A particle moves in the plane

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch Canonical transformations, generating functions and symplectic structure

Problem

A particle moves in the

xy

plane. Chapter 2 introduced plane polar coordinates

r=\sqrt{x^2+y^2}

\theta=\arctan(y/x)

as a point transformation of

x,y

; Chapter 13 defined the canonical momenta

p_x,p_y

conjugate to

x,y

. The generating function

F_2(x,y,P_r,P_\theta)=P_r\sqrt{x^2+y^2}+P_\theta\arctan(y/x)

is proposed to carry

(x,y,p_x,p_y)

to the polar phase-space variables

(r,\theta,P_r,P_\theta)

. ((a)) Using

p_x=\partial F_2/\partial x

and

p_y=\partial F_2/\partial y

, show that

P_r=\dfrac{xp_x+yp_y}{r}

and

P_\theta=xp_y-yp_x

. Evaluate

r

\theta

P_r

and

P_\theta

x=3.0\,\mathrm{m}

y=4.0\,\mathrm{m}

p_x=1.0\,\mathrm{kg\,m\,s^{-1}}

p_y=2.0\,\mathrm{kg\,m\,s^{-1}}

. ((b)) Using the Poisson bracket of Chapter 14,

\{f,g\}=\sum_k\left(\dfrac{\partial f}{\partial q_k}\dfrac{\partial g}{\partial p_k}-\dfrac{\partial f}{\partial p_k}\dfrac{\partial g}{\partial q_k}\right)

with

q_k=x,y

, evaluate

\{r,P_r\}

directly from

r(x,y)

and

P_r(x,y,p_x,p_y)

as functions of the old variables, at the same point, and say whether this one bracket is enough to show that the map is canonical.

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