Peeter Joot [email protected]

Change of variables in 2d phase space 1.1

Motivation

In  problem 2.2, it’s suggested to try a spherical change of vars to verify explicitly that phase space volume is preserved, and to explore some related ideas. As a first step let’s try a similar, but presumably easier change of variables, going from Cartesian to cylindrical phase spaces. 1.2

Canonical momenta and Hamiltonian

Our cylindrical velocity is ˆ ˙ r + r θ˙ θ, v = rˆ

(1.1)

1 L = mv2 2  1 = m r˙ 2 + r2 θ˙ 2 . 2

(1.2)

so a purely kinetic Lagrangian would be

Our canonical momenta are ∂L = mr˙ ∂r˙ ∂L ˙ pθ = = mr2 θ. ∂θ˙ pr =

(1.3a) (1.3b)

and our kinetic energy is 1 2 1 pr + p2 . (1.4) 2m 2mr2 θ Now we need to express our momenta in terms of the Cartesian coordinates. We have for the radial momentum H=L=

1

pr = mr˙ q d x 2 + y2 =m dt  1 2m = x x˙ + yy˙ 2 r or pr =

(1.5)

 1 xp x + ypy r

(1.6)

dθ dt y 2 d = mr atan . dt x

pθ = mr2

(1.7)

After some reduction (cyclindrialMomenta.nb), we find pθ = py x − p x y.

(1.8)

We can assemble these into a complete set of change of variable equations q r = x 2 + y2 y θ = atan x  1 xp x + ypy pr = p x 2 + y2 pθ = py x − p x y.

(1.9a) (1.9b) (1.9c) (1.9d)

Our phase space volume element change of variables is ∂(r, θ, pr , pθ ) dxdydp x dpy ∂(x, y, p x , py ) √x √ y2 2 0 x2 +y2 x +y y x − 2 2 0 x +y x2 +y2 = y(ypx − xpy ) x( xpy −ypx ) √ x2 2 3/2 (x2 +y2 )3/2 x +y ( x2 +y2 ) py − px −y

drdθdpr dpθ =

0 dxdydp x dpy √ y2 2 x +y x 0

x 2 + y2 x 2 + y2  3/2  1/2 x 2 + y2 x 2 + y2 = dxdydp x dpy .

=

We see explicitly that this point transformation has a unit Jacobian, preserving area.

2

(1.10)

Bibliography  RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996. 1.1

3

## Peeter Joot [email protected] Change of variables in 2d phase ...

In  problem 2.2, it's suggested to try a spherical change of vars to verify explicitly that phase space volume is preserved, and to explore some related ideas. As a first step let's try a similar, but presumably easier change of variables, going from Cartesian to cylindrical phase spaces. 1.2 Canonical momenta and Hamiltonian.

#### Recommend Documents

Peeter Joot [email protected] Velocity ... - Peeter Joot's Blog
... momentum space, and calculated the corresponding momentum space volume element. Here's that calculation. 1.2 Guts. We are working with a Hamiltonian.

The Role of Distal Variables in Behavior Change ...
Jan 1, 2004 - Between 1991 and 1999, the proportion of 8th, 10th-, and ... watched the ads (embedded in a television program) on laptop computers and.

16.09b Change of Variables Continued.pdf
16.09b Change of Variables Continued.pdf. 16.09b Change of Variables Continued.pdf. Open. Extract. Open with. Sign In. Main menu.

Phase Change Memory in Enterprise Storage Systems
form hard disk drives (HDDs) along a number of dimensions. When compared to ... 1 Gbit PCM device. PCM technology stores data bits by alternating the phase.

On the Value of Variables
rewriting rules at top level, and then taking their closure by evaluation contexts. A peculiar aspect of the LSC is that contexts are also used to define the rules at top level. Such a use of contexts is how locality on proof nets (the graphical lang

On the Use of Variables in Mathematical Discourse - Semantic Scholar
This is because symbols have a domain and scope ... In predicate logic, the symbol x is considered a free ... x has to be considered a free variable, given the ab-.

Ultrafast-laser-induced parallel phase-change ...
carrier concentration in the amorphous phase.8 By applying a voltage of 5 V on the .... Waser, Nanoelectronics and Information Technology: Advanced Elec-.