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

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

Phase-space Vector Diagram 3 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 11/21/2018 3 3 3

Stable & Unstable Fixed Points 4 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 11/21/2018 4 4 4

Other Types of Fixed Points with Damping/Excitation 5 Other Types of Fixed Points with Damping/Excitation 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 5 5 5

Linearization for Stability of a Fixed Point 6 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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 6 6 6

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 11/21/2018 7 7 7

Eigenvalues and Eigenvectors: Reminder 8 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

Non-superposition of Fixed Points 9 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 11/21/2018 9 9 9

10 10 Equipotential Curves 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 10 10 10 10 10 10

Time as degree-of-freedom 11 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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 11 11 11

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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 12 12 12

Equipotential Curves (cont.) 13 Equipotential Curves (cont.) Example: Synchrotron Motion / Pendulum Separatrix: 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 13 13 13

Higher Dimensional Equipotential 14 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 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/21/2018 14 14 14