"Calculus is so last century"? Hardly!

Two months ago a WSJ op-ed ('Calculus Is So Last Century') by Tianhul Michael Li caught the eyes of many mathematicians as well as physicists (and I can assure you, many space and solar physicists). In it Li basically argued that calculus is primarily of the last century as it "is the handmaiden of physics - invented by Newton to explain planetary and projectile motion."

Adding:

"While its place at the core of math education may have made sense for Cold War adversaries engaged in a missile and space race, 'Minute Man' and 'Apollo' no longer occupy the same prominent role in national security and prosperity they once did."

Which makes one wonder what alternative universe and planet this guy is living on? Surely not the one that I'm on!

A feature story in Saturday's NY Times, for example, expatiated on the latest developments including "hypersonic targeted warheads" not only under development by the U.S. but China and Russia as well. The U.S. has a non-nuclear version but the take is that the Russians and Chinese are developing nuclear warheads- independent of missiles- that can be hurled into space then come down on an enemy with little or no warning time. This is in contrast to Cold War ICBMs that general gave about 30 minutes advance notice, unless launched by submarines)

Most alarming is the drive by a number of nations, including the U.S., China and Russia, to develop low yield nukes which - as the article put it- could make their use much more likely than the "mutually assured destruction"  (MAD) model of the past allowed. Indeed, the Russian already have a military posture firmly in place that permits the use of tactical nukes (with 10- 25 kt warheads) in the event that NATO forces overwhelm Russian positions in Eastern Europe. (And cruise missiles are also being outfitted with low yield nukes.)

The current U.S. proposal to "modernize" its nuclear warheads is also seen as possibly destabilizing the existing system and rousing the Chinese and Russians to action in their own warhead development.

Far from national security having switched away from H-bomb Cold War worries, we are evidently entering a new era wherein nuclear "swords of Damocles" literally hang over our collective heads, in the form of targeted hypersonic warheads, or even nuclear-armed satellites,

None of these systems will use linear algebra (the math Li mainly advocates teaching) to reach their targets, but plain old brute differential calculus, of a form I described in earlier blog posts, e.g. from May 6, 2013 (the rocket problem).

Surely then, Calculus isn't passé if it is still central to the ultimate form of nullifying any national security or 'prosperity': nuclear war,

Meanwhile, in fields as diverse as astrophysics, astronomy, quantum mechanics and plasma physics - not to mention climate science -  differential calculus is as critical as ever. While it is true algorithms and numerical simulations, models have been developed in all these fields, it is also true that differential calculus underscores that development and continued tweaking of those models depends on the actual math, not just crunching numbers into numerical or statistical data sets.

Li is correct when he asks at the outset 'Can you remember the last time you did calculus?'  But that question could also be asked about linear algebra or finite mathematics, or a course in multivariate statistics.  The point is that any form of advanced learned in high school or college will fall by the wayside if it is not used in some way later on, irrespective of how it was taught.

Li argues that he isn't saying calculus shouldn't be taught, as it is "great mental training" which is most gracious of him. But if nuclear warheads are going to be set off in detached form and land on my head, I'd be curious as to the math underlying the dynamics. I don't want to just read about it in the NY Times.

Where I do agree with him more, is when he criticizes the "single drive toward calculus in high  school and college" which "displaces other topics more important for today's economy and society".  These include "statistics, linear algebra and algorithmic thinking" (such as revealed in finite math.)

I concur because I've always been skeptical of the high school AP Calculus curricula and whether students are really of sufficient mental maturity that the courses serve any useful purpose (apart from the usual academic 'feather' in the cap to expand their choice of college). And I never saw the reason for pre-med students to be taking calculus. In each case then, perhaps some finite math course or statistics would better serve these populations as opposed to differential and integral calculus.

Calculus then, ought to be here to stay, certainly for most college math and science majors and perhaps even some others (e.g. Philosophy) interested in exploring the role of quantum mechanics in modern expositions, say involving quantum nonlocality and entanglement.

Commemorating Hannes Alfven's Achievements (1)

As most researchers, workers in plasma physics are aware, this year marks the 70th anniversary of a ground breaking research letter ('Existence of electromagnetic -hydrodynamic waves') in Nature, that was to change the face of physics forever. Though only a half page long, it was the content that mattered, describing a new type of low frequency oscillation for a magnetized plasma.

As with many novel insights, the content was long disregarded (or openly dismissed) mainly because the new waves couldn't be demonstrated experimentally at the time. However, over decades and as experimental - lab techniques grew more refined (including experiments conducted in the space environment via satellites and on Space Shuttles) the novel waves we call "Alfven" soon began to occupy an important role in space and plasma physics.

It helps at this stage to examine a bit of the nature of Alfven waves, conceived by Hannes Alfven, and which were finally experimentally demonstrated in lab plasmas by 1960.

