1 Home exercise sheet 1.1 Generalized coordinates

1
1.1
Home exercise sheet
Generalized coordinates
Exercise 1.1: The thingy
Four massless rods of length L are connected at their ends to form a rhombus. A particle
of mass M is attached to each vertex. The opposite corners are joined by springs of spring
constant k. In the square configuration, the strings are unstretched. The motion is confined
to a plane, and the particles move only along the diagonals of the rhombus. How many
degrees of freedom are there in the system? Find appropriate generalized coordinates and
describe the kinetic and potential energy in terms of these.
1.2
Calculus of variation
Exercise 1.3: Geodesic on a sphere
A sphere of radius R is a set of points which satisfy the following:
x(θ, φ) = R sin θ cos φ, y(θ, φ) = R sin θ sin φ, z(θ) = R cos θ
1
(1)
Find the Geodesics on a sphere.
Note: a geodesic is a generalization of a straight line in curved space, namely geodesics are
(locally) the shortest path between points in the curved space.
Exercise 1.4: Shape of a bubble
Find the curve c passing through two given points A(x1 , y1 ) and B(x2 , y2 ) such that the
rotation of the curve c about the x-axis generates a surface of revolution having minimum
surface area.
Note: Start by thinking what you need to minimize and see how it relates to the infinitesimal
distance ds (which you know in Euclidean coordinates).
Note: No need to find the constants of integration
2