transformation_a

# transformation_a - NONLINEAR MECHANICAL SYSTEMS CANONICAL...

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NONLINEAR MECHANICAL SYSTEMS C ANONICAL T RANSFORMATION S AND N UMERICAL I NTEGRATION Jacobi Canonical Transformations A Jacobi canonical transformations yields a Hamiltonian that depends on only one of the conjugate variable sets. Assume dependence on new momentum alone. H(p*,q*) = K(p*) K(p*)/ q* = 0 Thus dp*/dt = e* dq*/dt = K(p*)/ p* – f* The simple relation between effort and the rate of change of momentum is recovered in the new coordinates. Mod. Sim. Dyn. Sys. Transforms & Numerical Integration page 1

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E XAMPLE : S IMPLE HARMONIC OSCILLATOR Hamiltonian H(p,q) = 2 1 (p 2 /I + q 2 /C) Hamilton's equations dq/dt = H/ p = p/I dp/dt = – H/ q = q/C Change variables from old (q,p) to new (P,Q) Define Z o = IC and the generating function S(q,Q) = Z o (q 2 /2) cotQ The transformation equations are p = S/ q = Z o q cotQ P = – S/ Q = Z o (q 2 /2)/sin 2 Q Mod. Sim. Dyn. Sys. Transforms & Numerical Integration page 2
Express the old variables in terms of the new p = 2P cosQ Z o q = 2P sinQ (1/ Z o ) Define ω o = 1/IC and the new Hamiltonian is H(P,Q) = ω o P = K(P) Hamilton’s equations in the new coordinates dQ/dt = K/ P = ω o dP/dt = – K/ Q = 0 Their solution is Q(t) = ω o t + constant P(t) = constant Mod. Sim. Dyn. Sys. Transforms & Numerical Integration page 3

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In essence this variable change has integrated the equations. As the product of P and Q has the units of action (energy by time) it is sometimes called a (simple harmonic) actional transformation. P HYSICAL INTERPRETATION : P is proportional to the total system energy. Its square root is proportional to oscillation amplitude. Q is the phase angle of the oscillations. Mod. Sim. Dyn. Sys. Transforms & Numerical Integration page 4
In general, finding Jacobi canonical transformations requires solving a non-trivial partial differential equation. A practical alternative is to separate the Hamiltonian into two parts, one with a known Jacobi canonical transform. H(p,q) = H j (p,q) + H n (p,q) Apply the known Jacobi canonical transformation H*(P,Q) = H j *(P) + H* n (P,Q) Mod. Sim. Dyn. Sys. Transforms & Numerical Integration page 5

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We may represent the second term as a set of canonical forces e*(P,Q) = – H* n / Q f*(P,Q) = – H* n / P The transformed equations become dP/dt = e*(P,Q) dQ/dt = H* j / P – f*(P,Q) An advantage of this change of variables is that, in
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transformation_a - NONLINEAR MECHANICAL SYSTEMS CANONICAL...

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