DIFFERENTIAL EQUATIONS AND OSCILLATIONS

<BODY LANG="EN" > <HR><center> <table border=1> <td><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></td> <td><img height=36 width=36 alt="" src="/UCESIcons/tools/up_fade.gif"></td> <td><A NAME="tex2html5" HREF="node1.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-diffeq.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H1>DIFFERENTIAL EQUATIONS AND OSCILLATIONS</H1> <P><IMG WIDTH=409 HEIGHT=291 ALIGN=BOTTOM ALT="figure4" SRC="img1.gif"><P> Many problems in physics are described by differential equations. This is due in part to the basic laws of nature (like Newton's second and Schrödinger equation) being differential in form, <BR><A NAME="SE"> </A><A NAME="newton"> </A><IMG WIDTH=500 HEIGHT=79 ALIGN=BOTTOM ALT="eqnarray10" SRC="img2.gif"><BR> The differential nature of these physical laws in turn may be a reflection of our use of continuous variables like position and probability. (The use of differential equations may also reflect traditionally-trained physicists viewing problems in differential forms) A differential equation can be transformed into an integral equation, but since integral equations are not how we usually learn physics, laws are hardly ever stated originally as integral equations. This may change in the future as computational techniques make integral equation straight forward to solve. See a latter Chapter. <P> The project in this chapter is formulated with differential equations. More specifically the <B>the physical problem</B> is: a <A HREF="node15.html#rel"> non relativistic particle</A> with mass m moves in one dimension under the influence of a <A HREF="node15.html#cons">conservative force</A> F in an <A HREF="node15.html#infr">inertial frame of reference</A>. We want to obtain the displacement <I>x</I>(<I>t</I>), the velocity <IMG WIDTH=75 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline1258" SRC="img3.gif"> and the acceleration <IMG WIDTH=74 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline1260" SRC="img4.gif"> as functions of time. At the begining we will solve the linear oscillator (p=2): <BR><IMG WIDTH=500 HEIGHT=37 ALIGN=BOTTOM ALT="equation24" SRC="img5.gif"><BR> <P> and later the anharmonic( p>2) oscillator. <P> <B>The model</B> consists in using discrete methods to solve differential equations. <P> <B>The method</B> consists of numerical techniques for the integration of differential equations. <P> <B>The implementation</B> is achieved with a program in C language (you can use another language like FORTRAN or Pascal, or programs like Maple or Mathematica). <P> <B>An investigation</B> is proposed for further work on the topic. <P> <P CENTER> <UL> <LI> A linear oscillator <UL> <LI> A linear damped oscillator </UL> <LI> A nonlinear oscillator </UL> <P> <B>The methodology and objectives</B> of this project were designed according to the <A NAME="tex2html1" HREF="/UCES/">(UCES)</A> paradigm. <P> Before starting with the solution of the differential equation for the oscillator, some semantics are introduced. <HR> <B> Author: </B> <A HREF="http://www.physics.orst.edu/~wwwnuc/faculty/Landau.html"> Rubin H. Landau </A> <BR> For more information e-mail: rubin@physics.orst.edu <P> <B>Coauthor</B>: Manuel J. Páez, mpaez@graphy.physics.orst.edu <P> </P><HR><center> <table border=1> <td><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></td> <td><img height=36 width=36 alt="" src="/UCESIcons/tools/up_fade.gif"></td> <td><A NAME="tex2html5" HREF="node1.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-diffeq.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> </BODY>