5. Exercise in ”Theoretical Physics I: Classical Mechanics”
23.11.10/25.11.10, University of Vienna
17. Consider a system of N particles of equal mass m with coordinates ~r1 , . . . ~rN , where there
are no external forces present and where the potential energy is given by two-particle
interactions,
N
X
V (~r1 , . . . , ~rN ) =
U (|~rj − ~rk |).
j<k
We study the behaviour of the system under the Galilean boost coordinate transformation
BV~ :
~t
~rj 7→ BV~ (~rj ) = ~rj + V
∀j = 1, . . . , N,
~ . It describes the change into a coordinate system that
with a fixed arbitrary velocity V
~ with respect to the former coordinate system. The
is constantly moving with velocity V
origins of the old and the new coordinate system coincide at t = 0.
a) Calculate how the Lagrange function of the system transforms under the boost and show
that the equations of motion are invariant under the transformation.
b) Derive the conserved quantity associated to the Galilean boost transformation by looking
at how the Lagrange function transforms under infinitesimal boosts. Hint: An infinitesimal
~ = ε~n with an arbitrarily small ε and an arbitrary
boost is given by a boost parameter V
direction vector ~n. Look at the total derivative of the transformed Lagrange function with
respect to ε in the limit ε → 0.
18. Consider a particle moving under the influence of a conservative force field related to one
of the following geometries. What motional quantities of the particle are conserved and
what are the associated symmetry transformations, i.e. the coordinate transformations that
leave the equations of motion invariant? You are not required to give explicit expressions
for the Lagrange functions.
a) Force field of (i) an infinite homogeneous xy-plane , (ii) an infinite homogeneous cylinder,
(iii) a homogeneous circular cone.
b) Force field of an infinite homogeneous circular helix with the radius R and the pitch d
per full turn. Hint: Choose appropriate coordinates and find a combination of elementary
coordinate transformations that leaves the system invariant.
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