Alfven waves, by the way, are the most important waves propagating in the solar atmosphere, as well as the Earth’s magnetosphere (underpinning the coupling between it and the ionosphere). They are important in that they efficiently carry energy and momentum along the magnetic field.


One way to get a handle (of sorts) on Alfven waves is to look at the analogy with mechanical waves – say propagating along a string put under tension. Consider the reference frame or coordinate system:


^ y
!
!
!
!----------------------------------------->.x

Say x marks the direction of propagation in the above coordinate system, and y is the direction of transverse (wave) displacement. Then the vertical force component is:

F_y = - T(@y/ @x)
where T is the tension and the bracketed quantity is the partial of y with respect to x. Thus, just as the restoring force for a mechanical wave is the string tension T, the restoring force for an Alfven wave is the magnetic tension. This magnetic version of “tension” accelerates the plasma and is opposed by the inertia of the ions (mainly from proton masses m(p))

Now, the wave speed on a string is related to u (mass per unit length), and T such that:

v = (T/ u) ^½

and as we can see,  increasing the string tension increases the wave speed in an analogous way to what magnetic tension does for the Alfven wave. The magnetic tension analog can be expressed (as we shall see) as: T(M) = B^2/ u_o

where B is the magnetic induction and u_o is the magnetic permeability for free space, u_o = 4 π x 10^-7 H/m)

Examining the origin of these waves always starts with setting out the basic equations for what we call “ideal MHD”, e.g. one of the equations is: @B / @t = Curl (v X B). We then introduce small perturbed quantitites (e.g. imagine introducing a small perturbation into the plasma velocity such that v_o -> v_1, which will also subject the mass density, fluid pressure and magnetic field to perturbation), such that:


rho = rho_o + rho_1

v = v_1
B = B_o + B_1

p = p_o + p_1


and we substitute these back into the original ideal MHD equations  Then, after using a LOT more math (which I will spare readers) we end up with the general Alfven wave quation:
 
 w^2 v_1 – c_s^2 (kv_x)k* + B_o/ u_o rho_o [k X k* X (v_1 X B_o)] = 0


where w denotes the plasma frequency, k is the wave number vector (k* the vector orientation) and the other quantities are as before, and c_s is the ion sound speed.  One will then take the preceding equation and resolve it into x, y components, viz.

x-component of wave:

w^2 v_x – c_s^2 k^2 v_x + B_z k^2/ u_o rho_o [v_zB_x – v_xB_z] = 0


y-component of wave:

w^2 v_y - B_x^2 k^2 v_y/ u_o rho_o = 0


or simply:

w^2 = [B_x^2/ u_o rho_o] k^2

where the quantity in brackets is the Alfven velocity or alternatively written:


v(A) = [w/ k] = B_x / [u_o rho_o]^½
    or

v(A) = B_o/ [u_o rho_o]^½

since B_o is in the x –direction

A more refined and useful form is obtained by incorporating the x and z-components and solving the resulting simultaneous equations, then doing some simplification to get:

w^2 = ½[(c_s^2 + v(A)^2k^2 +/- [(c_s^2 + v(A)^2 k^4 – 4 c_s^2 v(A)^2 cos^2(Θ) k^4]^1/2


Now, if one plots the preceding using (for the vertical axis ): c_s^2 + v(A)^2  and for the horizontal B_o (e.g. x) one will get what is called “Friedrich’s diagram”


!
!
-----------------
!
!
!


Visualize superimposed on the above axes, the following graphs:

1) a small “dumb bell” or figure-8 shaped graph centered at the origin. This will be for what we call “slow mode” waves

2) a single larger lobe that envelopes the smaller right lobe of the dumb bell. This will be for Alfven waves proper.

3) A circle- shaped graph surrounding both 1, 2 above. This will be for what we call the “fast MHD” mode.

The critical thing to note here is that the fast mode is the only MHD wave able to carry energy perpendicular to the magnetic field. This has important ramifications for solar flares, as well as magnetospheric effects (such as the aurora). Meanwhile, the phase velocity (w/k) of the slow mode wave perpendicular to the magnetic field is always zero. In the limit where the sound speed c_s^2 < < v(A)^2, and the Alfven speed v(A)^2 << c_s^2, the slow wave disappears. (Which you can easily validate and confirm for the equation in w^2)

Other properties, points to note:

- the velocity perturbation v_1 is orthogonal to B_o

- the wave is incompressible since DIV v_1 = ik.v_1 = 0

- the magnetic field perturbation (B_1) is aligned with the velocity perturbation. Since both are perpendicular to k and B_o

- the current density perturbation (J_1) exists as a current perturbation perpendicular to k and B_o e.g.

