Does Mathematical Physics Truly Mimic Reality?
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Obtaining an answer to that question starts with a simple description of a physical situation. With that description in mind we calculate a description of some other feature of the situation and in doing so we extract a new law of physics that did not seem clearly a part of the original description. To that end, the best example that I can conceive starts with the mathematical description of a bead sliding without friction on a rigid circular loop that we can expand or contract.
For this analysis cylindrical polar coordinates (r,θ,z) provide the most convenient description for our manipulations. We assert that the center of the loop coincides with the origin of the coordinate grid, that the loop itself lies in the plane defined by z=0, and that the loop is somehow anchored to a body so massive that the motion of the bead will not noticeably move the loop. In that system the positive radial direction (the r-direction) points away from the origin in all directions perpendicular to the z-axis and the positive z-direction points in the direction that we can call north for convenience. We define the positive θ-direction (the longitudinal direction) by way of the right-hand rule: we take motion to the east to define motion in the positive θ-direction, so when the fingers of your right hand curl eastward your right thumb points (more or less) in the direction of the positive longitudinal unit vector. That is, the unit vector for the θ-direction coincides with the unit vector for the z-direction. That fact gives us a consistent basis for calculating sums and differences of longitudinal displacements.
In the situation under consideration we describe the location of the bead in space as (r,θ), in which r represents the radius of the loop and θ represents the longitudinal displacement of the bead eastward from some arbitrarily defined prime meridian (θ=0) on the loop. That formulation instructs us to find the location of the bead by beginning at the origin of the coordinate grid, going a distance r along the prime meridian, and then going an angular distance θ to the east (to the west if θ is a negative number). Note that if we measure the angular displacements in radians, the distance that we move along the loop equals rθ.
We can also describe the location of the bead with the vector
The lower-case ee with a hat represents a unit vector, a vector of unit length that simply points in a direction parallel to an axis on a coordinate grid. In this case the unit vectors differ from the standard Cartesian unit vectors in two important respects. The pure radius vector always points away from the z-axis, so the radial unit vector points in a direction determined by the longitude of the point at which we place it (i.e. êr=êr(θ)). And the longitudinal unit vector lies on and points along the z-axis, so we must describe actual displacement or motion in the longitudinal direction with a cross product of that unit vector onto the radial unit vector. We could thus rewrite Equation 1 as
as long as we remember that the second term on the right side of the equality sign describes a circular arc.
To calculate the velocity of the bead in space, we differentiate Equation 2 with respect to the elapse of time to get
In that equationω=dθ/dt. The derivative of the radial unit vector with respect to a change in longitude tells us how the radial unit vector changes its direction due to a change in longitude. That derivative points in the direction shown by the vector cross product êθxêr and its magnitude appears to be indeterminate.
We can determine the magnitude of that derivative by going back to the fundamental definition of a derivative and applying it to the unit vector. Imagine drawing the unit vector and another unit vector displaced from the first by a minuscule amount of longitude (dθ), then imagine so moving one of those vectors that the two vectors meet base to base and then calculate the difference between the two of them. That difference comes out as a vector extending from the head of one unit vector to the head of the other and it thus has a magnitude equal to the length of either of the unit vectors multiplied by the angular displacement between them; that is, |êθ|dθ=dθ. We get the magnitude of the derivative when we divide that result by the displacement, so we have the derivative itself as
Substituting that expression for the derivative into Equation 3 gives us the velocity of the bead as
In the last term in that equation I made use of the vector identity
So we have four contributions to the velocity of the bead in space:
1. The first contribution, the first term on the right side of the equality sign, gives us the radial component of the bead’s velocity due to the expansion or contraction of the loop.
2. The second contribution gives us a velocity oriented along the loop. We expect the magnitude of that velocity to equal the rate at which a radius vector reaching to the loop turns as a result of the angular velocity in the formula, but we actually get twice as much. One half of that amount comes from multiplying the bead’s distance from the z-axis by the rate at which its longitudinal displacement changes, as we expect. The other half comes from multiplying the radial distance by the temporal derivative of the radial unit vector. The latter does not conform to any of our intuitions about this situation.
3. The third contribution describes the motion of the bead due to the shortening or lengthening of the arc between the bead and the prime meridian due to the expansion or contraction of the loop. Again, this conforms to our intuitive picture of a bead sliding on the loop.
4. The fourth contribution also defies our spatial intuitions. We have the longitudinal velocity multiplied by the longitude applied in the negative radial direction. That looks like something that would produce a radial motion of the bead, even if the loop is not changing its radius. That inference makes no sense, so we must assert that the fourth term on the right side of Equation 5 equals zero.
The fourth term and half of the second term on the right side of Equation 5 don’t belong there. We only have them because we inferred that the longitudinal derivative of the radial unit vector does not equal zero. If that derivative does equal zero, then those improper terms drop out of the equation and we have our description of the velocity of the bead as
That makes perfect sense, so we infer that we simply misinterpreted the nature of the radial unit vector.
