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.
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.
Related questions
Question
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.
Transcribed Image Text:1. Using Lagrangian formalism, find the equation of motion for a particle of mass m sliding on a
frictionless vertical elliptical wire, with horizontal principal axis a and vertical principal axis b,
in gravity g. Use the angle a giving the Cartesian coordinates of the mass as x(t) = a cos a(t),
y(t) = b sin a(t) to describe the motion.
(a) Verify that with the given parametrization, the bead lies on the ellipse for all time, so
that the constrained dynamics can be described by the single variable a(t).
(b) Derive the differential equation of motion for a.
(c) You should find a term in the equation of motion that is proportional to à?, as well as
the 'obvious' ä acceleration term. Show that the term proportional to a² vanishes in
the circular limit (i.e. a = b)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
