Conservative Fields: Examples

In my previous post on the basic theory behind conservative vector fields, I described some of the mathematical properties these fields will have:

- A conservative field has no curl:    

- By extension, a vector of 3 components (Mi + Nj + Pk) can be checked for conservancy with a component test:

            

- A conservative field can be 'integrated' to find a potential function. Conversely, any gradient of a function is conservative: 

This is all good stuff, but I like to see visual examples whenever possible. When we're talking graphs of vector fields and functions in space, this is entirely possible! I'm going to go through some examples now, using the graphs to emphasize the math.

Conservative Fields: Theory

Next stop: Vector Calculus!

To continue in the present direction--a discussion of Energy methods in Particle Dynamics--it’s necessary to take a quick stop over in mathematician territory. An unfriendly place, it is. Look alive.

The basic idea that will recur in this discussion is that, in certain types of vector fields, the net work done on an object moving from one point to another is independent of the path taken. This is called path-independence, and is a very important property of conservative fields. 

Derivation of the Work/Energy Principle

In my previous post on methods for particle dynamics, I stated that, by integrating Newton’s Second Law over a distance, we could make use of ‘Work/Energy’ methods. This is true as far as it goes, but is certainly not the whole truth. It’s important to realize what’s actually happening, lest we fall into a trap of some sort.

Newton’s Second Law, is, of course, a vector equation:

 So I have to be a little more specific when I say 'Integrate over a distance.' Start tossing around vectors willy-nilly, and you'll end up with results that don't make any sense. Anyway, there are two main questions regarding what needs to happen here: