1. Mechanics

2. Cartesian Coordinates
Normal space has three coordinates.
x1, x2, x3
Replace x, y, z
Usual right-handed system
A vector can be expressed in coordinates, or from a basis.
Unit vectors form a basis x3 x2 x1
Summation convention

3. Cartesian Algebra Vector algebra requires vector multiplication.
Wedge product
Usual 3D cross product
The dot product gives a scalar from Cartesian vectors.
Kronecker delta:
dij = 1, i = j
dij = 0, i ≠ j
Permutation epsilon:
eijk = 0, any i, j, k the same
eijk = 1, if i, j, k an even permutation of 1, 2, 3
eijk = -1, if i, j, k an odd permutation of 1, 2, 3

4. Coordinate Transformation
A vector can be described by many Cartesian coordinate systems.
Transform from one system to another
Transformation matrix M x3 x2 x1
A physical property that transforms like this is a Cartesian vector.

5. Systems A system of particles has f = 3N coordinates.
Each Cartesian coordinate has two indices: xil
i =1 of N particles
l =1 of 3 coordinate indices
A set of generalized coordinates can be used to replace the Cartesian coordinates.
qm = qm(x11,…, xN3, t)
xil = xil(q1, …, qf, t)
Generalized coordinates need not be distances

6. General Transformation
Coordinate transformations can be expressed for small changes.
The partial derivatives can be expressed as a transformation matrix.
Jacobian matrix
A non-zero determinant of the transformation matrix guarantees an inverse transformation.

7. Generalized Velocity Velocity is considered independent of position.
Differentials dqm do not depend on qm
The complete derivative may be time dependent.
A general rule allows the cancellation of time in the partial derivative.
The total kinetic energy comes from a sum over velocities.
time fixed
time varying
general identity

8. Generalized Force Conservative force derives from a potential V.
Generalized force derives from the same potential.

9. Lagrangian A purely conservative force depends only on position.
Zero velocity derivatives
Non-conservative forces kept separately
A Lagrangian function is defined: L = T - V.
The Euler-Lagrange equations express Newton’s laws of motion.

10. Generalized Momentum
The generalized momentum is defined from the Lagrangian.
The Euler-Lagrange equations can be written in terms of p.
The Jacobian integral E is used to define the Hamiltonian.
Constant when time not explicit

11. Canonical Equations
The independence from velocity defines a new function.
The Hamiltonian functional H(q, p, t)
These are Hamilton’s canonical conjugate equations.

12. Space Trajectory
Motion along a trajectory is described by position and momentum.
Position uses an origin
References the trajectory
Momentum points along the trajectory.
Tangent to the trajectory
The two vectors describe the motion with 6 coordinates.
Can be generalized x3 x2 x1

13. Phase Trajectory
Ellipse for simple harmonic
Spiral for damped harmonic
The generalized position and momentum are conjugate variables.
6N-dimensional G-space
A trajectory is the intersection of 6N-1 constraints.
The product of the conjugate variables is a phase space volume.
Equivalent to action
Undamped
Damped

14. Pendulum Space The trajectory of a pendulum is on a circle.
Configuration space
Velocity tangent at each point
Together the phase space is 2-dimensional.
A tangent bundle
1-d position, 1-d velocity V1 S1 V1 S1 q

15. Phase Portrait
A series of phase curves corresponding to different energies make up a phase portrait.
Velocity for Lagrangian system
Momentum for Hamiltonian system
A simple pendulum forms a series of curves.
Potential energy normalized to be 1 at horizontal
E > 2
E = 2
E < 2

16. Phase Flow A region of phase space will evolve over time.
Large set of points
Consider conservative system
The region can be characterized by a phase space density. p q

17. Differential Flow
The change in phase space can be viewed from the flow.
Flow in
Flow out
Sum the net flow over all variables. p q

18. Liouville’s Theorem
Hamilton’s equations can be combined to simplify the phase space expression.
This gives the total time derivative of the phase space density.
Conserved over time

19. Ergodic Hypothesis
The phase trajectories for the pendulum form closed curves in G-space.
The curve consists of all points at the same energy.
A system whose phase trajectory covers all points at an energy is ergodic.
Energy defines all states of the system
Defines dynamic equilibrium
E > 2
E = 2
E < 2

20. Spherical Pendulum
A spherical pendulum has a spherical configuration space.
Trajectory is a closed curve
The phase space is a set of all possible velocities.
Each in a 2-d tangent plane
Complete 4-d G-space
The energy surface is 3-d.
Phase trajectories don’t cross
Don’t span the surface S2 S2 x V2

21. Non-Ergodic Systems The spherical pendulum is non-ergodic.
A phase trajectory does not reach all energy points
Two-dimensional harmonic oscillator with commensurate periods is non-ergodic.
Many simple systems in multiple dimensions are non-ergodic.
Energy is insufficient to define all states of a system.

22. Quasi-Ergodic Hypothesis
Equilibrium of the distribution of states of a system required ergodicity.
A revised definition only requires the phase trajectory to come arbitrarily close to any point at an energy.
This defines a quasi-ergodic system.

23. Quasi-Ergodic Definition
Define a phase trajectory on an energy (hyper)surface.
Point g(pi, qi) on the trajectory
Arbitrary point g’ on the surface
The difference is arbitrarily small.
Zero for ergodic system

24. Coarse Grain
A probability density r can be translated to a probability P.
Defined at each point
Based on volume Dl
The difference only matters if the properties are significantly different.
Relevance depends on ei, di
A coarse-grain approach becomes nearly quasi-ergodic.
Integrals become sums