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Transcript

Action-Angle Variables based on FW-36 Hamilton-Jacobi theory can be used to calculate frequencies of various motions without completely solving the problem if the motion of the system is both separable and periodic. libration e.g. harmonic oscillator rotation e.g. pendulum going over the top 192 Let’s define the action variables of the system: libration se space a h p n i od area one peri r e v o n take rotation they are constants of the motion: we assume the relations are invertible we already know one constant: 193 we can use action variables as the integration constants 's in S: just a name This generates following transformation: S satisfies Hamilton-Jacobi equation: 194 Let’s define the angle variables of the system: “frequency”, a constant of the motion the angle variables increase linearly with time We will need the differential of the angle variables: depends on coordinates and constants definition separability 195 If the motion of the system is periodic, then in one period of the entire system all the degrees of freedom execute some integral number of individual periods. The corresponding change in the angle variables: but also indeed it is frequency fundamental frequencies of the system: follow directly from the Hamiltonian expressed as a function of the action variables 196 Example (harmonic oscillator in two dimensions with different spring constants): total energy Hamilton-Jacobi equation for Hamilton’s characteristic function separability 197 Example (continued): The action variables are then: can be inverted and we get the hamiltonian as a function of the action variables: and the fundamental frequencies of the system: 198 Poisson brackets based on FW-37 Let’s define the Poisson bracket of two functions, F and G: it is obviously antisymmetric Poisson bracket of a function F and the Hamiltonian: Hamilton’s equations: Poisson bracket formulation of classical mechanics! 199 PB is equivalent formulations of classical mechanics: Hamilton’s equations: applied to q and p PB of coordinates and canonical momenta: Canonical transformation to a new set of coordinates and momenta, Q and P, preserves Hamilton equations and thus Poisson brackets. A canonical transformation can be defined as one that preserves Poisson-bracket description of mechanics. 200 Transition to quantum mechanics: Define the commutator of two quantities: Canonical quantization prescription: non-commuting operators acting on a Hilbert space canonical commutation relations Equation of motion: Heisenberg operator equation of motion 201