1. Topics in Phase-Space Jeffrey Eldred1 1
Topics in Phase-Space
Jeffrey Eldred
Classical Mechanics and Electromagnetism
June 2018 USPAS at MSU 1 1 1 1 1 1

2. 2 2
Fixed Points 2
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
11/21/2018 2 2 2 2 2 2

3. Phase-space Vector Diagram3
Phase-space Vector Diagram
For a coupled pair of first-order time-independent differential equations, we can create a 2D vector diagram of the phase-space.
The fixed points, or equilibrium points, are where the derivatives of the coordinates are zero. 3
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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4. Stable & Unstable Fixed Points4
Stable & Unstable Fixed Points
For a stable fixed point, the motion near the fixed point is cyclic and bounded.
For an unstable fixed point, the motion diverges. 4
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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5. Other Types of Fixed Points with Damping/Excitation5
Other Types of Fixed Points with Damping/Excitation 5
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6. Linearization for Stability of a Fixed Point6
Linearization for Stability of a Fixed Point
We can determine the stability of a fixed point by studying the motion of trajectories near the fixed point. 6
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7. Linearization for Stability of a Fixed Point (cont.)7
Linearization for Stability of a Fixed Point (cont.)
The deviation from the fixed point can be written as a linear combination of the eigenvectors for A.
The instantaneous trajectory can be solved for an eigenvector:
The fixed point is unstable if and only if the real part of any of its eigenvalues is positive
The exponentiated matrix exp(A) is called the local Jacobian matrix.
The eigenvalues λ are called local Lyapunov coefficients. 7
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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8. Eigenvalues and Eigenvectors: Reminder8
Eigenvalues and Eigenvectors: Reminder
The eigenvalues λ and eigenvectors v of a matrix M fulfill:
The eigenvalues are given by the n solutions to the equation:
The eigenvector vi for the ith eigenvalue λi is given by solving:
Usually, the eigenvectors are then normalized by:
If there are n distinct eigenvalues there will be n orthogonal eigenvectors, but this is not necessarily guaranteed. 8
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
11/21/2018 8 8 8

9. Non-superposition of Fixed Points9
Non-superposition of Fixed Points
If I have separatrices and fixed points for system H1 and if I have separatrices and fixed points for system H2, what do I know about the separatrices and fixed points for H= H1 + H2? 9
Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
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10. 10 10
Equipotential Curves 10
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11. Time as degree-of-freedom11
Time as degree-of-freedom
If the Hamiltonian is not time-independent, we can find a time-independent Hamiltonian by defining a new spatial component
Example: Driven Harmonic Oscillator
As a time-dependent system,
As a time-independent system, 11
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11/21/2018 11 11 11

12. 12
Equipotential Curves
Recall that if a Hamiltonian is time-independent, than the Hamiltonian is conserved.
We can use the value of the Hamiltonian to specify a particular particle trajectory.
Particle trajectories do not cross.
For a fixed value of the Hamiltonian, we can write one coordinate in terms of the other along that trajectory. 12
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13. Equipotential Curves (cont.)13
Equipotential Curves (cont.)
Example: Synchrotron Motion / Pendulum
Separatrix: 13
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14. Higher Dimensional Equipotential14
Higher Dimensional Equipotential
In the higher dimensional space, particle trajectories still do not cross and we can still find equipotential subspaces for a fixed value of the Hamiltonian. 14
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