PHY5200 F06

Chapter 4: Energy

Reading

Taylor 4.8-4.9 (today) and 4.9-4.10 (Friday).

Recall

Spherical (Polar) Coordinates

Spherical coordinates are extremely useful in central force problems: planets orbiting the sun, 3-D harmonic oscillator, or electron bound to a proton. Spherical coordinates define a location by its

radius from the origin, r;
polar angle from the z axis, θ, similar to latitude on earth; and
azimuthal angle, φ, the angle from the x axis to the projection of the location onto the x-y plane, similar to longitude on earth.

We can relate (r, θ, φ) to (x, y, z) using geometry. First, it is easy to see that z = r cosθ. Then the x and y components are projections of the length projected onto the x-y plane, r sinθ. We find that x = r sinθ cosφ and y = r sinθ sinφ.

The surface of the earth is mapped with spherical coordinates with r approximately constant, so usually neglected. The latitude is a measure of the angle from the equator (positive for north and negative for south). This is an azimuthal angle, but measured from the equator rather than the north pole; θ = 90^o - latitude. The longitude is just like our azimuthal angle φ, where for earth we arbitrarily choose a particular direction for the x axis (φ=0).

A spherically symmetrical function f(r) means that f depends only on the distance from the origin, r, not the directions θ or φ, meaning that f(r) = f(r). Since r, θ, and φ are independent coordinates (we can change one without changing the others), then ∂f(r)/∂θ = ∂f(r)/∂φ = 0. Only the partial derivative with respect to r is non-zero.

The unit vectors are defined as the direction in which the corresponding coordinate increases while the other two don't change. The unit vectors are mutually orthogonal, so the dot product of two vectors in spherical coordinates is still the sum of the products of the corresponding coordinates, that is, if a = a_rrhat + a_θθhat + a_φφhat and b = b_rrhat + b_θθhat + b_φφhat then

a⋅b = a_rb_r + a_θb_θ + a_φb_φ.

The Gradient in Spherical Coordinates

The unit vectors rhat, θhat, and φhat aren't fixed, but change at different locations. Therefore, care must be used when taking derivatives of vectors in spherical coordinates. We must include the fact that the unit vectors change as well as the coordinates. We went through this exercise for polar coordinates, and let's do it again for spherical coordinates.

Conservative and Spherically Symmetric, Central Forces

Energy of Interaction of Two Particles

I have covered this in my earlier discussion of extending kinetic and potential energies to multiparticle systems. Please read over, paying attention to the arguments about relative coordinates.

Elastic Collisions

Collisions are an important tool in physics. Much of our understanding of the subatomic world comes from the study of collisions between subatomic particles. Many aspects of the motion of comets and man-made satellites can be understood as "collisions" with planets and stars. And even the physics of semiconductors and superconductors is phrased in the language of collisions.