Mechanics
Mechanics. Cartesian Coordinates Normal space has three coordinates. x 1, x 2, x 3 Replace x, y, z Usual right-handed system A vector can be expressed.
Dec 20, 2015
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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 x1
x2
x3
1e
ii
ii
i exexexexexr
3
13
32
21
1
),,( 321 xxxr
Summation convention
r
Cartesian Algebra
Vector algebra requires vector multiplication.
Wedge product Usual 3D cross product
The dot product gives a scalar from Cartesian vectors.
Kronecker delta: ij = 1, i = j
ij = 0, i ≠ j
Permutation epsilon: ijk = 0, any i, j, k the same
ijk = 1, if i, j, k an even permutation of 1, 2, 3
ijk = -1, if i, j, k an odd permutation of 1, 2, 3
kjiijk ebaba
jlimjmilklmijk iibaba
Coordinate Transformation
A vector can be described by many Cartesian coordinate systems.
Transform from one system to another
Transformation matrix M
x1
x2
x3
jij
i xMx
iij
j xMx 1x
2x
3x
A physical property that transforms A physical property that transforms like this is a Cartesian vector.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(x1
1,…, xN3, t)
xil = xi
l(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.
mm
lil
i qq
xx
lil
i
mm x
x
0
m
li
q
x
m
li
q
xJ
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.
t
xq
q
xx
lim
m
lil
i
mm
lil
i qq
xx
m
li
m
li
q
x
q
x
time fixed
time varying
general identity
j
jj qmT 2
21 )(
Generalized Force
Conservative force derives from a potential V.
Generalized force derives from the same potential.
i
m
li
li
m q
x
x
VQ
li
ilx
VF
i
m
li
ilm q
xFQ
mm q
VQ
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.
mmmm q
VQ
q
T
q
T
dt
d
0
mmmm q
V
q
T
q
V
dt
d
q
T
dt
d
0)()(
mm q
VT
q
VT
dt
d
0
mm q
L
q
L
dt
d
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
jjj
j q
Lqqp
),(
jjj q
L
q
L
dt
dp
Lqq
LE j
j
LqpLqq
LH j
jj
j
Canonical Equations
The independence from velocity defines a new function. The Hamiltonian functional H(q, p, t)
These are Hamilton’s canonical conjugate equations.
dt
dq
p
H j
j
dt
dp
q
H j
j
t
L
t
H
LqpH jj
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
x1
x2
x3
p
r
Phase Trajectory
Ellipse for simple harmonic Spiral for damped harmonic
q
p
Undamped
Damped
The generalized position and momentum are conjugate variables.
6N-dimensional -space
A trajectory is the intersection of 6N-1 constraints.
The product of the conjugate variables is a phase space volume.
Equivalent to action jj pqS
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
Phase Portrait
A series of phase curves corresponding to different energies make up a phase portrait.
Velocity for Lagrangian system Momentum for Hamiltonian system
q
pq,
E < 2 E = 2
E > 2
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
The region can be characterized by a phase space density.
dVN
2tt 1tt
q
p
jjj
dpdqdV
Differential Flow
jj
jj
in qdt
dpp
dt
dq
jj
j
j
j
jjj pq
p
p
q
qpq
t
The change in phase space can be viewed from the flow.
Flow in Flow out
Sum the net flow over all variables.
jj
j
jjjj
j
jj
out
qpp
pppq
q
0
j j
jj
jj
jj
j p
pp
pq
qt
q
p
jq
jp
jpjq
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
jj
pq
H
jj
qp
H
0
j
j
j
j
q
q
p
p
0
jj
jj
j
pp
qqt
0dt
d
Ergodic Hypothesis
The phase trajectories for the pendulum form closed curves in -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
q
pq,
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 -space
The energy surface is 3-d. Phase trajectories don’t cross Don’t span the surface
S2
xV2
S2
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 (pi, qi) on the trajectory Arbitrary point ’ on the
surface
The difference is arbitrarily small.
Zero for ergodic system
iip
),( iiii qqpp
iiq
),( iiii qqpp ),( ii qp
Coarse Grain
A probability density can be translated to a probability P.
Defined at each point Based on volume
The difference only matters if the properties are significantly different.
Relevance depends on i, i
A coarse-grain approach becomes nearly quasi-ergodic.
Integrals become sums
)(P
jjjiiii qpqpAqpA
),(),(
)(P
dqpAqpA iiii ),(),(
Related Documents
