What is the significance of Calculus of Variations in Classical Mechanics?

[evinda]

Hello! (Wave)

Could you give me some information about the subject Calculus of variations?
What is it about? What backround is needed?

 

[Ackbach]

Ah, I love CoV! The subject got off the ground historically with the brachistochrone problem: given two points, one not directly above the other, in normal gravity, what is the shape of the curve of fastest descent? (The word "brachistochrone" is from brachistos and chronos, Greek words meaning shortest time.) One of the Bernoulli brothers posed this question during the time of Newton. Newton solved it, but didn't attach his name to the solution. When the Bernoulli brother saw the solution, he said, "I see the paw of the lion" - meaning Newton.

But Newton used methods that were not so capable of generalization. The brachistochrone problem involves finding the curve $y(x)$ that minimizes the integral
$$t=\int_{x_1}^{x_2}\sqrt{\frac{1+(y')^2}{2gy}} \, dx.$$
The answer is an inverted cycloid.

Euler and Lagrange made extremely important contributions, producing the Euler-Lagrange equation:
$$\pd{f}{y}-\frac{d}{dx} \left( \pd{f}{y'} \right)=0,$$
which is of fundamental importance in the subject. This differential equation produces the extremal for the functional
$$\int_{x_0}^{x_1} f(x,y,y') \, dx.$$

CoV has been applied ferociously to classical mechanics, both in the Lagrangian formulation and in the Hamiltonian formulation. One of the more interesting applications of CoV in classical mechanics is the problem of the spinning top. You can predict the precession properties of the top using CoV.

It's also used extensively in control problems - particularly optimal control problems. While in many engineering applications you simply slap a PID controller at the problem, such a controller is almost never going to be optimal. You might not need the optimal controller, but if you do, there will likely be some CoV in your future.

It's a beautiful area of mathematics, with ongoing research.

If you're interesting in studying it, I would recommend Troutman's book - it's a good intro and uses convexity very cleverly to get some important early results.

The background required varies depending on the level of the book you're studying. Troutman requires up through multivariable calculus and linear algebra, and I would recommend mathematical maturity as well. A book like Ewing requires functional analysis and graduate-level real analysis!

 

[Ackbach]

Oh, and Differential Equations is essential, as well. The typical sophomore-level course is just fine.

 

[evinda]

Is it related to physics and graphs?

 

[Ackbach]

It's strongly related to physics. A junior-level physics course in classical mechanics, e.g., will be all over CoV in order to do Lagrangian dynamics.

As for graphs, if you mean graph theory, it would be more indirectly related. You could probably dream up some problem where they relate, but the basic issue is that graph theory is much more in the discrete math line, whereas CoV is squarely in the continuous math line.

 

[RLBrown]

Hello! (Wave)

Could you give me some information about the subject Calculus of variations?
What is it about? What backround is needed?

Here is a link to lesson 2 of a Classical Mechanics on-line course at Stanford.
At location 1:26:15 Lenny defines COV and introduces it's importance to Classical Mechanics. That application is calculation of "least action" in finding the trajectory of point masses in a system.

This shows an example of the importance of COV to physics. If you find it useful, go back and watch the whole lesson.