Classical Dynamics of Particles and Systems · Problem 7-8

Central-Force Motion — Marion & Thornton

Physics — Mechanics · lagrangian-snells-law-analogy

Problem

Consider a region of space divided by a plane. The potential energy of a particle in region 1 is

U_1

and in region 2 it is

U_2

. If a particle of mass

m

and with speed

v_1

in region 1 passes from region 1 to region 2 such that its path in region 1 makes an angle

\theta_1

with the normal to the plane of separation and an angle

\theta_2

with the normal when in region 2, show that

\frac{\sin\theta_1}{\sin\theta_2} = \left(1 + \frac{U_1 - U_2}{T_1}\right)^{1/2}

where

T_1 = \frac{1}{2}mv_1^2

. What is the optical analog of this problem?