math4248 weeks 12-13 1 topics: review of rigid body motion, legendre transformations, derivation of...

MATH4248 Weeks 12-13

1

Topics: review of rigid body motion, Legendre transformations, derivation of Hamilton’s equations of motion, phase space and Liouville’s theorem and Poincare’s recurrence theorem, examples including free fall, harmonic oscillator, and pendulums - plane, spherical and rotational, survey of advanced topics including Poisson brackets, Hamilton-Jacobi equation, action-angle variables, integrable systems and chaos

Objectives: To derive and understand Hamilton’s equations - in particular, their deep connections with symplectic geometry and topological dynamics

RIGID BODY MOTION-REVIEW

2

)Bdet()Adet()ABdet(

Definition The rotation group SO(3) consists of 3 x 3 matrices whose determinant equals 1 and whose inverses equal their transposes.

TTT AB)AB( 111 AB)AB(

Problem Show that SO(3) is a group under matrix multiplication using the following identities that hold for all 3 x 3 matrices A and B


3

z

y

x

100

0cossin

0sincos

z

y

x

Example Multiplication of a vector by the following matrix rotates the vector about the z-axis by angle

r)(Rr z

Example Show that rr


4

Example A rotational motion about the z-axis can be described by choosing the to be a function of t

r))t((R)t(r z

It is convenient to rewrite this equation as

)0(r))t((R)t(r z

where Tzzz ))0(θ(R))t(θ(R))t(θ(R

Problem Show that

Rt),3(SO))t((R z

id))0((R z

Problem Show that

(identity matrix)


5

Now, we can dispense with the primes to obtain thefollowing result: if a rigid body is rotating around the z-axis then there exists a function

)0(r))t((R)t(r z

that satisfies

Problem Show that the velocity of the particle satisfies

Rt),t(

body, the motion of that particle is described by the motion of its position vector by the following equation

0)0(

)t(r)t(S)t(r z

such that for any particle in

Tzzz ))t((R))t((R)t(S

where the matrix

is skew-symmetric


6

where

Problem Show that

)t(

0

0

)t(z

)t(r)t()t(r)t(S)t(r zz

hence

In this case we write

0

x

y

)t()t(r

)t(S)]t([ zz

Problem Analyse motion about the x and y axes


7

Problem Show that for general rotational motion ofa rigid body about the origin, there exists a functionO(t) with values O(t) in SO(3) such that O(0) = id (the identity matrix) and such that the motion of each particle in the rigid body is described by the equation

)0(r)t(O)t(r

Problem Differentiate this equation to obtain

)]t([)t(S

to show that S(t) is skew-symmetric then show thatid

T)t(O)t(O then differentiate the equation

T)t(O)t(O)t(S )t(r)t(S)t(r where

there exists )t(

(angular velocity in space) such that

hence )t(r)t()t(r

LEGENDRE TRANSFORMS

8

Definition Let f : R R be convex and differentiable.The Legendre transform of f is the function g : R R

Rx|)x(fpxmax)p(g

Example 1 2mx)x(f 2 0mxp|)x(fpx maxmaxxxdx

d m2p)p(gmpx 2

max Problem 1What is the Legendre transform of g ?

Problem 2 Interpret f and p if x is replaced by

x

LEGENDRE TRANSFORMS

9

Theorem 1 If

dRx|)x(fxpmax)p(g

is convex then itsRR:f d

is also a convex function and the Legendre transform of g equals f.

Theorem 2 If

p

Legendre transform, defined by

f is strictly convex, then maxx

determined from by the equation

maxxx|pxf

is

fgradRemarkxf

LEGENDRE TRANSFORMS

10

Example 2 Let G be a positive definite symmetric matrix and construct the function

xGp)x(fxpx

RR:f d

Then

pGx 1

max

pGp)p(g 1T

21

by xGx)x(f T

21

dd

HAMILTONIAN AS A LEGENDRE TRANSFORM

11

We will assume hereafter that the kinetic energy, and therefore the Lagrangian

RRRR:H ff Definition The Hamiltonian

)t,p,q(H

is the Legendre transformation of

with respect to

RRRR:L ff )t,q,q(L

is a strictly convex function of q

for all t,q

)t,q,q(L

q

HAMILTONIAN AND MOMENTUM

12

Is our new definition of the Hamiltonian, as a function of related to our old definition on page 16 of Weeks 8-9 vufoils as a function of ?

t,p,q

where

Answer Yes

is uniquely chosen to satisfy q

)t,q,q(pqL

LqLq)t,p,q(H

i ii

t,q,q

New Old

HAMILTON’s EQUATIONS

13

with dH computed from the old definition of H

Compare dH computed from the chain rule

to obtain Hamilton’s equations

dtpdqddHtH

pH

qH

dLpdqpqddH

dtqdpqddLtL

qL

tL

tH

qH

qL

pH ,p,q

LIOUVILLE’S THEOREM

14

Theorem Assume that L, and therefore H, does not depend explicitly on t. Then Hamilton’s system of 2f equations defines a volume preserving flow on the 2f-dimensional phase space with coordinates

