7th Asia-Pacific Conference on Plasma Physics, 12-17, Nov. 2023, Nagoya, Japan.

Modification of Lie transform for the systemwith a fast-varying coordinate
Linjin Zheng
Institute for Fusion Studies, University of Texas at Austin∗ E-mail: lzheng@austin.utexas.edu
The Lie transform perturbation theory ofHamiltonian systems has been widely used in thefield of plasma physics, for example, for studying
the guided center motion of charged particles ina magnetic field and developing the nonlinear gyrokinetic theory, etc. Nowadays, plasma simulation has become an important tool, which leadsthe Lie transform perturbation theory receive further attention. This is because, unlike the standard
gyrokinetic theory, the Lie transform formulation
preserves Liouville’s theorem and the energy conservation law for time-independent systems. These
properties are especially important for long-periodsimulations. Besides, the Lie transform is a near
identity transformation. It allows the expansion
generator to be determined in the order-by-orderanalyses. Also, the backward transformation canbe obtained rather easily from the forward transformation. The developments are extensively reviewed
recently for example in Refs. [1] and [2], as well as
in the book in Ref. [3].
In this work, we show that the conventional Lietransform perturbation theory needs to be modified when considering a system with a fast-varying
coordinate for ordering consistency [4,3]. For theguiding center motion of a charged particle in amagnetic field, the gyrophase is this type of coordinates. This is related to the discrepancy of the results obtained by the direct expansion method andthe Lie transform formulation. The direct method
gives the Lagrangian to the first order as follows [2] emcdΓ =A + ub · dX +µdζmc
 2 eue−+ µB + φ dt,(1)2m

As shown in [4], two reasons cause this discrepancy. First, the ordering difference between thetemporal variation of gyrophase and that of theother phase space coordinates needs to be taken
into account. Second, it is also important to notethat the limit and derivative cannot be commutedin general. When these facts are taken into account,the near identity transformation rule for one form
related to the Lagrangian becomes(Lg Γ)µ=g ν (∂ν γµ − ∂µ γν )
−g ν (∂ν γδ ) ∂µ g1δ − (∂µ γδ ) ∂ν g1δ .

Here, γµ and Γµ are the one form Lagrangians before and after the near-identity transformation, andg ν is the near-identity transformation generator.
The contribution of the second term on the rightis new. When coupled with the fast-varying coordinate dζ, the second term on the right gives riseto the otherwise missed term (mc/e)µdζ, which isactually of the same order as that of the first term
and can only be obtained in the tedious higher-order
perturbation analyses in the conventional theory.
This resolves the discrepancy between the directand Lie transform treatments in the Lagrangian
perturbation theory for charged particle motion ina magnetic field.
The correction to the near identity transformation rule pointed out in this work [4] affects generally the Lie transform framework in the classicalmechanics for the system with a fast-varying coordinate. Especially, it modifies the derivation of thenonlinear gyrokinetic equations, which will be addressed as well.
References[1] J. R. Cary and R. G. Littlejohn, Annals Physics
while the conventional Lie transform theory yields 151, 1 (1983).
in the same order [5,6][2] J. R. Cary and A. J. Brizard, Rev. Mod. Phys.
 2
81, 693 (2009).
 e

ued
Γ=A + ub · dX −+ µB + φ dt. (2) [3] Linjin Zheng, Magnetically confined fusion plasmc
2mmas, volume 3, kinetic theory (IOP Publishing,Bristol, UK, 2022).
Here, u is parallel velocity, µ is magnetic moment,[4] Linjin Zheng, Phys. Plasmas 30, 042515 (2023).
X is the guiding center position, and ζ denotes thegyrophase. Comparing Eqs. (1) and (2) one can see [5] R. G. Littlejohn, J. Plasma Phys. 29, 111
that the gyromotion contribution “(mc/e)µdζ” is (1983).
missing in the Lie trnsform formulation, although [6] A. J. Brizard, Phys. Plasmas 2 459 (1998).
it is of the same order as the bounce motion contri- * This research is supported by Department of Energy Grants DE-FG02-04ER54742 .
bution “ub · dX”.