PHYS 381:  STUDY GUIDE

MOVING COORDINATE FRAMES

LAGRANGIAN AND HAMILTONIAN MECHANICS

Outline:

Chapter 7 of Symon

1.  Moving Coordinate Frames

2.  Rotating Coordinate Frames

3.  Laws of motion on rotating earth: Foucoult pendulum

4.  Larmour's Theorem

Chapter 9 of Symon

5.  Generalized coordinates

a)  concept

b)  example: polar coordinates

c)  example: rotating polar coordinates

6.  Generalized velocities

a)  generalized velocities

b)  generalized momenta

c)  kinetic energies

d)  orthogonal coordinates and kinetic energy

e)  virtual work

7.  Lagrange's equations

a)  derivation

b)  the Lagrangian

c)  examples

8.  Constraints

a)  definition

b)  examples

c)  ignorable coordinates

d)  constants of the motion

9.  Hamilton's equations

a)  the Hamiltonian

b)  the equation

Homework:

Problem #1:  (Problem 7‑2 in Symon):

A mass, m, is attached to a support by a spring with spring constant, k.  The mass is hanging down from the spring, so there is a gravitational force on the mass as well.  Neglect any resistive or frictional force.  The support is then oscillated with an amplitude of A and at a frequency of ωa .
a)  Find the motion of the mass relative to the support.
b)  Find the motion of the mass relative to the lab (inertial frame).

HINT:  use y* as the distance the mass stretches the spring from the support.

Problem #2:  (Problem 7‑7 in Symon):

For the rotation of the earth (1 revolution per day with the sun rising in the East and setting in the West with the earth’s radius being 6.4 x 106 m), for an object dropped from a height of 100 meters,

a)  find the direction and the amount of deflection the object will undergo due to the Coriolis effect (that is, disregard the “Centifugal force” effects).

b)  explain in words why your result for part a is the way it is.

Problem #3:

a)  Calculate the mass of the earth based upon the fact that the Moon orbits the earth with a period of 27 days, 7 hours and 43 minutes at a distance of 384,400 km.

b)  Using the above mass for the earth and given Rearth = 6,378 km, calculate the magnitude of the true gravitational acceleration, g.

c)  Using the fact that Memphis is close to sea level at latitude 35° North, and given that the period for the earth's spin is 24 hours,  calculate the magnitude and direction of the term -(mw´w´r) at Memphis.

d)  Using the results of part c, compare g and ge.

Problem #4:  (Problem 7‑21 in Symon):

Consider two identical planets, each of mass, M; and radius, R; that are separated by a distance, a.

a) Neglecting the orbital motion of the planets about each other, what speed would you need to escape from one of the planets to reach the center point between the planets (so you could fall down to the other planet)?

b) Now include the orbiting motion of the two planets assuming they are orbiting each other in a circle, and find the speed you would need.

Problem #5:  (Problem 9‑1 in Symon)

Consider the coordinates u and w that are defined in terms of the polar coordinates r and θ:

u = ln(r/a) – θ cot(ζ)   and    w = ln(r/a) + θ tan(ζ)

where a and ζ are constants.

Sketch the curves of constant u and w:  this is already done for you - see notes on the Lagrangian.

a) Find the kinetic energy for a particle of mass m in terms of u, w, du/dt, and dw/dt .

b) Find expressions for Qu and Qw in terms of the polar force components Fr and Fθ .

c) Find pu and pw .

**  d) extra credit: find the forces Qu and Qw required to make the particle move with constant
speed, ds/dt, along a spiral of constant u=uo .

HINT:  Since the transformation equations are in terms of r and θ (polar form), you can start with the expressions for KE, Qi and pi in polar form rather than in rectangular form.

Problem #6:  (Problem 9‑3 in Symon)

For plane (2-D) parabolic coordinates f and h defined such that:     x = f – h   and   y = 2(fh)1/2
(see the excel spreadsheet accessed from  the course web page to see an image);
a)  find the expression for the kinetic energy in terms of f, h, f’, and h’;

b)  find the expression for the momenta:  pf   and  ph ;

c)  write out the Lagrange equations in these coordinates if we assume the particle is not acted on by any force.

Problem #7:  (Problem 9‑28 in Symon)

Start with the Lagrangian for the spherical pendulum (R = constant):  L = T – V where

T = kinetic energy = ½ mvθ2 + ½ mvφ2 = ½m(Rθ’)2 + ½m(R sin(θ) φ’)2, and

V = mgR cos(θ).  Remember that the Hamiltonian is a function only of the pi and qi .

a)  Write down the Hamiltonian function for the spherical pendulum.

b)  Write down the Hamiltonian equations of motion.

c)  Derive from them the equation:  E = ½ mR2θ’2 + pφ2/[2mR2sin2(θ)]  + mgRcosθ .

Problem #8:  (Problem 9‑22 in Symon)

a) Show that a uniform magnetic field B in the z direction can be represented in cylindrical coordinates by the vector potential
A = ½Bρφ (that is, show that Ñ×A = Bz .

b)  Write out the Lagrangian function for a particle in such a field.

c)  Write down the equations of motion (in cylindrical form).

d)  Find three constants of motion (momenta and energy).

Study Questions:

1.   a)  Write the vector equation of motion (Newton's 2nd law)

for an object as viewed from the surface of the rotating earth.

b)  Identify each term in the equation.

c)  Calculate the amount and direction of the "ficticious"

forces on an object of mass m moving in a direction d with a speed of v at a latitude of L  (m, d, v, and L will be specified in the test problem).

2.  Distinguish between cyclotron and Larmour frequencies.

3.  Express a generalized momentum in generalized coordinates, and use
spherical coordinates as an example of what they are.

4.  Express the kinetic energy in generalized coordinates, and use spherical coordinates as an example of what this looks like.

5.  Define a generalized force, and again use spherical coordinates as
an example of what this looks like.

6.  Be able to discuss the concept of VIRTUAL work.

7.  Define the Lagrangian, and write down Lagrange's equations.  Use spherical coordinates as an example of what this looks like in a specific case.

8.  Show what a constraint is by using an example.

9.  Define the Hamiltonian and write down Hamilton's equations.