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

2 2 Poisson Brackets 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 2 2 2 2 2 2

Poisson Brackets Are there any other invariants of motion? 3 Poisson Brackets Are there any other invariants of motion? Poisson Brackets: 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 3 3 3

Poisson Brackets Identities 4 Poisson Brackets Identities Properties of Poisson Brackets Poisson Brackets for Canonical Variables: 4 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 4 4 4

Poisson Brackets: Example 5 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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 5 5 5

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 6 6 6

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 7 7 7

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 8 8 8

Poisson Brackets: Example (cont.) 9 Poisson Brackets: Example (cont.) How these invariants appear for ε=0.2 9 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 9 9 9

Poisson Brackets: Example (cont.) 10 Poisson Brackets: Example (cont.) How these invariants appear for ε=2.0 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 10 10 10

Canonical Transformations 11 11 Canonical Transformations 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 11 11 11 11 11 11

Canonical Transformations 12 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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 12 12 12

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 13 13 13

Canonical Transformations (1D proof cont.) 14 Canonical Transformations (1D proof cont.) Use old Hamilton equations: Solve the system of equations: 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 14 14 14

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 15 15 15

Generating Functions q, Q independent q, P independent 16 Generating Functions q, Q independent q, P independent p, Q independent p, P independent 16 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 16 16 16

Simple Rescaling Example 17 Simple Rescaling Example Notice that phase-space area has to be conserved: 17 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 17 17 17

Jacobian (single variable transformation) 18 Jacobian (single variable transformation) (Sometimes this is called J for Jacobian, but I’ll leave it as M.) 18 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 18 18 18

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 19 19 19

20 20 Liouville’s Theorem 20 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 20 20 20 20 20 20

Propagation in time as a Canonical Transformation 21 Propagation in time as a Canonical Transformation Propagate in time : Calculate the Jacobian matrix: 21 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 21 21 21

Volume Conservation Conservation of Phase-space Volume 22 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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 22 22 22

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 23 23 23

24 Filamentation 24 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 24 24 24

Derivation of Action-Angle 25 25 Derivation of Action-Angle Coordinates 25 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 25 25 25 25 25 25

Action-Angle Coordinates 26 Action-Angle Coordinates 26 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 26 26 26

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 27 27 27

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 28 28 28

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 29 29 29

Two methods for Action-Angle Variables 30 Two methods for Action-Angle Variables Method 1 (Position-Momentum Form): Method 2 (Time-Energy Form): 30 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 30 30 30

31 31 Action-Angle Example 31 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 31 31 31 31 31 31

Example: Triangle Well 32 Example: Triangle Well Given the Potential: Calculate the Action: 32 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 32 32 32

Triangle Well (cont.) Calculate Generating Function: Calculate Angle: 33 Triangle Well (cont.) Calculate Generating Function: Calculate Angle: by using by using 33 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 33 33 33

34 Triangle Well (final) After some algebra, coordinates from action-angle: Calculate angular frequency from action: We have found the trajectories: by using 34 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/15/2018 34 34 34