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Mechanics
Cartesian Coordinates
 Normal space has three
coordinates.
 x1, x2, x3
 Replace x, y, z
 Usual right-handed system
x3

r

e1
 A vector can be expressed in
coordinates, or from a basis.
 Unit vectors form a basis

r  ( x1 , x 2 , x 3 )
3



1
2
3
r  x e1  x e2  x e3   x i ei  x i ei
i 1
Summation convention
x1
x2
Cartesian Algebra
 Vector algebra requires vector
multiplication.
 Wedge product
 Usual 3D cross product
 

a  b  e ijk ai b j ek
 The dot product gives a scalar
from Cartesian vectors.
 
a  b  ai bi
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
e ijke klm  d ild jm  d imd jl
Coordinate Transformation
 A vector can be described by
many Cartesian coordinate
systems.
x3
x3
x2
 Transform from one system to
another
 Transformation matrix M
x2
x1
x1
x j  M ij xi
xi  M ij x j
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.
xi
dxi  m dq m
q
 The partial derivatives can be
expressed as a transformation
matrix.
 xi l 
J  m
 q 
l
l
 Jacobian matrix
xi
0
m
q
l
 A non-zero determinant of the
transformation matrix
guarantees an inverse
transformation.
q m l
dq  l dxi
xi
m
Generalized Velocity
 Velocity is considered
independent of position.
 Differentials dqm do not
depend on qm
x
dxi  im dq m
q
l
l
time fixed
 The complete derivative may
be time dependent.
x
x
xi  im q m  i
q
t
 A general rule allows the
cancellation of time in the
partial derivative.
xi
xi

q m q m
 The total kinetic energy comes
from a sum over velocities.
T
l
l
l
l
time varying
l
1
2
 m (q
j
j
general identity
j 2
)
Generalized Force
 Conservative force derives
from a potential V.
V
Fil   l
xi
x
Qm   Fil im
q
i
l
 Generalized force derives from
the same potential.
V x
Qm   l im
i xi q
l
Qm  
V
q m
Lagrangian
d T
T
V


Q


m
dt q m q m
q m
d T d V
T
V



0
m
m
m
m
dt q
dt q
q
q
 A purely conservative force
depends only on position.
 Zero velocity derivatives
 Non-conservative forces kept
separately
d (T  V ) (T  V )

0
m
m
dt q
q
 A Lagrangian function is
defined: L = T  V.
d L
L

0
m
m
dt q
q
 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
p j (q j , q j ) 
p j 
L
q j
d L
L

dt q j q j
E
L j
q  L
j

q
H
L j
q  L  p j q j  L
q j
Canonical Equations
 The independence from velocity defines a new function.
 The Hamiltonian functional H(q, p, t)
H  p j q j  L
 These are Hamilton’s canonical conjugate equations.
dp j
H

j
q
dt
H dq j

p j
dt
H
L

t
t
Space Trajectory
 Motion along a trajectory is
described by position and
momentum.
x3
 Position uses an origin
 References the trajectory

p
 Momentum points along the
trajectory.
 Tangent to the trajectory
 The two vectors describe the
motion with 6 coordinates.
 Can be generalized

r
x1
x2
Phase Trajectory
 The generalized position and
momentum are conjugate
variables.
 Ellipse for simple harmonic
 Spiral for damped harmonic
 6N-dimensional G-space
p
 A trajectory is the intersection
of 6N-1 constraints.
q
Undamped
Damped
 The product of the conjugate
variables is a phase space
volume.
 Equivalent to action S  q j p j
Pendulum Space
 The trajectory of a pendulum is
on a circle.
 Configuration space
 Velocity tangent at each point
S1
V1
q
 Together the phase space is 2dimensional.
 A tangent bundle
 1-d position, 1-d velocity
V1
S1
Phase Portrait
 A series of phase curves corresponding to different energies
make up a phase portrait.
 Velocity for Lagrangian system
 Momentum for Hamiltonian system
 p
q,
E>2
E<2
E=2
q
 A simple pendulum
forms a series of
curves.
 Potential energy
normalized to be 1
at horizontal
Phase Flow
 A region of phase space will
evolve over time.
 Large set of points
 Consider conservative system
p
t  t2
t  t1
 The region can be
characterized by a phase
space density.
q
N  dV
dV  dq j dp j 
j
Differential Flow
 in  
dq j
dt
dp j  
dp j
dt
dq j
 out 




 q j 
 p j 
 q j 



dq j dp j   p j 
dp j dq j

q j
p j





  q j   p j 

dq jdp j   

dq jdp j
t
p j 
 q j
 
q j 
p j 


q j  

p j  
0
t
q j p j
p j 
j 
 q j
 The change in phase space
can be viewed from the flow.
 Flow in
 Flow out
 Sum the net flow over all
variables.
p
q j dp
j
dq j
p j
q
Liouville’s Theorem
 Hamilton’s equations can be
combined to simplify the phase
space expression.
H
  p j
q j
p j
p j
 This gives the total time
derivative of the phase space
density.
 Conserved over time
H
 q j
p j

q j
q j
0
 




q j 
p j   0
t
p j 
j 
 q j
d
0
dt
Ergodic Hypothesis
 p
q,
E>2
E<2
E=2
q
 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
Spherical Pendulum
 A spherical pendulum has a
spherical configuration space.
S2
 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
S2
 The energy surface is 3-d.
 Phase trajectories don’t cross
 Don’t span the surface
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 nonergodic.
 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.
g ( pi  pi , qi  qi )
pi  e i
qi  d i
 Zero for ergodic system
g ( pi , qi )
g ( pi  pi , qi  qi )
Coarse Grain
 A probability density  can be
translated to a probability P.
 Defined at each point
 Based on volume l
P(g )  l
P(g )  l 
 The difference only matters if
the properties are significantly
different.
l
 Relevance depends on ei, di
 A coarse-grain approach
becomes nearly quasi-ergodic.
 Integrals become sums
l 
A    ( pi , qi ) A( pi , qi )dl
A    ( pi , qi ) A( pi , qi ) p j q j
g 
j