Physicist Amok

Last night it occurred to me that, although I knew Landau’s derivation of conservation of energy in Lagrangian mechanics, I had never seen it done from Newton’s third law. A few minutes of scribbling produced a proof that might be of use to some poor physics student somewhere.

The overarching theme is Noether’s theorem. Consider some problem described by a differential equation (or an action principle, but for now let’s stay with differential equations), which has some continuous symmetry. The rotations that leave a square unchanged are a discrete symmetry. All displacements in time of a system which does not depend on any particular position in time are a continuous symmetry. Essentially, if we can construct a continuous function from the transformations to the real numbers, then it is a continuous symmetry.

Our program: change coordinates in the differential equation until the symmetry transformations we’re interested in are represented by a displacement in a single coordinate. For time displacement, this is simple. For spatial displacement, we use Cartesian coordinates with one axis along the direction of displacement. For rotation, we use polar coordinates.

Shift the coordinate infinitesimally, and Taylor expand to first order. For any solution of the differential equation, the zeroth order expansion cancels. Taylor’s remainder theorem makes the second and higher orders smaller than the first order, so to have invariance, the first order must vanish. The invariance must hold for different infinitesimal displacements, so the first order coefficient must vanish, and we may multiply this by something nonzero to get a cleaner form at the end. The integral of the first order coefficient with respect to the coordinate is a constant, then we integrate everything in sight by parts and occasionally use the original differential equation to substitute for things until we find a convenient form. The last part sounds vague, and it is. I don’t know if there is some algebraic structure that will yield an algorithm for getting clean, useable forms for conservation laws. How did I find this particular set of integrations and multipliers? I played around until it worked.

Conservation of energy comes from invariance under a displacement in time. Begin with Newton’s third law, F(x(t),t) = m D^2 x(t) . I use for the total time derivative. This is not the same as the partials, as $DF(x(t),t) = \partial_0 F(x(t),t) \cdot Dx(t) + \partial_1 F(x(t), t)$ (where $\partial_n$ is the partial derivative with respect to the $n^{\textrm{th}}$ argument). Let $t\rightarrow t + \delta t$ .

$mD^2 x(t+\delta t)$		$F(x(t+\delta t), t+\delta t)$
$mD^2 x(t) + m\delta t D^3 x(t)$		$F(x(t), t) + \delta t DF(x(t), t)$

The first terms on each side are equal by the original differential equation, and they cancel. Now we multiply by sides by x(t) and rearrange to get

$\delta t [ m x(t) \cdot D^3 x(t) - x(t) \cdot DF(x(t), t) ] = 0$

This must hold for different, nonzero values of $\delta t$ , so the quantity in brackets must be zero.

Aside: There is a subtlety to note: it is also zero before we threw in the x(t) , so it remains zero after we multiply by x(t) , so long as the particle isn’t at infinity. If we were not already garunteed this, then we would have to set up the coordinates where x(t) was never zero. This requires symmetry under spatial displacement (which yields conservation of momentum), and the fact that in any finite time interval there are points which are not visited by the path. The whole of the path over infinite time may visit all the points in the space (this is called ergodicity), but we can break the whole of time into connected subsets, each of which have an unvisited point, prove it on each subset, and then stitch them together. Spaces where we can do this slicing and stitching are called manifolds, and we’ll use this trick later.

Now we integrate with respect to time. Integration by parts on the first term yields (everything is multiplied by , but we’ll just put it back in later):

$\int x(t) \cdot D^3 x(t) dt$		$\int D(x(t) \cdot D^2 x(t)) dt – \int Dx(t) \cdot D^2 x(t) dt$
		$x(t) \cdot D^2 x(t) – \int D( \frac{1}{2}(Dx(t) \cdot Dx(t)) )$
		$x(t)\cdot D^2 x(t) – \frac{1}{2} Dx(t) \cdot Dx(t)$

There’s kinetic energy. The other term will cancel after we analyze the second integral.

$\int x(t) \cdot DF(x(t), t) dt$		$\int D(x(t) \cdot F(x(t),t)) dt – \int Dx(t) \cdot F(x(t), t) dt$
		$\int D(x(t) \cdot m D^2 x(t)) dt – \int Dx(t) \cdot F(x(t), t) dt$
		$m x(t) \cdot D^2 x(t) – \int Dx(t) \cdot F(x(t), t) dt$

where the next to last line follows by replacing the force with the mass times accelleration from Newton’s third law. Now we combine these, putting that we neglected in the first part back in. The [Unparseable or potentially dangerous latex formula. Error 5 : 573x40]

There’s conservation of energy. Now a few remarks to tie things up:

Remark. The second term looks unfamiliar. Try changing variables so that position is independent and time depends on it. This must be done carefully! If the path x(t) intersects itself, we break the integration into a sum of integrals over intervals of time where the path does not self-intersect, and change variables separately in each one so that t(x) is well defined. On any such interval , we get a term that looks like $\int_I x \cdot F(x, t(x)) dx$ , which is more familiar.

Remark. If our force has the form $F(x, t) = – \partial_0 \mathcal{U}(x,t)$ , then the second integral takes the form $-\mathcal{U}(x(t), t) + \int \partial_1 \mathcal{U}(x(t), t) dt$ . For potentials $\mathcal{U}$ which are independent of time, this is particularly useful as the second integral vanishes. For many oscillating potentials, the integral takes on an average value over long times.

For extra credit, go learn the quantum calculus analog of Taylor expansion (John Baez has a nice description) and do the same thing for discrete symmetries.

madhadron :: Oct.18.2007 :: Uncategorized :: Comments Off

There’s been a lot of controversy around James Watson over the last couple of days, to the point where emails have been flying around the Rockefeller University talking about boycotting his receipt of the Lewis Thomas Prize for Writing. I haven’t read his book, I find his statements somewhat ridiculous, and worse, I don’t think he actually made that big a contribution, even aside from stealing data. I think two points are enough:

1. Schrodinger wrote What is life? in 1944, nine years before the Watson and Crick paper, and laid out what was necessary for a molecule that carried hereditary structure. With the discovery of DNA methylation and other modifications, it’s become more and more clear that the pure double helix isn’t the sole carrier of hereditary character. Yes, it was nice to actually know what the molecule was, but it wasn’t one of the most important discoveries of the 20th century. The evolutionary synthesis was. The new quantum mechanics was. The structure of this molecule was not. It was a technical tour de force at the time, but Watson and Crick didn’t even do that.

2. I see far too many descriptions of figuring out the rough structure of DNA as “cracking the secrets of life.” It did no such thing. Before Watson and Crick, we knew that it was a polymer (Levene, 1919); we knew that it carried heritable structure (Avery, 1943); we knew that heritable structure was laid out linearly along the polymer (Morgan, no later than 1915). Crick laid out the “central dogma” of molecular biology in 1957, and everyone dug in and figured out how the encodings worked. The structure of DNA was a pleasant thing to have in your head for doing this, but the classical geneticists had been doing very well without such structures for many years. Unfortunately, having such a structure immediately made lots of people confuse genes (heritable traits, alleles of which are selected in evolution) with ORFs (pieces of DNA transcribed into RNA).

madhadron :: Oct.18.2007 :: Uncategorized :: 4 Comments »

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Conservation of Energy

James Watson

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