1. Canonical Transformations1 1
Canonical Transformations
Jeffrey Eldred
Classical Mechanics and Electromagnetism
June 2018 USPAS at MSU 1 1 1 1 1 1

2. 2 2
Poisson Brackets 2
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3. Poisson Brackets Are there any other invariants of motion?3
Poisson Brackets
Are there any other invariants of motion?
Poisson Brackets: 3
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4. Poisson Brackets Identities4
Poisson Brackets Identities
Properties of Poisson Brackets
Poisson Brackets for Canonical Variables: 4
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5. Poisson Brackets: Example5
Poisson Brackets: Example
This Hamiltonian is symmetric in x,y,z:
We might look for invariants of the form:
In fact, there are three invariants of the form: 5
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6. Poisson Brackets: Example (cont.)6
Poisson Brackets: Example (cont.)
Compute the Poisson Bracket:
fxy is an invariant.
By xyz-symmetry, fyz and fzx are also invariants. 6
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7. Poisson Brackets: Example (cont.)7
Poisson Brackets: Example (cont.)
We can use Poisson brackets of the invariants to find a new invariant:
We have found g, which is also an invariant!
Because: 7
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8. Poisson Brackets: Example (cont.)8
Poisson Brackets: Example (cont.)
Are there even more invariants?
But for an N-degree-of-freedom system, at most 2N-1 invariants.
We already have 5, so these new invariants hxy hyz hzx must be some function of the invariants we already have: 8
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9. Poisson Brackets: Example (cont.)9
Poisson Brackets: Example (cont.)
How these invariants appear for ε=0.2 9
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10. Poisson Brackets: Example (cont.)10
Poisson Brackets: Example (cont.)
How these invariants appear for ε=2.0 10
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11. Canonical Transformations11 11
Canonical Transformations 11
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12. Canonical Transformations12
Canonical Transformations
Consider a Hamiltonian system:
Make the change to a new set of coordinates:
Those new set of coordinates should have a new Hamiltonian, and will still follow Hamilton’s equations:
How do we ensure it is a “good transformation” so that the relationship between (qi,pi) and (Qi,Pi) remains the same? 12
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13. Canonical Transformations (1D proof)13
Canonical Transformations (1D proof)
Hamilton’s equations for the new coordinates,
Apply the chain rule to H:
Apply the chain rule to Q,P: 13
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14. Canonical Transformations (1D proof cont.)14
Canonical Transformations (1D proof cont.)
Use old Hamilton equations:
Solve the system of equations: 14
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15. Canonical Transformations (1D proof cont.)15
Canonical Transformations (1D proof cont.)
This condition is upheld if
And one of the following is also true
Because then the expression can be written 15
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16. Generating Functions q, Q independent q, P independent16
Generating Functions
q, Q independent
q, P independent
p, Q independent
p, P independent 16
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17. Simple Rescaling Example17
Simple Rescaling Example
Notice that phase-space area has to be conserved: 17
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18. Jacobian (single variable transformation)18
Jacobian (single variable transformation)
(Sometimes this is called J for Jacobian, but I’ll leave it as M.) 18
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19. Jacobian (multi-variate transformation)19
Jacobian (multi-variate transformation)
The physics convention is generally q1,p1,q2,p2….
Some mathematicians instead use q1,q2,…p1,p2… 19
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20. 20 20
Liouville’s Theorem 20
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21. Propagation in time as a Canonical Transformation21
Propagation in time as a Canonical Transformation
Propagate in time :
Calculate the Jacobian matrix: 21
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22. Volume Conservation Conservation of Phase-space Volume22
Volume Conservation
Conservation of Phase-space Volume
1) Canonical transformations conserve phase-space volume.
2) Propagation in time is itself a canonical transformation.
1&2) Phase-space volume is conserved over time. 22
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23. Liouville’s Theorem In a time-independent Hamiltonian system:23
Liouville’s Theorem
In a time-independent Hamiltonian system:
3) Particles that begin within a phase-space volume remain within that phase-space volume.
1&2) Phase-space volume is conserved over time.
From 1-3: Phase-space density is conserved.
Liouville's Theorem does not hold if:
A) There is a dissipative/excitative force
B) Particles are added/removed from the system.
C) The phase-space boundary of the particles
becomes more complex then the model allows 23
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24. 24
Filamentation 24
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25. Derivation of Action-Angle25 25
Derivation of Action-Angle
Coordinates 25
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26. Action-Angle Coordinates26
Action-Angle Coordinates 26
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27. Action-Angle (1D System)27
Action-Angle (1D System)
The formula for action is given by:
But let’s derive it. Consider a generic 1D Hamiltonian system:
Invent a generating function to new coordinates:
For J, phi to be action angle, we require:
(for a harmonic oscillator ω = E / J.)
Now we can calculate J from the angular frequency: 27
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28. Action-Angle (Time-Energy Form)28
Action-Angle (Time-Energy Form)
We can calculate J from the angular frequency:
And we can calculate the angular frequency from:
That gives us J, we can then calculate Φ: 28
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29. Action-Angle (Position-Momentum Form)29
Action-Angle (Position-Momentum Form)
We can calculate J from the angular frequency:
We can rewrite this in the familiar form 29
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30. Two methods for Action-Angle Variables30
Two methods for Action-Angle Variables
Method 1 (Position-Momentum Form):
Method 2 (Time-Energy Form): 30
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31. 31 31
Action-Angle Example 31
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32. Example: Triangle Well32
Example: Triangle Well
Given the Potential:
Calculate the Action: 32
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33. Triangle Well (cont.) Calculate Generating Function: Calculate Angle:33
Triangle Well (cont.)
Calculate Generating Function:
Calculate Angle:
by using
by using 33
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34. 34
Triangle Well (final)
After some algebra, coordinates from action-angle:
Calculate angular frequency from action:
We have found the trajectories:
by using 34
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