{*}M$ of $T^{*}M.$ Suppose that an almost complex structure, a Liouville form and a 1-form on $T^{*}M$ are shown by $P^{*}$, $\lambda $ and $\omega $, respectively$.$ Then we have $$\begin{array}{l}
\omega =\frac{1}{2}(x^{i}dx^{i}+y^{i}\delta y^{i})
\end{array}
\label{4.1}$$ and $$\begin{array}{l}
\lambda =P^{*}(\omega )=\frac{1}{2}(-x^{i}\delta y^{i}+y^{i}dx^{i}).
\end{array}$$ It is known that if $\phi $ is a closed 2- form on $T^{*}M,$ then $\phi _{\mathbf{H}}$ is also a symplectic structure on ${}{}T^{*}M$. If Hamiltonian vector field $X_{\mathbf{H}}$ associated with Hamiltonian energy $\mathbf{H}$ is given by $$\begin{array}{l}
X_{\mathbf{H}}=X^{i}\frac{\delta }{\delta x^{i}}+Y^{i}\frac{\partial }{\partial y^{i}},
\end{array}
\label{4.3}$$ then we deduce $$\begin{array}{l}
\phi _{\mathbf{H}}=-d\lambda =\delta y^{i}\wedge dx^{i}
\end{array}$$ and $$\begin{array}{l}
i_{X_{\mathbf{H}}}\phi =Y^{i}dx^{i}-X^{i}\delta y^{i}.
\end{array}
\label{4.5}$$ Moreover, the differential of Hamiltonian energy is written as follows: $$\begin{array}{l}
d\mathbf{H}=\frac{\delta \mathbf{H}}{\delta x^{i}}dx^{i}+\frac{\partial
\mathbf{H}}{\partial y^{i}}\delta y^{i}.
\end{array}
\label{4.6}$$ By means of **Eq.**(\[1.1\]), using **Eqs.** (\[4.5\]) and (\[4.6\]), the Hamiltonian vector field is calculated to be $$\begin{array}{l}
X_{\mathbf{H}}=-\frac{\partial \mathbf{H}}{\partial y^{i}}\frac{\delta }{\delta x^{i}}+\frac{\delta \mathbf{H}}{\delta x^{i}}\frac{\partial }{\partial y^{i}}.
\end{array}
\label{4.7}$$ Suppose that a curve $$\begin{array}{l}
\alpha {:\,}I\subset \mathbf{R}\rightarrow T^{*}M
\end{array}
\label{4.8}$$ is an integral curve of the Hamiltonian vector field $X_{\mathbf{H}}$, i.e., $$\begin{array}{l}
X_{\mathbf{H}}(\alpha (t))=\frac{d\alpha (t)}{dt},\,\,t\in I.
\end{array}
\label{4.9}$$ In the local coordinates, if it is considered to be $$\begin{array}{l}
\alpha (t)=(x^{i}(t),y^{i}(t)),
\end{array}
\label{4.10}$$ we obtain $$\begin{array}{l}
\frac{d\alpha (t)}{dt}=\frac{dx^{i}}{dt}\frac{\delta }{\delta x^{i}}+\frac{dy^{i}}{dt}\frac{\partial }{\partial y^{i}}.
\end{array}
\label{4.11}$$ Taking into consideration **Eqs.** (\[4.8\]), (\[4.6\]) and (\[4.10\]), we have the equations $$\begin{array}{l}
\frac{dx^{i}}{dt}=-\frac{\partial \mathbf{H}}{\partial y^{i}},\frac{dy^{i}}{dt}=\frac{\delta \mathbf{H}}{\delta x^{i}},
\end{array}
\label{4.12}$$ which are named to be *Hamiltonian equations* which are deduced by means of an almost complex structure $F^{*}$ on the horizontal distribution ${}{}HT^{*}M$ and vertical distribution $VT^{*}M.$

Hence the triple $(T^{*}M,\phi _{\mathbf{H}},X_{\mathbf{H}})$ is shown to be a *Hamiltonian mechanical system* which* *are deduced by means of an almost complex structure $F^{*}$ and using of basis $\{\frac{\delta }{\delta x^{i}},\frac{\partial }{\partial y^{i}}\}$ on the distributions ${}HT^{*}M$ and $VT^{*}M$.

Conclusions
===========

This paper has obtained to exist physical proof of the both mathematical equality given by $TM=HTM\oplus VTM$ and its dual equality$.$ Lagrangian and Hamiltonian dynamics have intrinsically been described by means of an almost complex structure $F^{*}$ being functional on distributions of tangent and cotangent bundles $TM$ and $T^{*}M$ of manifold $M$. Also, we deduce that Euler-Lagrange equations given by (\[3.12\]) turn into Hamiltonian equations defined by (\[4.12\]) taking care into equalities $x^{i}=\frac{\delta L}{\delta x^{i}},$ $y^{i}=\frac{\partial L}{\partial y^{i}}$ and $H=-L,$ and vice versa.

**Discussions**
===============

As is well-known, geometry of Lagrangian and Hamiltonian formalisms give a model for Relativity, Gauge Theory and Electromagnetism in a very natural blending of the geometrical structures of the space with the characteristics properties of these physical fields. Therefore we consider that the equations (\[3.12\]) and (\[4.12\]) especially can be used in fields determined the above of physical.

[9]{} M. Crampin, J. Phys. A: Math. Gen. 14** **(1981) 2567.

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F. Etayo, R. Santanaría, U.R. Trías, Diff. Geom. and its Appl., 24 (2006) 33.

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M. Tekkoyun, Phys. Lett. A, 340 (2005) 7.

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[^1]: E-mail address: tekkoyun@pau.edu.tr; Tel: +902582953616; Fax: +902582953593
