Fermat's principle of least time and Kepler's second law

[theophoretos]

While studying the history of classical mechanics I noticed that the primary motor for most of the early equations is the principle of conservation framed firstly as the law of inertia and then as the principle of least time (Fermat) and then as... But while trying to regain for the principle of least time its status as a conservational principle I noticed its similarity with Kepler's second law of equal area in equal time. In fact the Snell's law seems to be "equal distance in equal time" except that the distance here is not the distance traveled by light but the distance on the y-axis only. But to prove it involves some algebra impossible to solve, hence only assumed. If someone is interested in this topic or in the origin of classical mechanics please visit my webpage on this at http://www.geocities.com/theophoretos/fermat.html
and maybe show me if I "assumed" correctly. Thank you everyone.

 

[ZapperZ]

While studying the history of classical mechanics I noticed that the primary motor for most of the early equations is the principle of conservation framed firstly as the law of inertia and then as the principle of least time (Fermat) and then as... But while trying to regain for the principle of least time its status as a conservational principle I noticed its similarity with Kepler's second law of equal area in equal time. In fact the Snell's law seems to be "equal distance in equal time" except that the distance here is not the distance traveled by light but the distance on the y-axis only. But to prove it involves some algebra impossible to solve, hence only assumed. If someone is interested in this topic or in the origin of classical mechanics please visit my webpage on this at http://www.geocities.com/theophoretos/fermat.html
and maybe show me if I "assumed" correctly. Thank you everyone.

What you are discovering is a separate formulation of classical mechanics called the Principle of Least Action. This formulation is different from Newtonian concept of "force". In the Principle of Least Action, the dynamics of the system is based on the energy state and the constraints of motion via calculus of variation. It is out of this formulation that we get the Lagrangian/Hamiltonian in classical and quantum mechanics.

Edwin Taylor of MIT has been a very strong advocate of teaching this principle at the very early stages of a physics undergraduate program, maybe even earlier than teaching them Newton's laws and force concepts. [See his homepage at http://www.eftaylor.com/leastaction.html ]

However, I wouldn't call this principle as the "origin" of classical mechanics, because this principle did not become a coherent concept till much later. BTW, your description of Kepler's law via this principle is something most physics undergrad will encounter sooner or later in their advance mechanics courses. Classical mechanics text such as Marion's has a rather complete derivation of this.

Zz.

 

[theophoretos]

a thanks to ZZ for the hinting forward at H's and L's least action. I'm still trying to understand Lagrange right now. and for the eftaylor.com, and -- the Kepler's law has been formulated already by a principle of least...? what's Marion's first name? much to learn.

 

[turin]

Edwin Taylor of MIT has been a very strong advocate of teaching this principle at the very early stages of a physics undergraduate program, maybe even earlier than teaching them Newton's laws and force concepts.

I think that's asking for trouble. Physics is abstract enough for most people as it is.

 

[ZapperZ]

I think that's asking for trouble. Physics is abstract enough for most people as it is.

But not according to him, if you read one of his AJP papers. He claims that the concept of "energy" is less abstract than the concept of "force". Not everyone, of course, agrees with that.

Zz.

 

[arildno]

But not according to him, if you read one of his AJP papers. He claims that the concept of "energy" is less abstract than the concept of "force". Not everyone, of course, agrees with that.

Zz.

But still, you would have to introduce the requisite maths of calculus of variations/concepts of function spaces.

Or does he wish to skimp on the maths, and only present the necessary results in a "that's just the way it is"-manner?

(BTW, I agree fully with him that the Hamiltonian formalism is by far the most elegant approach to physics)

 

[turin]

I agree that the Hamiltonian formalism is more elegant, but the issue is pedagogy, right? The most blatant example of the danger of introducing physics through the Hamiltonian formalism that comes to mind is that of electromagnetism.

Students can learn that the Hamiltonian is the total energy, for instance (which is how I learned it at first). But, to be more general, you have to obfuscate the Hamiltonian to an abstraction that is not necessarily the total energy. Then, even if one can come to accept that, one is still faced with taking the so called canonical variables as fundamental, when there is, often enough, no clear physical reason to do so.

Perhaps, if there were two different tracks for students of physics, one more emperical and experimental, and the other more theoretical and mathematical, then the introduction through the Hamiltonian may be elegant. But, as has been pointed out, you don't just jump into it in the first year of college unless you have a rather non-traditional mathematical background. I believe the actual consequence of this policy would be simply to replace physics courses with math courses, and that students would wind up encountering the Hamiltonian at the same time whether they're doing the experimental or theoretical track. If they come at it in a physical context, then they can appreciate it in terms of all the physically intuitive things that they have learned. If they come at it in a mathematical context, then they can appreciate the elegance.