Canonical Transformations and Liouville’s Theorem

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Presentation transcript:

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 qi are cyclic Then solution is trivial…

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:

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…

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

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.

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.

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…

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

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.”

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.