Understanding Jacobian in relation to physics

[cboyce]

I'm working with a game physics engine that uses Jacobians to resolve contact forces. It's been a few years since my physics and linear algebra classes (where we didn't get to Jacobian matrices), so what I'm reading about Jacobians is fairly overwhelming. Most of what I can find are fairly formal definitions, without any examples about what I'm specifically looking for. Can someone give me a couple simple examples about how Jacobians would apply in physics contact resolution, or point me to a resource that does? Or are they complex enough that I need to relearn all the linear algebra leading up to them to understand how to use them?

 

[log0]

I am not an expert in this topic, but what you are trying to do is to solve a constraint matrix by linear approximation(thus jacobian).

Here is an example:
http://www.gamedev.net/community/forums/topic.asp?topic_id=512994

 

[cboyce]

Thanks for the link, I think that gives me a good idea where I need to start to understand them.

 

[Studiot]

Don't know if this helps but if you have a simple function of a single variable
such as y = f(x) you can differentiate this to get the slope.
As there is only one direction involved, there is only one slope to chose from.
Variously we write f'(x) or dy/dx etc.

When you are dealing with a function of several variables, as you must be, there are many directions to chose from, all with different slopes available.

The Jacobian is a method of handling this, which is why the matrix contains an array of partial differential coeffiecients. If you like it is a method of resolving the slopes into as many suitable directions as are needed.
Yes linear algebra theory confirms that this is the same number as the number of independent variables.

 

[cboyce]

I've been doing quite a bit of reading, and I've been trying to get an intuitive sense of the constraints in play. Per the example of a holonomic function at http://en.wikipedia.org/wiki/Holonomic#Examples", I can intuitively understand that x^2+x^3 - L = 0 constrains a point to someplace on a circle. But then it says given

, L is the distance between the two positions r_i and r_j, but wouldn't r_i -r_j already give you the distance between the two positions?

 

[alxm]

But then it says given

, L is the distance between the two positions r_i and r_j, but wouldn't r_i -r_j already give you the distance between the two positions?

In one dimension, yes. But note the boldface: $$\vec{r_i}$$ and $$\vec{r_j}$$ are vectors here, so $$\vec{r} = \vec{r_i} - \vec{r_j}$$ is the vector specifying the difference between the two, squaring it here is the http://en.wikipedia.org/wiki/Dot_product" of the resulting vector and itself, i.e. the x-component squared plus the y component squared and so on. So this works for any number of dimensions, by the pythagorean theorem $$\sqrt{x^2 + y^2}$$ is the distance in 2d, $$\sqrt{x^2 + y^2 + z^2}$$ in 3d and so on.