aa r X i v : . [ phy s i c s . c l a ss - ph ] J un Hamilton’s Dynamics in Complex Phase Space
M. A. Shahzad
Department of Physics, Hazara University, Pakistan. ∗ (Dated: June 18, 2019)We present the basic formulation of Hamilton dynamics in complex phase space. We extendthe Hamilton’s function by including the imaginary part and find out the corresponding Hamil-ton’s canonical equation of motion. Example of simple harmonic motion are considered and thecorresponding trajectory are plotted on real and complex phase space. Let H ( q, p, t ) = U ( q, p, t ) + jV ( q, p, t ) be a complexHamilton’s function [1–3] in complex phase space, where q is the generalized coordinate and p the conjugate gener-alized momentum defined on the phase space ( q, p ). Thetotal differentiation of the Hamiltonian H ( q, p, t ) is givenby dH = ∂U∂q dq + j ∂V∂q dq + ∂U∂p dp + j ∂V∂p dp + ∂H∂t dt (1)The relation of Lagrangian function L ( q, ˙ q, t ) with Hamil-ton’s function is defined by the following equation H = p ˙ q − L ( q, ˙ q, t ) = p ˙ q − (cid:2) W ( q, ˙ q, t ) + jZ ( q, ˙ q, t ) (cid:3) (2)Taking total differentiation of equation (2), we obtain dH = pd ˙ q + ˙ qdp − ∂L∂ ˙ q dq − i ∂L∂q dq − ∂L∂t dt (3)Defining the generalized momentum conjugate to q as p = ∂L∂ ˙ q (4)and using ∂L∂q dq = ddt ∂L∂ ˙ q dq = ˙ pdq (5)we obtain dH = ˙ qdp − ˙ pdq − (cid:0) ∂L/∂t (cid:1) dt (6)Since, q and p are independent variables, the variationsof dq , dp and dt are mutually independent. As a resulttheir coefficients must be equal in Equation (1) and (6).Hence ˙ q = ∂U∂p + j ∂V∂p (7)˙ p = − ∂U∂q − j ∂V∂q (8)and ∂H∂t = − ∂L∂t (9) ∗ Email:[email protected]
These first-order differential equations are called asHamilton’s canonical equation of motion. By consider-ing a complex Hamilton function of the form H ( q, p, t ) = U ( q, p, t )+ jV ( q, p, t ), we obtained the Hamilton’s canon-ical equation of motion for the Hamilton’s function H ( q, p, t ) defined on complex phase space. In order to un-derstand the Hamilton dynamics in complex phase-space,we consider an example of simple harmonic motion de-fined by the following Hamilton’s function H ( q, p, t ) withimaginary part equal to zero, H ( q, p, t ) = U ( q, p, t ) + jV ( q, p, t ) = P m + k q From equation (7,8,9), we have˙ q = ∂H∂p = pm ˙ p = − ∂H∂q = − kq∂H∂t = 0Figure (1(a)) shown the phase space trajectory of simpleharmonic motion.Consider an example of simple harmonic motion definedby the following complex Hamilton’s function H ( q, p, t )with non-zero imaginary part, H ( q, p, t ) = U ( q, p, t ) + jV ( q, p, t ) = P m + j k q The corresponding equation of motion can be written as˙ q = ∂H∂p = pm ˙ p = − ∂H∂q = − jkq∂H∂t = 0Figure (1(b),1(c)) shown the trajectory of simple har-monic motion on complex phase space.Let F ( q, p ) and H ( q, p ), with H ( q, p ) = U ( q, p ) + jV ( q, p ), be two functions on phase space. Then thePoisson bracket is given by[ F, H ] = [