If we differentiate Equation 7 with respect to time, we get a description of the acceleration of the bead and if we multiply that acceleration by the mass of the bead, we get a description of the force that produces that acceleration. We have, then,
That looks just about perfect. The first term describes the radial acceleration of the bead due to changes in the rate at which the loop increases or decreases its radius. The fourth term describes the acceleration of the bead away from the prime meridian in the longitudinal direction due to the change in the rate at which the length of the intervening arc of the loop changes. The remaining three terms should astound us.
The second, third, and fifth terms in Equation 8 describe the acceleration of the bead along the loop due to a longitudinal force acting on the bead. If there is no such applied force, those terms must add up to zero, which fact necessitates that
If we divide that equation by rω and integrate the result, we get
in which lnC represents the constant of integration. The antilogarithm of that equation gives us
and multiplying that result by the mass of the bead turns it into a statement of the law of conservation of angular momentum.
Our description of a bead sliding on a circular loop did not explicitly contain that law. It came from our analysis as an emergent property of circular motion. In our analysis we have discovered what physicists usually call a fictitious force, but which we would more appropriately call an obligatory force; that is, we have discovered the Coriolis force,
As astonishing as that result may be, it is, nonetheless, incomplete. Our analysis has not given us the centrifugal force, the force that the bead exerts on the loop as it seeks to travel in a straight line, in accordance with Newton’s first law of motion. That observation tells us that there’s something wrong with our description of this situation, so we need to try a different description and/or analysis.
Let’s describe the location of the bead with the vector radius r=r(θ). We have the velocity of the bead as
which is simply the first and third terms from the right-hand side of Equation 3. That’s an easy intuitive description: it gives us the radial component of the motion and the longitudinal component of the motion and does so in such a way that we can represent the time derivative as a vector operator
If we apply that operator to the velocity, then we should get a good description of the bead’s acceleration. So we have
That equation gives us the direct radial acceleration, the torque and the Coriolis acceleration, and the centripetal acceleration. The last term describes the rate at which the bead must accelerate toward the center of the loop in order to remain on a circular path that coincides with the loop. The inertial reaction to that acceleration is what we call centrifugal force, another obligatory force usually mis-identified as a fictitious force.
But, again, the description is incomplete. In this case the derivation does not yield the components of longitudinal velocity and acceleration due to the change in the length of the loop between the location of the bead and the prime meridian. For those components we should have
which are the same as the third term in Equation 5 and the fourth term in Equation 8. We can also write those equations as
If we add the first of those equations to Equation 13, we get
That just gives us Equation 7 and we know that’s correct because we know that the bead has only three components to its velocity on the loop: 1) the direct radial motion due to the expansion or contraction of the loop, 2) the longitudinal motion relative to the loop and 3) the motion relative to the prime meridian due to the loop changing size.
If we add the second of Equations 17 to the first line of Equation 15, we get
Again, this is a clearly wrong result, if only because the last term is the same as the last term in Equation 5, the term that seems to have no meaning. Further, the Coriolis force comes out wrong, as does the rate at which the longitudinal speed due to the change in the arc length of the loop changes.
We have one other possibility available to us. We can accept Equations 13 and 15 as correct descriptions of the dynamic situation and interpret the missing terms as part of a coordinate transformation. In this case the transformation involves a simple rotation about the z-axis.
As a result of that rotation our system has two prime meridians, separated from each other by the angle theta, which we treat as a constant. The bead resides, temporarily, on one of those prime meridians and observers using that meridian also use Equations 13 and 15 to describe the motion of the bead. Observers using the other meridian must subtract Equations 16 from Equations 13 and 15 in order to get a correct description of the motion that they would measure of the bead on the loop. In that statement I have assumed that we measure the constant theta from the first meridian to the second. The corrections describe only the relative motion between the two prime meridians due to changes in the radius of the loop, so they don’t affect the dynamic terms, such as the centrifugal force: they merely add to or subtract from the motion of the bead calculated from the dynamic description.
Thus we obtain a correct description of the motion of a bead sliding freely on a circular loop. It seems such a simple task and yet we didn’t get the correct description immediately. That fact implies that mathematics does not inherently mimic Reality. On the other hand, however, when we apply a simple operator to a simple description of the bead’s location we get, first, a correct description of the bead’s velocity and, then, a correct description of the forces acting on the bead and of the conservation law that constrains circular motion. True, we had to add a component to the velocity and the acceleration as a coordinate transformation, but we still have mathematical logic yielding a correct description that did not, as far as we can tell, pre-exist in our description of the bead’s location or of the operator that converted that location into velocity and acceleration.
We thus infer that mathematics does, indeed, mimic Reality, but also that it includes other options. In order to describe Reality correctly we must augment our mathematical logic with verbal logic, which we use to select the correct options. With both kinds of logic working together we can build an axiomatic-deductive structure that mimics Reality perfectly. That structure is the Map of Physics.
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