Friday, November 7, 2008

The Calculus of Variations and Hamilton's Principle from the Top Down

This installment of “It’s Obvious. Not!” looks at:

Book: “Classical Dynamics of Particles and Systems”

Edition: third

Authors: Jerry B. Marion and Stephen T. Thornton

Publisher: Harcourt Brace Jovanovich

Page: 172-177

Chapter Five in the third edition is titled "The Calculus of Variations" The book does a great job of giving a very detailed bottom-up derivation of Euler's equation and a second form of Euler's equation. I had a much easier time with the material once I figured out that Euler's equation was actually the goal of the derivation and how Euler's equation is used. Since the top-down view made things simple for me, I decided to post it here for other top-down thinkers.

Chapter five is simply building a set of tools to be used in chapter six with regard to Hamilton's Principle and Lagrangian mechanics. So, maybe the first question should be, 'Why are Hamilton's Principle and Langrangian mechanics important?'

Newtonian Mechanics essentially says that given information about all the forces acting on a system, the equations of motion can be derived for that system. It doesn't, however, say that the derivation will be easy. Especially in systems where there are forces due to constraints, (a particle moving on the surface of a sphere for example), the derivation can be quite complex.

Very broadly stated, Hamilton's Principle says that given two states of a physical system, nature will choose the allowable path between the two states that minimizes the time integral of the difference between the potential and kinetic energies of the two states. As it turns out, this alternate view of mechanics makes solving for the equations of motion of constrained systems much simpler in some cases. But this alternative view leads to finding equations, y(x), that minimize integrals that look like:

where f depends on x, y as a function of x and y prime, the first derivative of y with respect to x.

The book launches into a rather detailed and abstract derivation regarding varying y(x) to find a maximum or minimum value for the above integral. It's easy to get caught up in the derivation and wander off into the weeds, and I did. But, the derivation is headed somewhere. It turns out that if you have a y(x) that will produce a max or min of the integral, then it obeys Euler's equation:

And that's the real crux of the matter! If you wind up with an integral in the form of
rather than actually solving the integral equation, you can plug f into Euler's equation and solve the resulting differential equation to find the y(x) that produces a minimum.

The book presents a concrete example where the minimum of the integral:

is found by substituting f:
into Euler's equation. I'll cover that example in more detail in my next post. But, for now I just wanted to reveal the point of the rather lengthy initial derivation from the book. Euler's equation makes solving integral equations of the form shown above simpler!

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