# Analytical Solution to Normal Forms of Hamiltonian Systems

## Abstract

**:**

## 1. Introduction

**J**is the $2n\times 2n$ Poisson matrix

**J**=$\left[\begin{array}{cc}0& {I}_{n}\\ -{I}_{n}& 0\end{array}\right]$ and $\mathbf{y}=(q,p)$. Furthermore, the Hamiltonian H possesses an equilibrium ${y}^{0}$ (i.e., $\frac{\partial H}{\partial \mathbf{y}}({y}^{0})=0$) at the origin in ${R}^{2n}$. If not, we make the shift $y=\widehat{y}+{y}^{0}$ zero. For many dynamical systems, the Hamiltonian H represents the energy in the system. Furthermore, the Hamiltonian H will be in the form $H(q,p)=T+V$, where T is the kinetic energy and V is the potential energy of the system and is a function of q alone. The energy is constant, if a Hamiltonian does not depend explicitly on the time t:

## 2. Methodology

- $A(v,v)=0$ .
- $A(v,w)=-A(w,v)$ .

#### 2.1. Generating Function

#### 2.2. Normal Form

#### 2.3. Computing a Generating Function W

## 3. Example

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Allahem, A.
Analytical Solution to Normal Forms of Hamiltonian Systems. *Math. Comput. Appl.* **2017**, *22*, 37.
https://doi.org/10.3390/mca22030037

**AMA Style**

Allahem A.
Analytical Solution to Normal Forms of Hamiltonian Systems. *Mathematical and Computational Applications*. 2017; 22(3):37.
https://doi.org/10.3390/mca22030037

**Chicago/Turabian Style**

Allahem, Ali.
2017. "Analytical Solution to Normal Forms of Hamiltonian Systems" *Mathematical and Computational Applications* 22, no. 3: 37.
https://doi.org/10.3390/mca22030037