Systems of second order ODE's

by peterbone
Tags: order, systems
peterbone is offline
May24-05, 07:25 AM
P: 13
Hello. First post here.

I'm trying to write a program (from scratch) to simulate a double inverted pendulum (a cart with 2 pendulums one on top of the other). The system is modelled with a system of 3 second order ODE's, which I need to solve numerically using Runge Kutta. I know how to solve a system of first order ODE's numerically but not a system of second order ODE's. The equations are shown in this paper (there's no point in me re-writing them here):
(equations 4 to 6)

So can anyone tell me how to go about solving this initial value problem numerically? I have looked in many books but can only find examples of systems of first order equations and single second order equations.


Peter Bone
Phys.Org News Partner Science news on
Review: With Galaxy S5, Samsung proves less can be more
Making graphene in your kitchen
Study casts doubt on climate benefit of biofuels from corn residue
HallsofIvy is online now
May24-05, 07:35 AM
Sci Advisor
PF Gold
P: 38,890
Each second order differential equation is equivalent to two first order equations so you could write this system as six first order equations.
peterbone is offline
May24-05, 08:59 AM
P: 13
Thanks, but I don't know how to go about reducing the order of coupled differential equations because the 3 unkown variables x, theta1 and theta2 all appear in the same equations.

arildno is offline
May24-05, 10:35 AM
Sci Advisor
HW Helper
PF Gold
P: 12,016

Systems of second order ODE's

Suppose you've got a second order diff.eq system:
[tex]\frac{d^{2}\vec{Y}}{dt^{2}}=\vec{F}(y_{1},..y_{n},\dot{y}_{1},....,\dot {y}_{n},t), \vec{Y}(t)=(y_{1}(t),....,y_{n}(t)),\dot{y}_{m}\equiv\frac{dy_{m}}{dt}, 1\leq{m}\leq{n}; m,n\in\mathbb{N}[/tex]

Now, define:
[tex]x_{i}=y_{i}, x_{n+i}=\frac{dy_{i}}{dt}=\frac{dx_{i}}{dt}, 1\leq{i}\leq{n}[/tex]
Thus, we may form the 1-order differential system of 2n equations:
[tex]G_{i}(\vec{X},t)=x_{n+i}, 1\leq{i}\leq{n}[/tex]
[tex]G_{i}(\vec{X},t)=F_{i-n}(\vec{X},t), n<i\leq{2n}[/tex]
peterbone is offline
May25-05, 06:06 AM
P: 13
Thankyou, that was helpful.
I also found this site which explains the whole process of simulating a single inverted pendulum and includes the reduction stage.
I should be able to use the same method for the double inverted pendulum.

Register to reply

Related Discussions
Solving Systems of ODE's Calculus & Beyond Homework 4
Differential Equations - First Order Systems Calculus & Beyond Homework 6
system of ODE's Linear & Abstract Algebra 1
dirst order ODE's with x & expontials Differential Equations 3
ebook on ODE's? Science & Math Textbook Listings 0