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 

E-mail: 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 x-y 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 mass-spring 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 mass-spring 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 y-v 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 y-v 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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