Why is kinetic energy conserved?

[voko]

Interesting about Huygens. Underrated. His collision dynamics was clearly a lot better than that of Descartes!

Yes, he is clearly underrated. It was a very clever examination of a collision from two frames moving with respect to each other; very modern. I think his proof of conservation of energy was the first time ever proof of that. His other works are also first rate. It is a bit of tragedy that Newton wrote such an excellent work on dynamics that it overshadowed everything that existed just before it. It should, however, be noted that Huygens was among the very few names mentioned in Newton's work, so Newton clearly realized the Huygens's contribution to science.

Regrading your proof, one problem with your approach is that you treat velocities as scalars. That is OK when you form first and seconds order invariants because that can be trivially re-written in a vectorial form. But as you go to third and higher degrees, there is no standard interpretation of such invariants. This actually might be a fundamental reason why such invariants are not known - they simply cannot be formulated in a physically meaningful way.

 

[Philip Wood]

An unusually generous acknowledgment by Newton?

I'm treating a head-on, straight line collision, and my velocities are actually velocity components (dot products with the unit vector in one direction along the straight line). As such they can be raised to any power, but I do see the point of your remark. It's all very well to cube a velocity component, but cubing the velocity itself isn't a defined operation. I think this may well go to the heart of the matter.

 

[ChrisVer]

One more problem the OP should solve for him/her self is that in classical mechanics, the Lagrangian and its formalism is equivalent to the Newton's law... It's just another way of seeing it (since by Hamilton's principle you'll get the Newton's laws), which sometimes can be more simple or more complicated-depending on the problem.

So "why or what is the Lagrangian?" and feeling the explanations coming from it as "weird", while on the other hand accepts the Newtonian approach, shows exactly that.

So in this case, I'd refer you into studying Lagrangian from classical mechanics books, and see how they deal with it, and how the Newton's laws come naturally out of it.

 

[rkmurtyp]

voko. Profuse apologies. The notation is a bit naff. Sigma mu is the sum of initial momenta, Sigma mv is the final momenta. w is the velocity of the new frame relative to the old.

Interesting about Huygens. Underrated. His collision dynamics was clearly a lot better than that of Descartes!

I read about Huygens analysis of elastic collisions. I understand that Descartes analysed the collision process using conservation of 'quantity of motion' (product of mass and speed - a scalar quantity), and Huygens corrected it by changing conservation of quantity of motion to conservation of moementum. Some say that Huygens solution applies to one dimensional collisions only.

In the light of the above would you please let me know:

1. Where we go wrong by using Descartes analysis instead of Huygens analysis, with a numerical example?
2. Does Huygens solution not apply to multidimentional elastic collisions?

 

[voko]

Huygens used "quantity of motion" as well. He, however, proved that the quantity of motion can change in a collision, while Descartes posited it would be constant. He also stated, although I am not sure whether a proof was ever published, that momentum is conserved, even though he did not use the term "momentum"; specifically he stated that the common centre of gravity of two or more bodies moves uniformly in a straight lines before and after collision, and he stated that he could prove that for spherical bodies, for elastic and inelastic collisions, for head-on and oblique collisions.

For a particular example, say there is a stationary 2 kg ball, and a 1k ball moving toward the first one at 2 m/s. The total quantity of motion is 2 kg.m/s before collision. According to Descartes, it should stay that way after the collision.

Huygens described a method of determining the speeds after the collision: $$ {2 \ \text{kg} + 1 \ \text{kg} \over 2 \cdot 1 \ \text{kg} } = {2 \ \text{m/s} \over v} $$ so v = 4/3 m/s, where v is the speed the 2 kg ball after the collision; the speed of the 1 kg ball is such that the relative speeds before and after the collision be equal, so it is 2/3 m/s. The resultant speeds are consistent with conservation of energy and momentum.

The total quantity of motion after collision is 10/3 kg.m/s, which is greater than the initial quantity 2 kg.m/s.

A remark on terminology. Some languages now use "quantity of motion" in sense of "momentum". This message uses the historic definition, where "speed" is strictly the magnitude of velocity and is always positive.