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:

Kerbal Space Program: Day 1

So, from time to time, certain gaming blogs write 'Game Diaries' documenting their experiences playing, say, Minecraft or Civilization. They're a lot of fun to read, and usually get me in the mood for playing the game in question. I recently discovered an interesting little gem called Kerbal Space Program, which seems like a cartoonish rocket-building game but is actually an incredibly detailed simulation of orbital mechanics. Its misadventures and smiling green spacemen lend themselves to storytelling, so here we go:

Acceleration, Energy, and Momentum

In any Dynamics class, 3 main methods of analysis will be taught -- Acceleration, Energy Methods, and Impulse/Momentum. Within the class, it’s pretty easy to figure out which method the professor wants: If we’re in Chapter 4, it must be Energy, right?

What about later on? While I’m still only through the first-level Dynamics course in my curriculum, I do assist others with the class occasionally (I’m a walk-in tutor for some of the 200/300 level engineering courses here), and haven’t yet developed a good set of rules to delineate between the methods. Here’s a first effort, hopefully I can add more in the future.

Introduction

Hello! My name is Joe, and I'm an Engineering Mechanics student at the University of Wisconsin - Madison. I'm here to discuss the physical and mathematical concepts present in my coursework, with the aim of gaining a better understanding for myself and those who make their way here. With any luck, I'll be able to carry this forward into my professional career, should such a thing ever manifest itself. Until then, look for helpful posts on such banalities as:

- The history and origins of calculus
- Operations in vector fields
- The cross product, and why it's calculated as a determinant

Interesting stuff, right? We'll see. Thanks for stopping by!