Mechanical Expressions Beta Extended
 Mechanical Expressions is a new program which allows you to create explicit mathematical models of mechanical phenomena.
Based
on feedback from beta testers, we have added a number of features,
including Matlab support, export of differential equations, forces
aligned with vectors and additional capabilities for our HTML5
JavaScript export.
We have built these into a new Beta which is available for download here. The Beta period is extended to the end of March 2014.


How to... embed a Euclid's Muse model in a web page
 People
have asked us about the simulations which appear on the Mechanical
Expressions web pages. For example, the double pendulum on the features page.
Yes, this is a simulation created with Mechanical Expressions and it is embedded using an iFrame.
Now these simulations live on our server, but it is possible to create an iFrame which links to a simulation on Euclid's Muse. A double pendulum simulation, for example, is here.
There
are 2 methods of embedding it in an iFrame, but before you do, check
the licensing for the simulation and make sure your use conforms. Most
models on Euclid's Muse are licensed as Creative Commons Attribution,
which means you can do what you want with them, so long as you attribute
the author. The license type is shown in a small icon beside the
app's author. Click on it to get a description of the license. Method 1: Click the Embed button and copy the iframe html into your webpage. This will create an iFrame linking to the Euclid's Muse version.
Method 2: Click the Download App
button. This will give you the full html for the simulation.
You can put this on your website and link to it from within an
iFrame in your document.

Problem of the Month 
This month, a geometry problem which dates back to Archimedes.
CD is the diameter of a circle, and FB is perpendicular to CD. Tangents to the circle at D and B intersect at E.
Show that CE bisects BF.
Is there an analogous theorem for ellipses? How about hyperbolas and parabolas?
An app which lets you explore this problem is here.

A Solution to December's Problem

Points A, B and C lie on a smooth closed curve such that triangle ABC
has sides with maximal sum of squares. Show that the medians of
ABC are normal to the curve.
An app which lets you explore this problem is here.
We'll use a trick from Mark Levi's Mathematical Mechanic and Mechanical Expressions to solve this problem.
The
trick is this, if we insert springs of equal stiffness between AB, BC
and CA, then the potential energy of the springs is proportional to the
sum of squares of their extension. If we give our springs 0
natural length, then the potential energy is proportional to the sum of
squares of their lengths. When the mechanical system is in
equilibrium, the potential energy is a minimum (or maximum for an
unstable equilibrium). Hence if we can use physical arguments to
determine an equilibrium position, we will have solved our optimization
problem.
We can make a model in Mechanical Expressions. We
use a generic parametric curve (f(T),g(T)) and put three points at
parametric locations s, t and u on the curve. We add springs with 0
natural length and stiffness k. Mechanical Expressions will
compute the reaction force in the parameters s,t and u. At
equilibrium the forces should all be 0.
Examining
the reaction in s (above) we see that it is proportional to the dot
product of the tangent at B with the sum of the vectors BA and BC. Hence
the median of the triangle (which is (BA+BC)/2) is perpendicular to the
tangent.
This is clearly also an equilibrium position for the same set of springs attached to the sides of the tangent triangle.
The problem of finding the point with minimum sum of squares to the sides of a triangle is solved by the Symmedian Point, the reflection of the medians in the angle bisectors. 



