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Hamiltonian Systems

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INSITTUTO DE ESTUDIOS SUPERIORES DE TAMAPULIPAS

M?XICO MMVI

Hamiltonian Systems

Maple 10 Document

Prepared by

Prof. David Macias Ferrer

E-mail: david.macias@iest.edu.mx

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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