The moving trihedron and all that

Transcription

1 MATH Harrell The moving trihedron and all that B Lecture 5 T N Copyright 2013 by Evans M. Harrell II.

2 This week s learning plan You will be tested on the mathematics of curves. You will think about the physics of moving particles You will start understanding functions where vectors go in and scalars come out.

3 Reminders and Clarifications TEST TOMORROW (yikes!)

4 Clicker quiz Find a normal vector to t i + t 2 j at (2,4) A 4 i j! B i + 2 j C 2 i - j D 1 E (2 i j)/5 1/2 F none of the above

5 The moving trihedron The curve s preferred coordinate system is oriented along (T,N,B), not some Cartesian system (i,j,k) in the sky.

6 The moving trihedron A spaceship doesn t see a big Cartesian grid in the sky. Looked at from the inside, a better basis for vectors will use the unit tangent T, the principal normal N., and the binormal B.

7 The trihedron of unit vectors 1. T(t) = rʹ (t)/ rʹ (t) or just... dr/ds. Because of the chain rule, since the speed rʹ (t) is ds/dt.

8 The trihedron of unit vectors 1. T(s) = dr/ds. 2. N(s) = (dt/ds)/κ, where κ = dt/ds What you need to remember: dt/ds determines the direction of N. Its magnitude defines the curvature. dt/ds = (dt/dt) / (ds/dt).

9 The trihedron of unit vectors 1. T(s) = dr/ds. 2. N(s) = (dt/ds)/κ, where κ = dt/ds is our definition of the curvature in 3D. 3. B(s) = T(s) N(s).

10 Example What s the formula, for example for c(t) = (sin(πt),cos(2πt),t) at (0,1,1)? 1. Parametric form

11 Example What s the formula, for example for c(t) = (sin(πt),cos(2πt),t) at (0,1,1)? 2. Single equation

12 Just what is curvature? How do you know a curve is curving? And how much? The answer should depend just on the shape of the curve, not on the speed at which it is drawn. So it connects with arclength s, not with a timeparameter t.

13 Just what is curvature? HOW ABOUT HERE? WHICH CURVES MORE?

14 Just what is curvature? HOW ABOUT HERE? WHICH CURVES MORE?

15 Just what is curvature? HOW ABOUT HERE? WHICH CURVES MORE?

16 Just what is curvature? HOW ABOUT HERE? WHICH CURVES MORE?

17 Just what is curvature? HOW ABOUT HERE? WHICH CURVES MORE?

18 How rapidly do T and N change? T N N T T T N T N N T N N T

19 Just what is curvature? And let s be quantitative about it! 2D: How about dφ/ds, where φ is the direction of T with respect to the x-axis? To get started, notice that the direction of T is the same as that of the tangent line. That is, tan φ = dy/dx = (dy/ds)/(dx/ds) (fasten seatbelts for the next slide!)

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21 Different expressions for κ κ = dφ/ds κ = (dφ/dt)/(ds/dt) κ = xʹ (s) yʹ ʹ (s) - yʹ (s) xʹ ʹ (s) κ = xʹ (t) yʹ ʹ (t) - yʹ (t) xʹ ʹ (t) (xʹ (t)) 2 + (yʹ (t)) 2 3/2 Huh??

22 Example Circle of radius 5. No calculus needed! If you move distance Δs along the perimeter, the change in angle is Δs/5. So Δφ/Δs = 1/5. The general rule for a circle is that the curvature is the reciprocal of the radius.

23 Example Spiral: The formula for curvature is complicated, but the spiral is simple, so the curvature should be simple. Still, we ll be lazy and use Mathematica:

24 Example

25 Dimensional analysis What units do you use to measure curvature? Answer: 1/distance, for instance 1/cm. 1/κ is known as the radius of curvature. It s the radius of the circle that best matches the curve at a given contact point.

26 The curve equations of Frenet and Serret The first of these is dt/ds = κ N

27 The curve equations of The first of these is dt/ds = κ N So what is dn/ds? Frenet and Serret

28 The curve equations of The first of these is dt/ds = κ N So what is dn/ds? Frenet and Serret Like all vectors, it must be of the form T + N + B

29 N Nʹ = 0 Because it is ½ the derivative of N N = 1 (constant).

30 The curve equations of Frenet and Serret The first of these is dt/ds = κ N So what is dn/ds? 0, because N N = 1. Well, N T = 0, so Nʹ T.+ N Tʹ = 0. Like all vectors, it must be of the form T + N + B Therefore Nʹ T.= - N Tʹ = - N κnʹ = - κ. Nʹ = _- κ_ T + 0 N + τ B.

31 The curve equations of Frenet and Serret The first of these is dt/ds = κ N Next: dn/ds = -κ T + τ B What does torsion tell us?

32 The curve equations of The first of these is dt/ds = κ N Next: dn/ds = -κ T + τ B Finally, db/ds = - τ N Frenet and Serret

33 The curve equations of Frenet and Serret d T N =(κb + τt) T N ds B B

34 CULTURE BREAK

35 Motion in 3 D Remember that a curve s favorite coordinate system is based on the moving trihedron (T,N,B). What happens to a moving particle in this moving frame? velocity acceleration

36 Motion in 3 D v = v T + 0 N + 0 B. So what s the acceleration in the local frame?

37 Motion in 3 D magnitude direction

38 Motion in 3 D + 0 B

39 Selected applications of vector calculus to physics Angular momentum, L := r p = r m rʹ. How does this change in time? (This is called the torque.)

40 Selected applications of vector calculus to physics Magnetic motion, F := (q/c) v B, Lorentz force law. Funny font because the magnetic field is not the same as the binormal.

41 Selected applications to physics Magnetic motion, F = (q/c) v B Suppose for now that B is a constant vector. rʹ ʹ = (q/cm) rʹ B

42 Selected applications to physics But if rʹ ʹ = (q/cm) rʹ B and the initial velocity rʹ (t) happens to be perpendicular to B, then rʹ (t) and rʹ ʹ (t) will both remain perpendicular to the magnetic field B. The entire trajectory is therefore in a plane perpendicular to B, which must be parallel to B after all! Moreover, rʹ ʹ and rʹ are perpendicular, so rʹ is constant, as we have seen. The velocities must be of the form rʹ (t) = A cos(qt/cm - φ) i ± A sin(qt/cm - φ) j

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