Proof Let

f2f1f1 R)p,p,q,q()p,q(x

)x(g tbe the flow RtRRg ,

f2f2:

t defined by Hamilton’s equations so thatis the solution of the equations with initial value xLet D be any region in the phase space and define

RR:D by )Dg(volume)t( tD

It suffices to prove that

Rt,0)t(D


15

If h be the vector field )x(hx

tttt ggg

Now

then for small t,

Therefore )t(otidxh

xxg t

)t(ot)x(hx)x(g t

implies

txhtrace1dx

xxg

det)t( D

t

D

)t()t( DtgD and D was arbitrary, therefore it suffices to prove that

0)0(D


16

hdivtrace)0(xh

D

and the result follows since

Therefore

f1f1 qH

qH

pH

pH ,,,,,div

hdiv

0iiii q

Hf

1ipp

Hf

1iq

POINCARE’S RECURRENCE THEOREM

17

Proof We consider the images of U under iterates of g

Theorem Let g be a volume preserving continuous one-to-one mapping that maps a bounded region D of Euclidean space onto itself : gD = D. Then in any nonempty open subset U of D there exists a point x which returns to U, i.e. for some Uxg

n .0n

,Ug,,Ug,gU,U n2

If they were pairwise disjoint then D would have infinite volume, contradicting our bounded assumptionhence for some

0UgUg,0jkkj

Ugxggx jkkj

POINCARE’S RECURRENCE THEOREM

18

Corollary Poincare’s theorem shows that the flow is ergodic, that is the phase space can not be divided into two subsets that are invariant under g and have positive volume. The ergodic theorem implies that almost all points return will be recurrent

Many mechanical systems have this property. A particle moving in a cup will (with probability one) return arbitrarily close to itself even though, if the cup is unsymmetric, the motion is unpredictably chaotic!

These qualitative properties are studied in topological dynamics, a field motivated by the three body problem

PLANE PENDULUM

19

Lagrangian

θ g

θcosmgθmL 22

21

momentum θmθLp 2θ

Hamiltonian

θcosmgLpθH2

2θ

θm2

p

equationsθsinmg

θ

Hθp,

m

θp

p

Hθ

2θ

PLANE PENDULUM

20

Phase space should be thought of as a cylinder

Orbits are oscillations clockwise around (q,p)=(0,0)

mgEmg

mgE

EmgOrbits are fixed points with pendulum down

Orbits are either separatrices (infinitely long swing), or unstable fixed points with pendulum up

mgEOrbits are rotations clockwise (top), or counterclockwise (bottom)

SPHERICAL PENDULUM

21

θcosmgθ2sin2m2

p

2m2

pH

22θ

Lagrangian

θ g

m

cosmg)sin(m21L

2222

θmθp2

θsinmp22

Hamiltonian

Momenta

2θ mp θsinmg

θsinm

cospp

22

2

θ

θsinm

p

22

2

0p

ROTATING PENDULUM

22

θcosmg2

θ2sin22m

m2

pH -

2

2θ

Lagrangian

θ g

m

cosmg)sin(m21L

2222

θmθp2

Hamiltonian

Momenta

θsinmgcosθsinmp22

θ 2

θ mp

POISSON BRACKETS

23

Theorem A transformation is canonical (satisfies Hamilton’s equations for someHamiltonian K(Q,P) iff the Poisson bracket

1]P,Q[qP

pQ

pP

qQ

p,q

dpp

QPdq

q

QPpPdQpdq

or equivalently the following differential form is exact

)p,q(PP),p,q(QQ

i.e. equals dF for some generating function F= F(q,Q)

CANONICAL TRANSFORMATIONS

24

Example (f=1) Let Q, P be ‘polar’ coordinates, so that

PH

Problem Show that this is a canonical transformationAnd compute its generating function

2

2q2mm2

2p1P,qmp1

cotQ

Problem Show that the Hamiltonian for the harmonicoscillator becomes and therefore

P,tQ0P,Q

P, Q is an action variable, angle variable They give

)tcos(m2p),tsin(m2q

HAMILTON-JACOBI THEORY INTEGRABLE SYSTEMS AND CHAOS

25

HJ theory provides the most powerful method to solvemechanical problems. Hamilton’s principle function

is the minimal action

0t

S)t,

q

S,q(H

is a generating function from (q,p) to

q

0qLdtmin)t,q;t,q(S 00H

)p,q( 00

It can be computed from Jacobi’s complete integral S

by the Hamilton Jacobi eqn

and the eqn )t,,q(S)t,,q(S)t,q;t,q(S 000H

A system is completely integrable if f-pairs of action-angle variables exist, else it tends to exhibit chaos

math4248 weeks 12-13 1 topics: review of rigid body motion, legendre transformations, derivation of...

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