A1 Dynamical Astronomy
Proof of Keplers first law from Newtonian dynamics
A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse.
It would be a pity to have a course on dynamical as- as r P is the component of v perpendicular to r
Chem 3890 HW3
1. Orthogonal polynomials
0 = A0;
1 = A1 * x;
3 = A3 * (x ^ 3 - B1 * x);
We know that each of these functions is normalized and that each distinct pair is orthogonal. This gives
in principle six equations for 4 unknow
Chem 3890 HW0
Clear variable values each time the .nb is run.
Question from Getting Started
How many distinct variables in
ab + a*b + cosa +Log[a*b] +Exp[a] + sin(ab) ?
The distinct variables are ab, a, b, cosa, and sin, so 5. No
Chem 3890 Assignment 4
1. Uncertainty relation for x and K. See handwritten sheet.
2. Check result from Q1 for a particular case
$Assumptions = cfw_b > 0, k > 0, > 0, m > 0
cfw_b > 0, k > 0, > 0, m > 0
Chem 3890 Assignment 6
1. Particle-in-a-bowl solved by oscillating the pup
(i) SE in dimensionless variables-see handwritten page
(ii) Plot harmonic and anharmonic potential energies
PE made dimensionless by dividing by
VHO = ^ cfw_2 / 2
Each radial function should be multiplied by
[m*omega/hbar]^cfw_3/2 to have the correct units of
1. 3D particle-on-a-spring-see handwritten pages
(iii) Some derivatives
Fnl = A * ^ cfw_n * Exp[- ^ cfw_2 / 2
Chem 3890 HW 7
1. Average potential and kinetic energy for a 1s electron
See handwritten sheet for details.
Vbar = - 8 * I1 * Integrate[ * Exp[- 2 * ], cfw_, 0, Infinity]
- 2 I1
See handwritten work.
Chem 3890 Assignment 5
1. Numerical solution of the SE for the HO at and near the ground-state energy.
(i) Schrdinger equation for HO with dimensionless coordinate at the ground state energy. Here is
related to x in Eq. 2.71 of Griffiths, and the Schrding
Chem 3890 Assignment 1
1. Ordinary derivatives
X[x_] := B * Sin[n * Pi * x / l]
(a) Take the difference
deltaX = X[x2] - X[x1]
- B Sin
+ B Sin
(b) Use the derivative
dX = (D[X[x], x] /. x x1) * (x2 - x1)
The equation of motion
We consider the motion of a point mass under the influence of a gravitational field
created by point mass that is fixed at the origin.
F IGU RE
Newtons laws give the basic equation of motion for such a system. We denote
Keplers Laws and Gravity
Observations of a supernova.
Tycho made detailed observations
of the supernova of 1572 now
called Tychos supernova.
It was initially brighter than Venus
and was visible for 18 months
before fading from view.
Teaching the Kepler laws for freshmen.
Maris van Haandel
RSG Pantarijn, Wageningen, the Netherlands
IMAPP, Radboud University, Nijmegen, the Netherlands
(Dated: November 12, 2007)
We present a natural proof of Keplers law of ellipses in terms