Spring 2006 2 Mech

A mechanical system consists of two particles, one of which moves in three dimensions, the other of which is confined to a plane. The particle masses are m1 and m2 , respectively. The potential energy of the system is

U (x, y, z, ρ, φ) = V (u, v),

where
u ≡ αx + βy + γz v ≡ y + aφ .

Thus, the potential U depends only on two linear combinations of the five degrees of freedom.

(a) Write down the Lagrangian and the equations of motion.

(b) Noether’s theorem says that to each continuous one-parameter family of transformations of the generalized coordinates, there corresponds an associated conserved quantity. For this system, identify all such one-parameter families and conserved quantities.

(c) Is anything else conserved by the dynamics?

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Answer

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