a novel method for determining caustics in classical trajectories

A NOVEL METHOD A NOVEL METHOD FOR DETERMINING FOR DETERMINING

CAUSTICS IN CAUSTICS IN CLASSICAL CLASSICAL

TRAJECTORIES.TRAJECTORIES.P. OLOYEDEP. OLOYEDE1,21,2, G.V. MIL’NIKOV, G.V. MIL’NIKOV22, ,

H. NAKAMURAH. NAKAMURA1,21,2,,

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1.GRADUATE UNIVERSITY FOR 1.GRADUATE UNIVERSITY FOR ADVANCED STUDIES &ADVANCED STUDIES &

2.INSTITUTE FOR MOLECULAR 2.INSTITUTE FOR MOLECULAR SCIENCE, OKAZAKI.SCIENCE, OKAZAKI.

MOTIVATIONMOTIVATION QUANTUM MECHANICAL QUANTUM MECHANICAL

CALCULATIONS ? ACCURATECALCULATIONS ? ACCURATE

BUT DIFFICULT AND EXPENSIVEBUT DIFFICULT AND EXPENSIVE CLASSICAL MECHANICAL CLASSICAL MECHANICAL

SIMULATIONS. EASY AND CHEAPSIMULATIONS. EASY AND CHEAP

BUT INACCURATE.BUT INACCURATE.NEXT BEST THING ?NEXT BEST THING ? ADD QUANTUM EFFECTS TO ADD QUANTUM EFFECTS TO

CLASSICAL SIMULATIONS.CLASSICAL SIMULATIONS. HOW ?HOW ?

HOW…?HOW…? NEED TO DETERMINE ENVELOPE NEED TO DETERMINE ENVELOPE

OF FAMILY OF TRAJECTORY.OF FAMILY OF TRAJECTORY.

(CAUSTICS).(CAUSTICS).

AT TURNING POINT OF AT TURNING POINT OF TRAJECTORY, CAUSTICS OCCUR.TRAJECTORY, CAUSTICS OCCUR.

CAUSTICS: |∂p/∂q| = ∞ OR |∂q/∂p|= CAUSTICS: |∂p/∂q| = ∞ OR |∂q/∂p|= 0.0.

BASIC EQUATIONSBASIC EQUATIONS ((Mil’nikov, Nakamura, JCP ,115, Mil’nikov, Nakamura, JCP ,115,

6881,2001)6881,2001)

ji

2

C

ji

2

R

ji

2

L

ji

2

0

j

i

*C*R**L0

pp

HB

pq

HB

qp

HB

qq

HB

q

pA

;ABABAABBA

EQUATION IS EQUATION IS PROGATED PROGATED ALONG WITH ALONG WITH TRAJECTORY.TRAJECTORY.

DIFFERENTIADIFFERENTIALL EQUATION EQUATION DIVERGES AT DIVERGES AT TURNING TURNING POINT.POINT.

AVOIDABLE ?! AVOIDABLE ?!

AVOIDING DIVERGENCEAVOIDING DIVERGENCE IN 1-D,IN 1-D, ∂p/ ∂q IS INVERTIBLE AND PROPAGATION ∂p/ ∂q IS INVERTIBLE AND PROPAGATION

CONTINUES.CONTINUES.

FOR MULTI-DIMENSIONAL CASE:FOR MULTI-DIMENSIONAL CASE: D(pD(p11,p,p22,…p,…pNN)) INVERTIBLE ?INVERTIBLE ?

D(qD(q11,q,q22,…q,…qNN)) DIRECTION IS NON-INTUITIVE.DIRECTION IS NON-INTUITIVE.

WHAT CAN BE LEARNT FROM THE PROJECTION OF WHAT CAN BE LEARNT FROM THE PROJECTION OF TRAJECTORY FROM PHASE SPACE TO ORDINARY TRAJECTORY FROM PHASE SPACE TO ORDINARY MOMENTUM- AND COORDINATE- SPACE ?MOMENTUM- AND COORDINATE- SPACE ?

PHASE SPACE PHASE SPACE PROJECTIONPROJECTION

P-SPACEP-SPACE

Q-SPACEQ-SPACE MIXED SPACE.MIXED SPACE.

P-Q SPACEP-Q SPACE

TRANSFORMATIONS.TRANSFORMATIONS. SEQUENCE OF CANONICAL SEQUENCE OF CANONICAL

TRANSFORMATIONS.TRANSFORMATIONS. AT POINT OF TRANSFORMATION,AT POINT OF TRANSFORMATION,

TRANSFORM A : STRANSFORM A : STT*A*S, S – *A*S, S – EIGENVECTOR. EIGENVECTOR.

- - THUS, DIVERGING ELEMENT IS THUS, DIVERGING ELEMENT IS ALWAYS AT POSITION (N,N).ALWAYS AT POSITION (N,N).

INVERT ONLY ELEMENT (N,N).INVERT ONLY ELEMENT (N,N). COODRINATES AND MOMENTUM HAVE COODRINATES AND MOMENTUM HAVE

BECOME MIXED IN Ã WHICH IS NOWBECOME MIXED IN Ã WHICH IS NOW D(PD(P11,P,P22,…P,…PNN)) D(QD(Q11,Q,Q22,…Q,…QNN))

TRANSFORMATIONS TRANSFORMATIONS (Contd.)(Contd.)where:where:

MOMENTUMMOMENTUM

PPii,. . ., P,. . ., PN-1N-1 ( NEW)= p ( NEW)= pii,. . ., p,. . ., pN-1N-1 (OLD) (OLD)

COORDINATESCOORDINATES

QQii,. . ., Q,. . ., QN-1N-1 ( NEW)= q ( NEW)= qii,. . ., q,. . ., qN-1N-1 (OLD) (OLD)

PPNN (NEW)= - q (NEW)= - qNN (OLD) (OLD)

QQNN (NEW)= p (NEW)= pNN (OLD) (OLD)

SIGN CHANGE ENSURES INVARIANCE OF HAMILTON’S SIGN CHANGE ENSURES INVARIANCE OF HAMILTON’S EOM.EOM.

