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Consider the uniform motion of a free particle of mass m. The Hamiltonian is a constant of the motion, and so is the quantity F(x, p, t) = x − pt/m. (a) Compare {H, F} with ∂F ∂t . Prove that F is also a constant of the motion. (b) Prove that the Poisson bracket of two constants of the motion (F(x, p, t) and G(x, p, t)) is itself a constant of the motion, even if the constants F(x, p, t) and G(x, p, t) depend explicitly on the time. (c) Show in general that if the Hamiltonian and a quantity F are constants of the motion then ∂F/∂t is a constant of the motion too.

Consider the uniform motion of a free particle of mass m. The Hamiltonian is a constant of the motion, and so is the quantity F(x, p, t) = x − pt/m. (a) Compare {H, F} with ∂F ∂t . Prove that F is also a constant of the motion. (b) Prove that the Poisson bracket of two constants of the motion (F(x, p, t) and G(x, p, t)) is itself a constant of the motion, even if the constants F(x, p, t) and G(x, p, t) depend explicitly on the time. (c) Show in general that if the Hamiltonian and a quantity F are constants of the motion then ∂F/∂t is a constant of the motion too.

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Consider the uniform motion of a free particle of mass m. The Hamiltonian is a constant of
the motion, and so is the quantity F(x, p, t) = x − pt/m.
(a) Compare {H, F} with ∂F

∂t . Prove that F is also a constant of the motion.

(b) Prove that the Poisson bracket of two constants of the motion (F(x, p, t) and G(x, p, t))
is itself a constant of the motion, even if the constants F(x, p, t) and G(x, p, t) depend explicitly on the time.
(c) Show in general that if the Hamiltonian and a quantity F are constants of the motion
then ∂F/∂t is a constant of the motion too.

Consider the uniform motion of a free particle of mass m. The Hamiltonian is a constant of the motion, and so is the quantity F(x,p,t) = x – pt/m. ƏF (a) Compare {H, F} with Prove that F is also a constant of the motion. Ət (b) Prove that the Poisson bracket of two constants of the motion (F(x,p,t) and G(x,p,t)) is itself a constant of the motion, even if the constants F(x,p,t) and G(x,p,t) depend explicitly on the time. (c) Show in general that if the Hamiltonian and a quantity F are constants of the motion ƏF is a constant of the motion too. Ət then
Transcribed Image Text:Consider the uniform motion of a free particle of mass m. The Hamiltonian is a constant of the motion, and so is the quantity F(x,p,t) = x – pt/m. ƏF (a) Compare {H, F} with Prove that F is also a constant of the motion. Ət (b) Prove that the Poisson bracket of two constants of the motion (F(x,p,t) and G(x,p,t)) is itself a constant of the motion, even if the constants F(x,p,t) and G(x,p,t) depend explicitly on the time. (c) Show in general that if the Hamiltonian and a quantity F are constants of the motion ƏF is a constant of the motion too. Ət then
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