INSITTUTO DE ESTUDIOS SUPERIORES DE TAMAPULIPAS
RED DE UNIVERSIDADES AN?HUAC
M?XICO MMVI
Hamiltonian Systems
Maple 10 Document
Prepared by
Prof. David Macias Ferrer
Email: david.macias@iest.edu.mx
Madero City, Mexico
URL: http://www.geocities.com/dmacias_iest/MyPage.html
Goals
 To show the principal characteristics of the Non Linear Differential Equations Systems, in particular: Hamiltonian Systems
 To use the Maple Tools to find Hamiltonian Function associated.
 To show the powerful Maple 10 graphics tools to visualize some trajectories on phase plane.
Introduction
We consider the following system:
where f and g are functions of two variables defined in a region Ω on xy plane. If the following relationship:
is completed then, the system is called Hamiltonian System. Additionally, exist a function such that:
or
in other words, this quantity is conserved.
Examples
Example 1

(3.1.1) 
For this system:

(3.1.2) 

(3.1.3) 
so that:
> 

therefore, the differential equations system is an Hamiltonian system.
Hamiltonian Function is given by:
In other hand, the phase portrait with one trajectory in are given by:
You can see that, exist two equilibrium points, in and , this is:

(3.1.4) 
A quantitative analysis is possible if they are analyzed for each equilibrium point, through the Jacobian matrices and its eigenvalues, this is
For :
Jacobian matrix is:

(3.1.5) 
eigenvalues are:

(3.1.6) 
as the eigenvalues are real and distinct then, the equilibrium point is a saddle point.
For :
Jacobian matrix is:

(3.1.7) 
eigenvalues are:

(3.1.8) 
as the eigenvalues are cojugate imaginary then, the equilibrium point can be a spiral source, a spiral sink or a center. In other words, the nature of this equilibrium point cannot be determined analytically.
Returning to the Hamiltonian function, the following graph shows the existence of periodic solutions around the point :
If we compare these graphs, we have:
You can see that, the phase portrait and level curves of the Hamiltonian surface are similar, and their graphs state a lot about the behavior of the solutions.
Level curves are:
Example 2
One important area of application of the Hamiltonian systems theory are in the field of Mechanical, in particular: Mechanical Vibrations. In this example we will show the free vibration of a massspring system.
If we consider that a 0.0625 slug is attached to a spring whose constant is 4 and suppose that there is no external force and no damping then, this phenomenon can be modeled through a second order linear differential equation, that is:
where represent the vertical displacements of the mass attached to the spring. This equation governs the motion of a vibrating mass at the end of a vertical spring. This phenomenon is known as Simple Harmonic Motion.
If then, the differential equation given can be replaced by:

(3.2.1) 
whose general solution is given by:

(3.2.2) 
Of the equation (2.12) we have:

(3.2.3) 

(3.2.4) 
so that:
therefore the massspring system is an Hamiltonian system.
Hamiltonian is given by:
In Physics, the expression given by (2.17) is called Energy Function.
You can see that, exist only one equilibrium point: in in yv plane.

(3.2.5) 
In this case, a quantitative analysis is possible if the equilibrium point can be analyzed through the Jacobian matrix and its eigenvalues, this is:
For the unique point we have:
Jacobian matrix is:

(3.2.6) 
eigenvalues are:

(3.2.7) 
as the eigenvalues are cojugate imaginary then, the equilibrium point can be a spiral source, a spiral sink or a center. In this case, is evident that the equilibrium point is a center.
Returning to the Hamiltonian function, the following graph shows the existence of periodic solutions around the point :
You can see that, the phase portrait and level curves of the Hamiltonian surface are similar
Level curves of the Energy Function is given by:
The contours lines are coincident with the trajectories on yv phase plane.
Bibliography
Blanchard P.,Devaney R.L.,Hall G.R.,"Differential Equations", First Edition, Brooks Cole Publishing and ITP Company, USA, 1998.
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