In the previous post in this series, we looked at how to model a stabilized inverted pendulum using the control systems design features in Mathematica 8. We were quickly able to simulate a linearly controlled cart-and-pendulum system, and show that it is stable against some fairly large perturbations. But what about a double (or triple or quadruple… ) pendulum? A general n-link pendulum is depicted below. In this post we'll see how to derive the equations of motions for this system, find out whether we can stabilize it with a linear control scheme, and produce some animations of the results.

Can you balance a ruler upright on the palm of your hand? If I concentrate, I can just barely manage it by constantly reacting to the small wobbles of the ruler. This challenge is analogous to a classic problem in the field of control systems design: stabilizing an upside-down (“inverted") pendulum. One of the best things about Mathematica is that it makes specialist areas like control systems accessible to non-specialists. This lets you freely combine and develop new ideas without needing to be an expert in everything. It also makes Mathematica a great platform for learning and exploring new areas. Using the new control systems features (one of several new application areas integrated into Mathematica 8), I've been experimenting with models of stabilized inverted pendulums. I'm no expert in control theory, but you'll see that one doesn't need to be.

During discussions at the International Mathematica User Conference 2009 with bioinformaticians using Mathematica, I learned a lot of very important things---like why protein folding isn't something you can order at the dry cleaner. I also learned that a lot of people seriously dig Mathematica's modeling and automatic interface construction capabilities, which make it easy for them to create interactive applications and simulations. Whether it is protein structure prediction using comparative modeling and fold recognition, or visualizing large-scale sequence alignments, Mathematica makes it fast and accurate. To make it easy for you to check out Mathematica's capabilities for this field, we have designed the Mathematica Solution for Bioinformatics portal. This website, which I researched and created, highlights Mathematica's capabilities and features several case studies, articles, and tutorials to help you get started. One of the cool things I enjoyed researching were the interactive Demonstrations. If you are like me and learn best by looking at an example, there's no better resource for learning how to create interactive applications in Mathematica, because the code used for creating the application is freely available right there.

The global H1N1 outbreak has researchers stepping up their efforts to build a mathematical model that health authorities can use to identify optimal medication strategies for emerging infectious diseases. Zhilan Feng, a mathematics professor at Purdue University, is one of those researchers. Feng, who's collaborating with the Centers for Disease Control and Prevention (CDC), is using Mathematica to develop and analyze a model of the dynamics and medication control of influenza. In this video, she demonstrates why Mathematica is the perfect tool for their work.

I am always intrigued by the many ways people use Mathematica. But it was even more exciting to be a part of the American Chemical Society Fall 2009 National Meeting and hear the true excitement and awe of our users' latest discoveries of what is possible in Mathematica. I also had a lot of fun introducing new users to our software! In August, we traveled to ACS in beautiful Washington, DC, USA. The ACS meeting brought together the largest scientific society and its members' families, colleagues, and students. It provided an ideal venue to demonstrate Mathematica's capabilities in chemistry and chemical engineering. We demonstrated a broad range of features, including Mathematica 7's fully curated chemical, genomic, and proteomic data, built-in parallelization capabilities, and unsurpassed modeling and visualization capabilities. The ability to visualize any data as well as update it on the fly has bridged a gap many researchers and scientists have had to work around when using other tools. You can even rapidly develop and test algorithms as well as generate accurate structural renderings in 2D and 3D using the integrated data, which is easy to retrieve programmatically.

Major growth in air traffic is forcing regulators and traffic management teams within the industry to create and study more efficient flight operations. Mike Ulrey, a member of Boeing's Advanced Air Traffic Management team, is tackling this problem with Mathematica. In this video, Ulrey describes how Mathematica's graphical and visualization capabilities play a crucial role in developing models to analyze and test the safety of new flight operations. "It puts the whole conversation of whether it's safe on a firm quantitative-model basis that enables people to make decisions about whether to go forward," says Ulrey. "They have much better insight and they have confidence in the results."

Judging elements is like choosing a favorite ice cream. Carbon and hydrogen are like vanilla and chocolate, the basis for so many other flavors, but too commonplace to claim as your preferred element. By using the load-on-demand information packages that are readily available in Mathematica, one can better investigate the popularity of the 118 elements available in ElementData by studying how often they occur in the 34,000 chemicals featured in ChemicalData. Of all the elements, hydrogen and carbon unsurprisingly occur most frequently, respectively in 94 and 93 percent of the chemicals. As an organic chemist, my focus has traditionally been on carbon-containing molecules, so I cannot help but view the periodic table from a carbon-centered perspective: how will certain elements affect the behavior of molecules to which they are bonded, and how will they interact with other molecules?

As readers of this blog might know, I have an unhealthy interest in the chemical elements. And Mathematica. The combination means that all my element-related projects are created using almost no tools other than Mathematica. It’s not that I use it because I feel like I should---I certainly use other programs when appropriate (Photoshop, for example). It’s just that Mathematica is the best available tool for many of the things I want to do. Really. My latest project is the website periodictable.com. Let me say right off the bat that this is a fabulous domain name. It took me months of convincing and no small amount of cash to acquire it. It really deserves to host the definitive online periodic table reference website, not just another run-of-the-mill online periodic table, of which there are thousands. Take a quick look at it here.