Function Reference

This section contains the description of the LTP functions. The functions and their purpose are summarized in the table below.

  
Acronyms 		 


CBH 		 - Campbell-Baker-Hausdorff formula


LB 		 - Lie bracket or product


LTP 		 - Lie Tools Package


PHB 		 - Philip Hall basis

Table 1: Main functions in LTP.

Function	Purpose
cbhexp	Calculates the exponent $Z_3\in\hat{L}(\bar{X}_m)$ resulting from the composition of exponential mappings in equation (2) via the CBH formula (including brackets up to a given order $k$).
createLBobjects	Declares the generators $\bar{X}_m$ of the Lie algebra $L_k(\bar{X}_m)$. If needed, it also permits to declare any number of linear combinations of these generators $\sum_{i=1}^{m}a_i X_i$ with symbolic coefficients $a_i$. The LTP assigns a name to each linear combination allowing it to be used by other LTP functions.
phb	Declares the generators $\bar{X}_m$ of the free nilpotent Lie algebra $L_k(\bar{X}_m)$ of degree $k$ and constructs a Hall basis for $L_k(\bar{X}_m)$.
phbize	Expresses any Lie monomial $X\in L_k(\bar{X}_m)$ in the Hall basis.
reduceLB	Reduces a general Lie polynomial $S\in L_k(\bar{X}_m)$ with symbolic coefficients to its simplest form in a given HB.
reduceLBT	Given a list of dependencies between the elements of the HB, reduces a general Lie polynomial $S\in L_k(\bar{X}_m)$ with symbolic coefficients to its simplest form.
regroupLB	Applies the distributivity properties (over addition and scalar multiplication) of the Lie product to an arbitrary Lie polynomial in $S\in L(\bar{X}_m)$ and collects its terms.
simpLB	Applies the distributivity over scalar multiplication property to a given Lie product $X\in L(\bar{X}_m)$ and returns the simplified product $\alpha Y = X$, together with its scalar symbolic component $\alpha$, and the Lie monomial $Y\in L(\bar{X}_m)$.
wner	Computes the right-hand side of equation (10) and expresses it in the HB, treating $\dot{\gamma}_i$ and $\gamma_i$, $i=1,\ldots,r$ as symbolic scalars.
wnde	Constructs the differential equation for the logarithmic coordinates $\gamma_i$ given by the Wei-Norman equation (12).

Table 2: Auxiliary functions in LTP.

Function	Purpose
ad	Calculates $(ad_X^n)Y$ for $X,Y\in L(\bar{X}_m)$.
bracketlen	Returns the length $l(G)$ of a Lie product $G\in L(\bar{X}_m)$.
calcLB	Given the symbolic expressions for two vector fields in the canonical coordinate system, calculates their Lie product.
calcLBdiffop	Given the symbolic expressions for two partial differential operators, calculates their Lie product.
codeCBHcf	Generates code in either Fortran or C for the evaluation of the scalar symbolic coefficients in a given Lie polynomial $S\in L(\bar{X}_m)$.
createSubsRel	Creates Maple substitution relations for the the symbolic evaluation of controls $u_i$, $i=0,\ldots,m$ in the dynamic system (15). These substitution relations can then be used to permit calculations involving systems with drift and to accommodate for piece-wise constant controls of arbitrary symbolic magnitude, as well as to allow the controls to switch at arbitrary symbolic moments in time.
ead	Computes the series expansion of $(e^X)Y(e^{-X})=(e^{ad_X})Y$. for $X,Y\in L(\bar{X}_m)$ including brackets up to a given order.
eadr	Computes the series expansion of $(e^X)Y(e^{-X})=(e^{ad_X})Y$. for $X,Y\in L_k(\bar{X}_m)$; re-expresses the result in the HB and further simplifies it according to a given list of dependencies involving the elements of the HB.
evalLB2expr	Returns a symbolic Maple expression for later evaluation of a Lie product of two vector fields, possibly containing symbolic scalars.
pead	Computes the product of exponentials $\prod_{i=1}^{n} e^{ad_{X_i}}X_{n+1}$ for $X,Y\in L(\bar{X}_m)$ including brackets up to a given order.
peadr	Computes the product of exponentials $\prod_{i=1}^{n} e^{ad_{X_i}}X_{n+1}$ for $X,Y\in L_k(\bar{X}_m)$; re-expresses the result in the HB and further simplifies it according to a given list of dependencies involving the elements of the HB.
posxinphb	Returns the position index $i$ of a Lie product $B_i$ in the HB.
selectLB	Extract, as a Maple symbolic expression for later use, the part of a given Lie polynomial $S\in L(\bar{X}_m)$ which contains brackets up to, greater than, or equal to a given order.