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Canonical Transformations and Liouville’s Theorem

Canonical Transformations and Liouville’s Theorem. Daniel Fulton P239D - 12 April, 2011. A Trivial Solution to Hamilton’s Equations. Consider: Hamiltonian is constant of motion, no explicit time dep. All coordinates q i are cyclic Then solution is trivial….

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Canonical Transformations and Liouville’s Theorem

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  1. Canonical TransformationsandLiouville’s Theorem Daniel Fulton P239D - 12 April, 2011

  2. A Trivial Solution to Hamilton’s Equations Consider: • Hamiltonian is constant of motion, no explicit time dep. • All coordinates qiare cyclic Then solution is trivial…

  3. Motivation for Canonical Transformations • Want coordinates with trivial solution (see previous). • Question: Given a set of canonical coord. q, p, t and Hamiltonian H(q,p,t), can we transform to some new canonical coord. Q, P, t with a transformed Hamiltonian K(Q,P,t) such that Q’s are cyclic, and K has no explicit time dependence? • Look for equations of tranformation of the form:

  4. Structure of Canonical Transformations Both sets of coordinates must be canonical, therefore they should both satisfy Hamilton’s principle. Integrands must be the same within a constant scaling and an additive derivative term… Can always have intermediate transformation…

  5. : Scale Transformations Suppose we just want to change units. The transformed Hamiltonian is then and the integrands are related by…

  6. F - Generating Function As long as F = F(q, p, Q, P,t), it won’t change value of the integral, however it does give information about relation between (q, p) and (Q, P). Example: (F is given) Since qi and Qi are independent, each coefficient must be zero separately. This gives 2n eqns relating q, p to Q, P.

  7. Four Basic Canonical Transformations If we work from eqs of trans back to F, might get… Note: It is possible to have mixed conditions. e.g.

  8. Example: 1D Harmonic Oscillator (i) Hamiltonian is The Hamiltonian suggests something of the form below, but we need to determine f(P) such that the transformation is canonical. Try a generating function…

  9. Example: 1D Harmonic Oscillator (ii) Immediately, write down the solutions…

  10. Statement of Liouville’s Theorem • The state of a system is represented by a single point in phase space. • In terms of large systems, it’s not realistic or practical to predict the dynamics exactly, so instead we use statistical mechanics… • … we have an ensemble of points in phase space, representing all possible states of the system, and we derive information by averaging over all systems in this ensemble. • Liouville’s Theorem: “The density of systems in the neighborhood of some given system in phase space remains constant in time.”

  11. Proof of Liouville’s Theorem (Goldstein) • Consider infinitesimal volume surrounding a point, bounded by neighboring points. • Over time, the shape of the volume is distorted as points move around in phase space but… • No point that is inside the volume can move out, because if it did it would have to intersect with another point, at which point they would always be together. • From a time t1 to t2, the movement of the system is simply a canonical transformation generated by the Hamiltonian. • Poincare’s integral invariant indicates that the volume element should not change. • dN and dV are constant, therefore D = dN/dV is constant.

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