Physical Dynamics (SPA5304) – Homework 4
Due Friday 10-Feb-2017 at 16:00. Attempt to answer all questions.
Problem 1 (Quick part A-type questions) [15 marks]
(i) Consider a system with generalized coordinates ~q = (q1 , . . . , qn ) and let the Lagrangian of
the system be L(~q, ~q˙, t). Now, consider a transformation h that maps ~q to ~q̃ = h~q. When
is h called a symmetry of the system? State the condition in terms of the Lagrangian
L(~q, ~q˙, t).
[3]
(ii) Assume that the infinitesimal transformation
~q → ~q + δ~q
leaves the Lagrangian L invariant, namely δL = 0. Noether’s theorem says what quantity
is conserved? Write down the expression for the conserved quantity I using L and δ~q.
(No derivation is necessary; just write down the expresssion.)
[3]
(iii) What is the conserved charge associated with translational symmetry (invariance)? [3]
(iv) In the central force problem in a plane, the Lagrangian is
L=
µ 2
(ṙ + r2 ϕ̇2 ) − V (r).
2
Find a cyclic coordinate of this system, and write down the expression for the conserved
conjugate momentum explicitly in terms of µ, r, ϕ̇.
[3]
(v) In the same central force problem as in (iv), we learned in class that the 2-dimensional
problem of finding motion in the r, ϕ coordinates can be reduced to a 1-dimensional
problem of finding motion in the r coordinate, with the effective potential
Veff (r) = V (r) +
l2
.
2µr2
What is the physical meaning of the second term? Give a short, one-line answer.
[3]
Problem 2 [15 marks]
State how many degrees of freedom the following systems have, and in each case indicate,
using a diagram, appropriate generalised coordinates which can be used to describe the
motion. (You cannot just say that the generalized coordinates are “x”, “θ” etc. You should
indicate what they are in the diagram.)
(i) A particle in three dimensions in gravitational field.
[3]
(ii) A particle constrained to move on a fixed plane.
[3]
1
(iii) A spring pendulum contained in the (x̂, ŷ) plane, where a mass is attached to an endpoint
of a spring of elastic constant k and equilibrium length l. The other endpoint of the
spring is fixed at the origin.
[3]
(iv) A system of two particles, both of which are constrained to move along a ring of radius
R. A spring of elastic constant k is stretched between two particles and the potential is
[3]
V = 21 kd2 where d is the distance between the two particles.
p
[3]
(v) A particle which is constrained to move on the cone z = x2 + y 2 .
Problem 3 [20 marks]
A conservative mechanical system consists of a mass m that is constrained to move along a
circle of radius R. The centre of the circle is at the origin O of the coordinate system. The
mass is connected to a point A along the x̂-axis at a distance 2R from the centre of circle
with a spring of elastic constant k, so that the corresponding elastic potential has the form
Vspring = (k/2)d2 , where d is the (varying) distance between the mass and point A. Gravity
acts, as usual, along the vertical direction. See the figure for a depiction of the system.
Figure 1: Figure for Problem 3. The spring is depicted in blue.
(i) How many degrees of freedom does the system have? Indicate generalised coordinates
to describe the motion of the system.
[3]
(ii) Write down the Lagrangian of the system.
[7]
(iii) Write down the Lagrange equations of the system.
[5]
(iv) Find the equilibrium positions of the system and discuss their stability.
[5]
2