A NOVEL METHOD FOR DETERMINING CAUSTICS IN CLASSICAL TRAJECTORIES.

1. A NOVEL METHOD FOR DETERMINING CAUSTICS IN CLASSICAL TRAJECTORIES. P. OLOYEDE1,2, G.V. MIL’NIKOV2, H. NAKAMURA1,2, ========= 1.GRADUATE UNIVERSITY FOR ADVANCED STUDIES & 2.INSTITUTE FOR MOLECULAR SCIENCE, OKAZAKI.

2. MOTIVATION • QUANTUM MECHANICAL CALCULATIONS ? ACCURATE BUT DIFFICULT AND EXPENSIVE • CLASSICAL MECHANICAL SIMULATIONS. EASY AND CHEAP BUT INACCURATE. NEXT BEST THING ? • ADD QUANTUM EFFECTS TO CLASSICAL SIMULATIONS. • HOW ?

3. HOW…? • NEED TO DETERMINE ENVELOPE OF FAMILY OF TRAJECTORY. (CAUSTICS). • AT TURNING POINT OF TRAJECTORY, CAUSTICS OCCUR. • CAUSTICS: |∂p/∂q| = ∞ OR |∂q/∂p|= 0.

4. BASIC EQUATIONS(Mil’nikov, Nakamura, JCP ,115, 6881,2001) • EQUATION IS PROGATED ALONG WITH TRAJECTORY. • DIFFERENTIAL EQUATION DIVERGES AT TURNING POINT. • AVOIDABLE ?!

5. AVOIDING DIVERGENCE • IN 1-D,∂p/ ∂q IS INVERTIBLE AND PROPAGATION CONTINUES. • FOR MULTI-DIMENSIONAL CASE: • D(p1,p2,…pN)INVERTIBLE ? D(q1,q2,…qN) • DIRECTION IS NON-INTUITIVE. • WHAT CAN BE LEARNT FROM THE PROJECTION OF TRAJECTORY FROM PHASE SPACE TO ORDINARY MOMENTUM- AND COORDINATE- SPACE ?

6. PHASE SPACE PROJECTION P-SPACE Q-SPACE • MIXED SPACE. • P-Q SPACE

7. TRANSFORMATIONS. • SEQUENCE OF CANONICAL TRANSFORMATIONS. • AT POINT OF TRANSFORMATION, • TRANSFORM A : ST*A*S, S – EIGENVECTOR. - THUS, DIVERGING ELEMENT IS ALWAYS AT POSITION (N,N). • INVERT ONLY ELEMENT (N,N). • COODRINATES AND MOMENTUM HAVE BECOME MIXED IN Ã WHICH IS NOW D(P1,P2,…PN) D(Q1,Q2,…QN)

8. TRANSFORMATIONS (Contd.) where: MOMENTUM Pi,. . ., PN-1 ( NEW)= pi,. . ., pN-1 (OLD) COORDINATES Qi,. . ., QN-1 ( NEW)= qi,. . ., qN-1 (OLD) • PN (NEW)= - qN (OLD) • QN (NEW)= pN (OLD) SIGN CHANGE ENSURES INVARIANCE OF HAMILTON’S EOM.

9. NEW REPRESENTATION • EQ. IN THE NEW REPRESENTATION: • NEW COEFFICIENTS HAVE BEEN DERIVED IN TERMS OF THE OLD ONES. e.g REPLACE qN BY -PN IN OLD B0 i.e OLD(∂2H/∂qN∂qN) = NEW(∂2H/∂PN∂PN) GIVING NEW (B0)NN AS OLD (Bc)NN

10. CLOSE-CAUSTICS • IN REACTION ZONE, CAUSTICS ARE NO LONGER PERIODIC. HERE PERIOD OF CAUSTICS CAN BE AS SMALL AS A TENTH OF THE PERIOD IN THE ASYMPTOTIC. • NICELY TREATED BY THIS METHOD AUTOMATICALLY USING SUCCESSIVE SERIES OF TRANSFORMATIONS.

11. NUMERICAL TEST (CHEMICAL REACTION, A+BC) • DIM POTENTIAL : A+BC COLLISION. TOTAL ANG. MOM., J= 0. COLLISION ENERGY 1.2eV, JROT=0, NV = 0. • REDUCES TO 4 X 4 DIMENSION SINCE Z, z = 0. • ANALYTICAL ∂pi/ ∂qj MATRIX USING CONSERVATION OF ENERGY AND MOMENTUM EQUATIONS.

12. INITIAL CONDITIONS

13. ASYMPTOTICS DIVERGENCES

14. REACTION ZONE DIVERGENCES

15. CAUSTICS & TURNING POINT.(Single Trajectory)

16. CAUSTICS FOR A FAMILY

17. INVARIANCE TO NUMBERTRANSFORMATIONS.

18. NUMERICAL TEST (CHAOTIC SYSTEM) • Henon- Heiles Potential. H=(1/2μ)*(px2 + py2)+(1/2)*(x2+y2)+λ(x2y-y3/3);λ,μ =1 • NO ANALYTICAL EXPRESSION. BUT AT TURNING POINT,∂q/∂p IS 0. • EQUATION IS ∂A/∂t =AHqqA + 1. A= ∂q/∂p • AFTER FEW STEPS, A IS INVERTED AND PROPAGATION PROCEEDS WITH ORIGINAL EQUATION.

19. HENON-HEILES POTENTIAL(POINCARE SURFACE)

20. HENON-HEILES POTENTIAL (CAUSTICS)

21. FUTURE DIRECTION. • APPLICATION OF THIS METHOD TO INCLUDE TUNNELING TRAJECTORY INTO MULTI-DIMENSIONAL REACTIONS. • FURTHER REFINEMENT IN ORDER TO GAIN INSIGHTS INTO ALL THE AREAS TO WHICH IT CAN BE APPLIED.