By Robert Eugene Marshak

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56) Solving for the ﬁnal velocities and allowing the second particle to be initially at rest in the lab (v2 = 0) frame gives the one-dimensional scattering formulas v1 = m1 − em2 v1 , m1 + m2 v2 = (1 + e)m1 v1 . 57) REVIEW OF NEWTONIAN PARTICLE MECHANICS 29 For e = 1, one gets the elastic scattering limit. If e = 0, then v1 = v2 and one gets the totally inelastic limit, where the two particles stick together m2 , after collision. Other cases worth examining are the limits when m1 m2 , and m1 = m2 : m1 m1 m2 : v1 = v1 , m1 m2 : v1 = −ev1 , m1 = m2 : v2 = (1 + e)v1 , (1 + e)m1 v2 = v1 m2 (1 + e) v2 = v1 .

110) The term “phase-space” without further qualiﬁcation will be reserved for the product space of positions and canonical momenta. 18 Phase-space plot for the plane pendulum. The stable point is marked with an x. The separatrix between oscillatory and rotational dynamics is indicated by the dashed line. The direction of ﬂow is clockwise about the stationary point. where ω0 = g/l and φ is a dimensionless measure of the time. 111) where the amplitude can be expressed as function of the energy E(θmax ) = Es (1 − cos θmax )/2.

112) A particle with energy 0 ≤ E < Es will oscillate, with an amplitude θmax , about the stable point. Traditionally, such integrals are solved by substitution of variables and/or the use of a good set of integral tables. com/. Hand tweaking of the solution, such as simplifying the results and validating that the correct branch has been returned, is often necessary. 44 CLASSICAL MECHANICS gives the exact solution for oscillatory motion in terms of an elliptical integral of the ﬁrst kind F(φ, k2 ) φ ω0 t = F(φ, k) = −1 1− k2 sin φ 2 dφ = sn−1 (sin φ, k2 ).