Hamilton's Dynamics in Complex Phase Space

Home , Simple harmonic motion

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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 total diﬀerential is deﬁned as [4]

′ ′ ′ ∂H (q,p) ∂H (q,p) dH = dq + dp ∂q ∂p

′

p Using H (q,p)= U(q,p)+ jV (q,p), we obtain

− ′ ∂U(q,p) ∂V (q,p) dH = dq + j dq − ∂q ∂q ∂U(q,p) ∂U(q,p) + dp + j dp − ∂p ∂p

− ∗ − − − − Since, z = q + jp and z = q − jp, we have q dz = dq + jdp (a) ∗ dz = dq − jdp

and

1 ∗ dq = (dz + dz ) 2 1 ∗ dp = (dz − dz ) 2j The total diﬀerential dH of a complex valued Hamilton

p function H(z) can be expressed as

∂H(z) ∂H(z) ∗ dH = dz + dz (11) − ∂z ∂z∗ where the Wirtinger derivatives [5] are deﬁned by − ∂ 1 ∂ ∂ − − − − = h − j i (12) iq ∂z 2 ∂q ∂p ∂ 1 ∂ ∂ (b) = + j (13) ∂z∗ 2h∂q ∂pi Using H = pq˙ − L, we have

∂L ∂L ∗ dH = pdq˙ +qdp ˙ − dz1 − dz1 (14) ∂z1 ∂z1∗

where

z1 = (q + jq˙) ∗ 1 ip z = (q − jq˙)

1 ∗ − dq = (dz1 + dz1 ) 2 − 1 ∗ dq˙ = (dz1 − dz1 ) − 2j

− Equation (14) becomes − − q 1 ∂ ∂ dH = pdq˙ +qdp ˙ − − j L dq + jdq˙ (c) 2∂q ∂q˙ 1 ∂ ∂ FIG. 1: Trajectories of simple harmonic motion on − + j L dq − jdq˙ (15) phase space (q,p), (jq,p) and (q,jp). 2 ∂q ∂q˙ 3

Using equation (4) and (5), equation (15) can be rewrit- with equation (11), we obtain ten as ∂ ∂ q˙ = j − H (19) ∂z ∂z∗ dH =qdp ˙ − pdq˙ (16) ∂ ∂ p˙ = − + H (20) ∂z ∂z∗ Using dq = (dz + dz∗)/2, and dp = (dz − dz∗)/2j, we Equation (19-20) are Hamilton equation of motion in have complex phase space. Substituting back the Wirtinger derivatives (equation (12,13)) in equation (19-20), we ob- q˙ ∗ p˙ ∗ dH = (dz − dz ) − (dz + dz ) (17) tain Hamilton equation of motion in real phase space; 2j 2 ∂H q˙ = (21) or ∂p ∂H p˙ = − (22) q˙ p˙ q˙ p˙ ∗ ∂q dH = − dz − + dz (18) 2j 2 2j 2 Equation (19-20) are basic Hamilton’s canonical equation of motion which can be used to understand the dynamics Comparing the coeﬃcient of dz and dz∗ in equation (18) of particles in complex phase space.

[1] William Rowan Hamilton, On a General Method in Dy- [4] P. Henrici, Applied and Computational Complex Analysis, namics, Philosophical Transactions of the Royal Society, Volume 1: Power Series Integration Conformal Mapping 247-308, 1834. Location of Zero, Wiley-Interscience, 704, 1988. [2] William Rowan Hamilton, Second Essay on a General [5] W. Wirtinger, Zur formalen Theorie der Funktionen von Method in Dynamics, Philosophical Transactions of the mehr komplexen Ver¨anderlichen, Mathematische Annalen, Royal Society, 95-144, 1835. Volume 97,357-375, 1927. [3] T. L. Chow, Classical Mechanics, 2nd Edition, Taylor and Francis Group, 2013.