# Proof of the Law of Conservation of Energy

1. Nov 29, 2009

### 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 dont 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/$$\partial$$Xi.$$\dot{}X$$ +$$\partial$$U/$$\partial$$Yi.$$\dot{}Y$$ (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

2. Nov 30, 2009

### vanesch

Staff Emeritus
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...)

3. Nov 30, 2009

### Hootenanny

Staff Emeritus
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.

4. Nov 30, 2009

### 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 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

Last edited: Nov 30, 2009
5. Nov 30, 2009

### Hootenanny

Staff Emeritus
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.

6. Nov 30, 2009

### dE_logics

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

7. Nov 30, 2009

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

8. Nov 30, 2009

### 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.

9. Nov 30, 2009

### superfahd

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!

10. Dec 1, 2009

### 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!