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11 July 2003

Off On a Tangent Vector

by George Hrabovsky, President of MAST

News from MAST

Welcome back!  I know it has been a long time since the last theory column.  Dianna and I have been moving her and my new step-daughter, Stephanie, to Madison from Upper Michigan and there has only been time to write the math column.  We have done a lot of storm chasing, and we have lots of video and still images to analyze.

The bad news is that Dianna starts a new job on the 21st of July and we will not be able to attend the Conference as we had hoped.  I have already sent a copy of my paper to Sheldon.

Four-Velocity

In normal Newtonian mechanics velocity is described as the time derivative of the position vector.  Another way of thinking about it is that the velocity vector is tangent to the position vector at every point.  In other words, velocity is the tangent vector of the position vector.

In our last substantive column I introduced the idea of four-vectors.  In the four-dimensional geometry of spacetime, we do not have trajectories in the classical sense, we have world lines.  We can thus define a different kind of velocity, the tangent vector to the world line.  This is called the four-velocity.

Spacelike, Null, and Timelike Vectors

With all of this weirdness regarding four-dimensional spacetime, it is useful to set forward some definitions.  If we take the inner product of a vector, say A with itself we have three possible outcomes,

A · A = {> 0 A is a spacelike vector . ... a null vector < 0 A is a timelike vector

Note that the spacetime interval is timelike, we can thus define a new quantity

τ^2 = -x^α · x^α .

We call this quantity the proper time.

Four-Velocity Revisited

Using the notion of proper time we can define the four-velocity,

U^α = x^α/τ .

Answer to the Theory Challenge from Last Time

The task was to add the result,

x = Re[A_0 ]/m [(ω_0^2 - ω ^2)^2   + 4γ^2 ω^2]^1/2  ^( (ω t + θ_0 + β))

to the solution for a damped oscillator,

x = A ^(-γ t) cos(ω_1 t + θ) .

We get

x = A ^(-γ t) cos(ω_1 t + θ) + Re[A_0 ]/m [(ω_0^2 - ω ^2)^2& ... bsp;  + 4γ^2 ω^2]^1/2  ^( (ω t + θ_0 + β))

which becomes

x = A ^(-γ t) cos(ω_1 t + θ) + Re[A_0 ]/m [(ω_0^2 - ω ^2)^2   + 4γ^2 ω^2]^1/2  sin(ω t + θ_0 + β) .

Note that the first term

A ^(-γ t) cos(ω_1 t + θ)

vanishes exponentially in time.  We call this the transient term.  The remaining term oscillates steadily,

Re[A_0 ]/m [(ω_0^2 - ω ^2)^2   + 4γ^2 ω^2]^1/2  sin(ω t + θ_0 + β),

and is called the steady-state term.

Theory Challenge

What work is done by an oscillator in the steady state?

Books That I Like

Bernard F. Schutz (1990), A First Course in General Relativity, Cambridge University Press. This has a nice introduction to spacetime diagrams and derives how to view other coordinate systems.

Ray D'Inverno (1992), Introducing Einstein's Relativity, Oxford University Press. This is my favorite introduction to relativity.

Online Resources

4-Vectors:

For a brief description go to:

http://www-physics.mps.ohio-state.edu/~cleo/hep/explain/4vector_1.html

For a slightly more advanced approach,

http://www.pact.cpes.sussex.ac.uk/users/markh/RQF1/node12.html

Forced Oscillators

Here is a fairly elementary site that presents the topic in general terms.

http://www.pinkmonkey.com/studyguides/subjects/physics/chap10/p1010601.asp

Here is a more challenging site that gets similar results to those above.

http://colos1.fri.uni-lj.si/~colos/COLOS/TUTORIALS/JAVA/JAVAXYZET/RESONANCE/HTML/resonance_6.html

Created by Mathematica  (July 11, 2003)