Mechanics

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

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

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.

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

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.

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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