F, U ] + j [ F, V ] (10)In particular, [ q i , q i + jp k ] = [ q i , q i ] + j [ q i , p k ] = jδ ik and [ q i , jp k ] = [ q,
0] + j [ q, p ] = jδ ik . With the help ofHamilton’s equations (7,8,9), we have dFdt = [ F, H ] + ∂F∂t = [
F, U ] + j [ F, V ] + ∂F∂t −8 −6 −4 −2 0 2 4 6 8q−8−6−4−202468 p (a) −40 −30 −20 −10 0 10 20 30 40iq−40−2002040 p (b) −400 −200 0 200 400q−100−75−50−250255075100 i p (c) FIG. 1: Trajectories of simple harmonic motion onphase space ( q, p ), ( jq, p ) and ( q, jp ). Consider a bivariate Hamilton function H ′ ( q, p ) as-sociated to the univariate Hamilton function H ( z ) via H ′ ( x, y ) = U ( x, y ) + jV ( x, y ) = H ( z ) | z = x + jy . The totaldifferential is defined as [4] dH ′ = ∂H ′ ( q, p ) ∂q dq + ∂H ′ ( q, p ) ∂p dp Using H ′ ( q, p ) = U ( q, p ) + jV ( q, p ), we obtain dH ′ = ∂U ( q, p ) ∂q dq + j ∂V ( q, p ) ∂q dq + ∂U ( q, p ) ∂p dp + j ∂U ( q, p ) ∂p dp Since, z = q + jp and z ∗ = q − jp , we have dz = dq + jdpdz ∗ = dq − jdp and dq = 12 ( dz + dz ∗ ) dp = 12 j ( dz − dz ∗ )The total differential dH of a complex valued Hamiltonfunction H ( z ) can be expressed as dH = ∂H ( z ) ∂z dz + ∂H ( z ) ∂z ∗ dz ∗ (11)where the Wirtinger derivatives [5] are defined by ∂∂z = 12 h ∂∂q − j ∂∂p i (12) ∂∂z ∗ = 12 h ∂∂q + j ∂∂p i (13)Using H = p ˙ q − L , we have dH = pd ˙ q + ˙ qdp − ∂L∂z dz − ∂L∂z ∗ dz ∗ (14)where z = ( q + j ˙ q ) z ∗ = ( q − j ˙ q ) dq = 12 ( dz + dz ∗ ) d ˙ q = 12 j ( dz − dz ∗ )Equation (14) becomes dH = pd ˙ q + ˙ qdp − (cid:16) ∂∂q − j ∂∂ ˙ q (cid:17) L (cid:0) dq + jd ˙ q (cid:1) − (cid:16) ∂∂q + j ∂∂ ˙ q (cid:17) L (cid:0) dq − jd ˙ q (cid:1) (15)Using equation (4) and (5), equation (15) can be rewrit-ten as dH = ˙ qdp − ˙ pdq (16)Using dq = ( dz + dz ∗ ) /
2, and dp = ( dz − dz ∗ ) / j , wehave dH = ˙ q j ( dz − dz ∗ ) − ˙ p dz + dz ∗ ) (17)or dH = (cid:16) ˙ q j − ˙ p (cid:17) dz − (cid:16) ˙ q j + ˙ p (cid:17) dz ∗ (18)Comparing the coefficient of dz and dz ∗ in equation (18) with equation (11), we obtain˙ q = j (cid:16) ∂∂z − ∂∂z ∗ (cid:17) H (19)˙ p = − (cid:16) ∂∂z + ∂∂z ∗ (cid:17) H (20)Equation (19-20) are Hamilton equation of motion incomplex phase space. Substituting back the Wirtingerderivatives (equation (12,13)) in equation (19-20), we ob-tain Hamilton equation of motion in real phase space;˙ q = ∂H∂p (21)˙ p = − ∂H∂q (22)Equation (19-20) are basic Hamilton’s canonical equationof motion which can be used to understand the dynamicsof particles in complex phase space. [1] William Rowan Hamilton, On a General Method in Dy-namics, Philosophical Transactions of the Royal Society ,247-308, 1834.[2] William Rowan Hamilton, Second Essay on a GeneralMethod in Dynamics,
Philosophical Transactions of theRoyal Society , 95-144, 1835.[3] T. L. Chow, Classical Mechanics, 2nd Edition, Taylor andFrancis Group, 2013. [4] P. Henrici, Applied and Computational Complex Analysis,Volume 1: Power Series Integration Conformal MappingLocation of Zero,
Wiley-Interscience , 704, 1988.[5] W. Wirtinger, Zur formalen Theorie der Funktionen vonmehr komplexen Ver¨anderlichen,