CONSTANTS of MOTION in Mechanics, a Constant of Motion Is

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CONSTANTS OF MOTION In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion… Common examples include energy, linear momentum, angular momentum and the Laplace-Runge- Lenz vector (for inverse-square force laws). In quantum mechanics, an observable quantity Q will be a constant of motion if it commutes with the hamiltonian, H, and it does not itself depend explicitly on time. Wikipedia Classical You've swung so far as to risk that top trill of your motion. There, poised, where beyond would kill, you're all potential. To move. Again, and when you do, down, it's all kinetic, and what drew you there compels you rush on. Don't stop, please. Equations No outer force, the push/pull of a father's dream, career jig. It's natural. a caress given a hug returned Neither reward, nor dissipation figure much in the meet equations of our motion. Quantum So the world plays tough -- torn menisci, nixed grant. And then you saunter by with simple gifts -- a touch, sweet love. I am. But now the test. Imagine it turned around: We fall in love, all settings "high," and then -- in just a trice things fall apart, shoes land, the world turns on its random wear and tear. Where are we, dear? Time rate change Together, still. But the equations, heartless, say stasis is not an option. Just move on, kids, through war bad knees, oh a lost child. To the parts that cry and muse, love is the sole constant of the motion. .

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1 the Basic Set-Up 2 Poisson Brackets
MATHEMATICS 7302 (Analytical Dynamics) YEAR 2016–2017, TERM 2 HANDOUT #12: THE HAMILTONIAN APPROACH TO MECHANICS These notes are intended to be read as a supplement to the handout from Gregory, Classical Mechanics, Chapter 14. 1 The basic set-up I assume that you have already studied Gregory, Sections 14.1–14.4. The following is intended only as a succinct summary. We are considering a system whose equations of motion are written in Hamiltonian form. This means that: 1. The phase space of the system is parametrized by canonical coordinates q =(q1,...,qn) and p =(p1,...,pn). 2. We are given a Hamiltonian function H(q, p, t). 3. The dynamics of the system is given by Hamilton’s equations of motion ∂H q˙i = (1a) ∂pi ∂H p˙i = − (1b) ∂qi for i =1,...,n. In these notes we will consider some deeper aspects of Hamiltonian dynamics. 2 Poisson brackets Let us start by considering an arbitrary function f(q, p, t). Then its time evolution is given by n df ∂f ∂f ∂f = q˙ + p˙ + (2a) dt ∂q i ∂p i ∂t i=1 i i X n ∂f ∂H ∂f ∂H ∂f = − + (2b) ∂q ∂p ∂p ∂q ∂t i=1 i i i i X 1 where the first equality used the definition of total time derivative together with the chain rule, and the second equality used Hamilton’s equations of motion. The formula (2b) suggests that we make a more general definition. Let f(q, p, t) and g(q, p, t) be any two functions; we then define their Poisson bracket {f,g} to be n def ∂f ∂g ∂f ∂g {f,g} = − .
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