NEW REPRESENTATIONNEW REPRESENTATION EQ. IN THE NEW REPRESENTATION: EQ. IN THE NEW REPRESENTATION:

NEW COEFFICIENTS HAVE BEEN DERIVED NEW COEFFICIENTS HAVE BEEN DERIVED IN TERMS OF THE OLD ONES.IN TERMS OF THE OLD ONES.

e.g REPLACE qe.g REPLACE qNN BY -P BY -PNN IN OLD B IN OLD B00

i.e OLD(∂i.e OLD(∂22H/∂qH/∂qNN∂q∂qNN) = NEW(∂) = NEW(∂22H/∂PH/∂PNN∂P∂PNN))

GIVING NEW (BGIVING NEW (B00))NNNN AS OLD (Bc) AS OLD (Bc)NNNN

Ã*B~

à B~

à ÃB~

B~

à C*R**L0

CLOSE-CAUSTICSCLOSE-CAUSTICS

IN REACTION ZONE, CAUSTICS IN REACTION ZONE, CAUSTICS ARE NO LONGER PERIODIC. HERE ARE NO LONGER PERIODIC. HERE PERIOD OF CAUSTICS CAN BE AS PERIOD OF CAUSTICS CAN BE AS SMALL AS A TENTH OF THE SMALL AS A TENTH OF THE PERIOD IN THE ASYMPTOTIC.PERIOD IN THE ASYMPTOTIC.

NICELY TREATED BY THIS NICELY TREATED BY THIS METHOD AUTOMATICALLY USING METHOD AUTOMATICALLY USING SUCCESSIVE SERIES OF SUCCESSIVE SERIES OF TRANSFORMATIONS.TRANSFORMATIONS.

NUMERICAL TEST NUMERICAL TEST (CHEMICAL REACTION, (CHEMICAL REACTION,

A+BC)A+BC) DIM POTENTIAL : A+BC DIM POTENTIAL : A+BC

COLLISION. TOTAL ANG. MOM., J= COLLISION. TOTAL ANG. MOM., J= 0. COLLISION ENERGY 1.2eV, 0. COLLISION ENERGY 1.2eV, JROT=0, NV = 0.JROT=0, NV = 0.

REDUCES TO 4 X 4 DIMENSION REDUCES TO 4 X 4 DIMENSION SINCE Z, z = 0.SINCE Z, z = 0.

ANALYTICAL ∂pANALYTICAL ∂pii/ ∂q/ ∂qjj MATRIX USING MATRIX USING CONSERVATION OF ENERGY AND CONSERVATION OF ENERGY AND MOMENTUM EQUATIONS.MOMENTUM EQUATIONS.

INITIAL CONDITIONSINITIAL CONDITIONS

ASYMPTOTICS ASYMPTOTICS DIVERGENCESDIVERGENCES

REACTION ZONE REACTION ZONE DIVERGENCESDIVERGENCES

CAUSTICS & TURNING CAUSTICS & TURNING POINT.POINT.

(Single Trajectory)(Single Trajectory)

CAUSTICS FOR A CAUSTICS FOR A FAMILYFAMILY

INVARIANCE TO NUMBERINVARIANCE TO NUMBERTRANSFORMATIONS.TRANSFORMATIONS.

NUMERICAL TEST NUMERICAL TEST (CHAOTIC SYSTEM)(CHAOTIC SYSTEM)

Henon- Heiles Potential.Henon- Heiles Potential.

H=(1/2H=(1/2μμ)*(p)*(pxx22 + p + pyy

22)+(1/2)*(x)+(1/2)*(x22+y+y22)+)+λλ(x(x22y-y-yy33/3);/3); λλ,,μμ =1 =1

NO ANALYTICAL EXPRESSION. BUT NO ANALYTICAL EXPRESSION. BUT AT TURNING POINT,∂q/∂p IS 0. AT TURNING POINT,∂q/∂p IS 0.

EQUATION IS ∂A/∂t =AHqqA + 1. A= EQUATION IS ∂A/∂t =AHqqA + 1. A= ∂q/∂p∂q/∂p

AFTER FEW STEPS, A IS INVERTED AFTER FEW STEPS, A IS INVERTED AND PROPAGATION PROCEEDS WITH AND PROPAGATION PROCEEDS WITH ORIGINAL EQUATION.ORIGINAL EQUATION.

HENON-HEILES HENON-HEILES POTENTIALPOTENTIAL

(POINCARE SURFACE)(POINCARE SURFACE)

HENON-HEILES HENON-HEILES POTENTIALPOTENTIAL

(CAUSTICS) (CAUSTICS)

FUTURE DIRECTION.FUTURE DIRECTION.

APPLICATION OF THIS METHOD APPLICATION OF THIS METHOD TO INCLUDE TUNNELING TO INCLUDE TUNNELING TRAJECTORY INTO MULTI-TRAJECTORY INTO MULTI-DIMENSIONAL REACTIONS. DIMENSIONAL REACTIONS.

FURTHER REFINEMENT IN FURTHER REFINEMENT IN ORDER TO GAIN INSIGHTS ORDER TO GAIN INSIGHTS INTO ALL THE AREAS TO INTO ALL THE AREAS TO WHICH IT CAN BE APPLIED.WHICH IT CAN BE APPLIED.