 By John H. Lowenstein

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Extra resources for Essentials of Hamiltonian Dynamics

Example text

The reader should be familiar with the case of a disk of infinitesimal thickness r , radius ρ, and density d : L= M = πdρ 2 r 1 1 I⊥ = πdρ 4 + πdρ 2r 2 = Mρ 2 + Mr 2 , 4 4 1 1 4 2 Ir = πdρ = Mρ . 17). The total moments of inertia can be calculated by integrating over r : M = πd 1 I⊥ = πd 4 1 Ir = πd 2 ρ(r )2 dr, ρ(r )4 dr + πd ρ(r )4 dr. 17 Decomposition of an axially symmetric rigid body into infinitesimal disks. For a sphere of radius R with O at the center, we can substitute ρ(r ) = R2 − r 2, −R ≤ r ≤ R, and do the integrals to obtain 4 M = π R 3 d, 3 I⊥ = Ir = 2 M R2.

29). Show that the three quantities H , pφ , and pψ are independent except on certain sub-manifolds of phase space. Describe the physical states to which these sub-manifolds correspond. 18. You will find the Mathematica function ListPlot helpful for plotting lists of data points generated by your numerical integrations. 2 The Hamiltonian formalism Having introduced the Hamiltonian formulation of classical dynamics, and applied it within a number of familiar contexts, we now turn to a systematic study of its salient features.

Suppose we have a canonical transformation generated by a type-2 generating function F2 (q, P, t). 9) generates the same transformation of coordinates. Here we assume that the coordinate transformations among the sets (q, p), (q, Q), (q, P), and (Q, P) are nonsingular for all relevant t. On differentiating F1 (q, Q, t) with respect to qk , with the remaining arguments fixed, we have ∂ F2 ∂ F2 ∂ P j ∂ Pj ∂ F1 = + − Q j. ∂qk ∂qk ∂ P j ∂qk ∂qk Upon inserting the derivative formulas for F2 , this reduces to ∂ F1 = pk .