J_1 = k X B_o

- when c_s^2 << v(A)^2 the fast mode wave becomes a compressional Alfven wave. This has a group velocity equal to its phase velocity w/k

Why the maddening complexity with these waves and being able to quantify them? Mainly because Alfven used both the electromagnetic theory of James Clerk Maxwell in combination with hydrodynamics (or fluid dynamics) which hitherto had been established as separate disciplines of physics. But by marrying them via the sort of steps outlined above (to do with Alfven waves) Alfven succeeded in opening up an entirely novel area called magnetohydrodynamics.

Oddly, and ironically, at the time they were conceived these waves had no practical basis, or applications. But by the 1950s, when thermonuclear research took off, they emerged as it became possible to generate high temperature plasmas artificially on Earth.  The rest, as they say, is history.

Next: Some further contributions of Hannes Alfven.

Solving Basic Plasma Physics Problems (2)

Before going on to solve another basic type of plasma physics problem, let's look at the previous one to do with plasma orbital theory and gyro-motion. Again, the question was:

A proton moves in a uniform electric and magnetic field, with fields given by:E = 10 V/m (x^) and B = 0.0001 T (z^)where '^' denotes vector direction.

a) Find the gyrofrequency and the gyro-radius

b) Find the proton's E X B drift speed

c)Find the gyration speed v(o) and compare to the drift speed

d)Find the gyro-period, gyration energy and magnetic moment of the proton.

Let's first work on (a). We need basic data also to solve this set, including the mass of the proton: m(p) = 1.7 x 10^-27 kg, and the proton charge: q = 1.6 x 10^-19 C

The proton's gyro-frequency is just: W(g) = qB/m(p)

W(g) = (1.6 x 10^-19C) (0.0001T)/ {1.7 x 10^-27 kg} = 9.4 x 10^3 s^-1

The gyro radius r, is defined: r = v(perp)/ W(g)

so we need to get v(perp) first.

For the proton, given the information provided in the blog, this can be obtained from the energy of gyration: E(g) = u(m) B, so v(perp) = [2u(m)B/ m(p)]^1/2  = [100 m/s]^1/2 = 10 m/s

Then, the radius of gyration is: r = (10 m/s)/ (9.4 x 10^3 s^-1) = 0.0011m or 0.1 cm

If u(m) is a constant of the motion, as we expect it to be, then the local perpendicular gyration velocity is just v(perp).

(b) The proton's (E X B) drift speed is just the magnitude [E/B]^y = 10^5 m/s (^y)

(c) and so is 3 orders of magnitude greater than the perpendicular gyration velocity.

(d) The gyro-period is T(g) = 2 pi/ W(g) =

2 pi/ (9.4 x 10^3 s^-1) = 3.5 x 10^-7 s

The gyration energy E(g) = u(m) B = 8.5 x 10^-26 J

Therefore, the magnetic moment of the proton is:

u(m) = E(g)/ B = (8.5 x 10^-26 J)/ (0.0001T) = 8.5 x 10^-22 J/T


--
Now, in the new problem we examine a plasma magnetic mirror (see the model shown)

Consider a plasma mirror machine of length 2L with a mirror ratio of 10 so that B(z=L) = B(z=-L) = 10 B(0). A group of N (N > 1) electrons with an isotropic velocity distribution is released at the center of the machine. Ignoring collisions and the effect of space charge, how many electrons escape?

The mathematical basis for a typical plasma mirror machine is shown, and note the end points (at z = ± L) are “pinched” and thus of narrower bore than at the midpoint. Then this yields higher magnetic induction, B, at those points which will be “B_max”.

The mirror ratio is (B_max/ B_min) = 10, meaning that the induction strength at those end points will be ten times the induction at the center point or apex of the magnetic loop or mirror machine.

We define what is called the “loss cone angle”:

sin (Θ)_L = ± [B_min/ B_max]^1/2

In the problem, B_min = B(0) or the "zero level" for the magnetic induction, say at position L = 0. This doesn't mean the induction is zero at that point literally, however.

To do the problem, one must understand he's really being asked for the fraction of electrons lost. A special condition obtains which applies to the angle - for which the electrons will be TRAPPED only provided:

Θ (O) > (Θ)_L

Thus, Θ (O) = (Θ)_L

is said to be the "loss cone" of the system or machine.

If an isotropic particle distribution (in this case, electrons) is introduced at a position L = 0, the fraction of particles that will be lost to the mirror system is:

f(L) = 1/ 2 π INT (0 to Θ(L)) 2 π sin(Θ) d Θ

= f(L) = INT (0 to Θ(L)) sin(Θ) d Θ

= 1 - cos (Θ)_L

Now, from the problem, if N denotes the total electrons released, then you will have to find the fraction lost from:f(L) = N - [1 - B(0)/ B(L)]^1/2

bearing in mind, B(z=L) = B(z= -L)

We shall finish solving this in the next instalment, but readers are invited to try their hand in the meantime!