Proof of the Law of Conservation of Energy

[superfahd]

Hi guys. I've decided to review my physics after a long time through Leonard Susskind's youtube lectures. I'm at lecture 2 and I'm already confused!

in the 1st half hour, he gives a proof of the law of conservation of energy. In the course of this proof he uses the formula: F = - $$\vec{}\nabla$$ U (x, y). Where U(x,y) is the potential energy of a particle at position (x,y). I don't remember any such formula from my secondary school classes. Can someone please explain to me how this formula comes out and what it even means?

Also he then writes: d U(x,y) / dt = $$\Sigma$$_i$$\partial$$U/$$\partial$$X_i.$$\dot{}X$$ +$$\partial$$U/$$\partial$$Y_i.$$\dot{}Y$$ (I'm not sure if I rendered the formula correctly. This latex thing is confusing). How does he arrive to this? I realize that differentiation concepts need a lot of review but I'm only doing this as a hobby so can someone explain it to me as such? Thanks a lot

 

[vanesch]

To be honest, I don't think that you've seen that in high school (depends maybe on the high school in question).

The inverted triangle stands for a vector differentiation operator. If you apply it to a function of 3 variables x, y and z, it produces a vector where the first component is the partial derivative of the function wrt x, the second component is the partial derivative of the same function, but wrt y this time, and the third component is the partial derivative wrt z.

For instance, if U(x,y,z) = x^3*y^2 + 5*y*z

then you get as a first component: 3 x^2 * y^2
as a second component: 2*x^3 * y + 5 * z
and as a third component: 5 * y

So this gives you a vector in every point in space. For instance, in the point (1,2,3) this becomes the vector with components (12,19,10)

(if I didn't make any silly mistake...)

 

[Hootenanny]

Whilst it may not be crucial for your understanding here, it is important to note that the quantity U is not the potential energy, but simply the potential. The two are very closely related, but not identical.

 

[rcgldr]

A simpler version of this, using a single dimension:

http://hyperphysics.phy-astr.gsu.edu/hbase/pegrav.html

I don't see how using math could 'prove' anything in physics though. All we can do is confirm that physics theories closely approximate (as best as we can measure) reality via experiments.

U is not the potential energy, but simply the potential.

U is normally used for potential energy, while Φ or V is used for potential. Wiki link, note the part about gravitational potential:

http://en.wikipedia.org/wiki/Potent...tween_potential_energy.2C_potential_and_force

 

[Hootenanny]

I don't see how using math could 'prove' anything in physics though. All we can do is confirm that physics theories closely approximate (as best as we can measure) reality via experiments.

I disagree. For example, Noether's theorem states that we can obtain a law of conservation if the action of a physical system has a differentiable symmetry. Admittedly, we have to start from somewhere, say from a physical law. However, if we assume that a physical law holds (and has some symmetry), then we can prove a conservation law, for example.

 

[dE_logics]

Proof of law of conservation of energy?...it's an assertion!...well that's what most books say.

 

[Dadface]

I disagree. For example, Noether's theorem states that we can obtain a law of conservation if the action of a physical system has a differentiable symmetry. Admittedly, we have to start from somewhere, say from a physical law. However, if we assume that a physical law holds (and has some symmetry), then we can prove a conservation law, for example.

But aren't the symmetries confirmed by experimental observations as are any physical laws.

 

[SystemTheory]

There was the famous observation which showed that gravity bends light, tending to validate one of Einstein's theories. This was celebrated as front page news the world over.

If I recall correctly, Einstein remarked, "A thousand experiments could not prove the theory, and a single experiment could disprove it. The theory, however, is quite correct."

Maybe my recollection is off but the principle of proving things is based upon models that are internally self-consitent, and open ended theories are tested repeatedly by gaining more and more empirical experience.

 

[superfahd]

A simpler version of this, using a single dimension:

http://hyperphysics.phy-astr.gsu.edu/hbase/pegrav.html

I don't see how using math could 'prove' anything in physics though. All we can do is confirm that physics theories closely approximate (as best as we can measure) reality via experiments.

U is normally used for potential energy, while Φ or V is used for potential. Wiki link, note the part about gravitational potential:

http://en.wikipedia.org/wiki/Potent...tween_potential_energy.2C_potential_and_force

Thanks a lot jeff. This was precisely the definition i was looking for but was having trouble finding. I mean how the heck do you google a formula anyway!

Also now that my curiosity has peaked to an annoying level, I'll probably be bugging you guys about a lot of such stuff as I go on with the lectures!

 

[SystemTheory]

Keyword search gets easier when you recognize more patterns. I usually google a term of art with "wiki" or "hyperphysics" or "nasa" and sometimes use the Image rather than text box. However when you're still learning a new area it can yield confusing information.

For example, in google, try "del operator". You'll get the Wiki and Hyperphysics search links right at the top!