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Using Lie group integrators to solve two and higher dimensional variational problems with symmetry

  • * Corresponding author: Elizabeth L. Mansfield

    * Corresponding author: Elizabeth L. Mansfield
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  • The theory of moving frames has been used successfully to solve one dimensional (1D) variational problems invariant under a Lie group symmetry. In the one dimensional case, Noether's laws give first integrals of the Euler–Lagrange equations. In higher dimensional problems, the conservation laws do not enable the exact integration of the Euler–Lagrange system. In this paper we use the theory of moving frames to help solve, numerically, some higher dimensional variational problems, which are invariant under a Lie group action. In order to find a solution to the variational problem, we need first to solve the Euler Lagrange equations for the relevant differential invariants, and then solve a system of linear, first order, compatible, coupled partial differential equations for a moving frame, evolving on the Lie group. We demonstrate that Lie group integrators may be used in this context. We show first that the Magnus expansions on which one dimensional Lie group integrators are based, may be taken sequentially in a well defined way, at least to order 5; that is, the exact result is independent of the order of integration. We then show that efficient implementations of these integrators give a numerical solution of the equations for the frame, which is independent of the order of integration, to high order, in a range of examples. Our running example is a variational problem invariant under a linear action of $ SU(2) $. We then consider variational problems for evolving curves which are invariant under the projective action of $ SL(2) $ and finally the standard affine action of $ SE(2) $.

    Mathematics Subject Classification: Primary: 58E12; Secondary: 65D30.


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  • Figure 1.  The two different paths $ \gamma_1 $, $ \gamma_2 $

    Figure 2.  2-Norm of the difference between the two moving frames

    Figure 3.  The imaginary component of $ u $

    Figure 4.  Plots of solutions to the variational problem defined by (49), computed integrating the two different ways; the plots look identical to the naked eye

    Figure 5.  Absolute value of the difference between the two surfaces in Figure 4

    Figure 6.  Plots of solutions to the variational problem defined by (53), computed integrating the two different ways; the plots look identical to the naked eye

    Figure 7.  Absolute value of the difference between the two surfaces in Figure 6

    Figure 8.  2–norm of the difference between the two moving frames

    Figure 9.  A plot of the minimiser as an evolving curve, $ (t, x(s,t), u(s,t)) $

    Table 1.  Table of Coefficients

    Order Monomial Coefficient
    2 hk R
    3 $h^2k$ $ \frac{1}{2} \partial_x R $
    4 $ h^3k $ $ \frac{1}{6} \partial_x^2 R -\frac{1}{12} {\rm ad}_{ \mathcal{Q}^x}( \partial_x R) + \frac{1}{12} {\rm ad}_{ \partial_x \mathcal{Q}^x}(R) $
    $ h^2k^2 $ $ \frac{1}{4} \partial_x \partial_y R -\frac{1}{6} {\rm ad}_{{\rm ad}_{ \mathcal{Q}^x}( \mathcal{Q}^y)}(R) - \frac{1}{12} {\rm ad}_{ \mathcal{Q}^y}({\rm ad}_{ \mathcal{Q}^x}(R)) $
    5 $ h^4k $ $ \frac{1}{24} \partial_x^3 R - \frac{1}{24} {\rm ad}_{ \mathcal{Q}^x}( \partial_x^2 R) + \frac{1}{24} {\rm ad}_{{ \partial_x^2 \mathcal{Q}^x}}(R) $
    $ h^3k^2 $ $ \frac{1}{12} \partial_x^2 \partial_y R - \frac{1}{24} {\rm ad}_{ \mathcal{Q}^x} ( \partial_x \partial_y R) -\frac{1}{24} {\rm ad}_{ \partial_x \mathcal{Q}^x} ( \partial_y R) $
    $ - \frac{1}{24} {\rm ad}_{ \mathcal{Q}^x} ({\rm ad}_{ \mathcal{Q}^y} ( \partial_x R)) -\frac{1}{12} {\rm ad}_R( \partial_x R) + \frac{1}{2} {\rm ad}_{ \partial_x \mathcal{Q}^y} ( \partial_x R) $
    $+ \frac{1}{6} {\rm ad}_{ \partial_x^2 \mathcal{Q}^y} (R) - \frac{1}{24} {\rm ad}_{ \mathcal{Q}^y}( {\rm ad}_{ \partial_y \mathcal{Q}^x}(R)) -\frac{1}{24}{\rm ad}_{ \partial_x \mathcal{Q}^y}({\rm ad}_{ \mathcal{Q}^x}(R)) $
    $ + \frac{1}{8}{\rm ad}_{\left[ \partial_x \mathcal{Q}^y, \mathcal{Q}^x\right]} (R)+\frac{1}{8}{\rm ad}_{\left[ \mathcal{Q}^y, \partial_x \mathcal{Q}^x\right]}(R) $
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