# American Institute of Mathematical Sciences

December  2019, 6(2): 485-511. doi: 10.3934/jcd.2019025

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

 SMSAS, University of Kent, Canterbury, CT2 7FS, UK

* Corresponding author: Elizabeth L. Mansfield

Received  March 2019 Revised  September 2019 Published  November 2019

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 $SE(2)$.

Citation: Michele Zadra, Elizabeth L. Mansfield. Using Lie group integrators to solve two and higher dimensional variational problems with symmetry. Journal of Computational Dynamics, 2019, 6 (2) : 485-511. doi: 10.3934/jcd.2019025
##### References:
 [1] S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Physics Reports, 470 (2009), 151-238.  doi: 10.1016/j.physrep.2008.11.001. [2] S. Blanes, F. Casas and J. Ros, Improved high order integrators based on the Magnus expansion, BIT Numerical Mathematics, 40 (2000), 434-450.  doi: 10.1023/A:1022311628317. [3] F. Casas and B. Owren, Cost efficient Lie group integrators in the RKMK class, BIT Numerical Mathematics, 43 (2003), 723-742.  doi: 10.1023/B:BITN.0000009959.29287.d4. [4] E. Celledoni, H. Marthinsen and B. Owren, An introduction to Lie group integrators-basics, new developments and applications, Journal of Computational Physics, 257 (2014), 1040-1061.  doi: 10.1016/j.jcp.2012.12.031. [5] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds., NIST Digital Library of Mathematical Functions, Release 1.0.24 of 2019-09-15, URL http://dlmf.nist.gov/. [6] E. B. Dynkin, Calculation of the coefficients in the Campbell-Hausdorff formula, Doklady Akademii Nauk SSSR (N.S.), 57 (1947), 323–326, English translation available at: http://people.math.umass.edu/ gunnells/S14/lie/dynkin-BCHfmla.pdf. [7] K. Engø, A. Marthinsen and H. Z. Munthe-Kaas, Diffman: An object-oriented MATLAB toolbox for solving differential equations on manifolds, Applied Numerical Mathematics, 39 (2001), 323-347.  doi: 10.1016/S0168-9274(00)00042-8. [8] T. M. N. Gonçalves and E. L. Mansfield, Moving frames and conservation laws for Euclidean invariant Lagrangians, Studies in Applied Mathematics, 130 (2012), 134-166.  doi: 10.1111/j.1467-9590.2012.00566.x. [9] T. M. N. Gonçalves and E. L. Mansfield, On moving frames and Noether's conservation laws, Studies in Applied Mathematics, 128 (2011), 1-29.  doi: 10.1111/j.1467-9590.2011.00522.x. [10] T. M. N. Gonçalves and E. L. Mansfield, Moving frames and Noether's conservation laws-the general case, Forum Math. Sigma, 4 (2016), e29, 55 pp. doi: 10.1017/fms.2016.24. [11] J. K. Hale, Ordinary Differential Equations, Second edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. [12] B. C. Hall, Lie Groups, Lie Algebras and Representations: An Elementary Introduction, Second edition, Graduate Texts in Mathematics, 222. Springer, Cham, 2015. doi: 10.1007/978-3-319-13467-3. [13] A. Iserles, H. Z. Munthe-Kaas, S. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, Acta Numer., Cambridge Univ. Press, Cambridge, 9 (2000), 215-365.  doi: 10.1017/S0962492900002154. [14] A. Iserles and S. Nørsett, On the solution of linear differential equations in Lie groups, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 983-1020.  doi: 10.1098/rsta.1999.0362. [15] A. Iserles, S. P. Nørsett and A. F. Rasmussen, Time symmetry and high-order Magnus methods, Applied Numerical Mathematics, 39 (2001), 379-401.  doi: 10.1016/S0168-9274(01)00088-5. [16] I. A. Kogan and P. J. Olver, Invariant Euler-Lagrange equations and the invariant variational bicomplex, Acta Applicandae Mathematicae, 76 (2003), 137-193.  doi: 10.1023/A:1022993616247. [17] E. L. Mansfield, A Practical Guide to the Invariant Calculus, Cambridge Monographs on Applied and Computational Mathematics, 26. Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511844621. [18] P. C. Moan and J. Niesen, Convergence of the Magnus series, J. Found Comput Math, 8 (2008), 291-301.  doi: 10.1007/s10208-007-9010-0. [19] H. Munthe-Kaas, Lie-Butcher theory for Runge-Kutta methods, BIT Numerical Mathematics, 35 (1995), 572-587.  doi: 10.1007/BF01739828. [20] H. Munthe-Kaas, Runge-Kutta methods on Lie groups, BIT Numerical Mathematics, 38 (1998), 92-111.  doi: 10.1007/BF02510919. [21] H. Munthe-Kaas, Higher order Runge-Kutta methods on manifolds, Appl. Numer. Math., 29 (1999), 115-127.  doi: 10.1016/S0168-9274(98)00030-0. [22] H. Munthe-Kaas and A. Zanna, Numerical integration of ordinary differential equations on homogenous manifolds, In: Cucker F., Shub M. (eds) Foundations of Computational Mathematics. Springer, Berlin, Heidelberg. (1997), 305-315. doi: 10.1007/978-3-642-60539-0_24. [23] M. Suzuki, On the convergence of exponential operators-the Zassenhaus formula, BCH formula and systematic approximants, Commun.Math. Phys., 57 (1977), 193-200.  doi: 10.1007/BF01614161. [24] G. M. Tuynman, The derivation of the exponential map of matrices, Amer. Math. Monthly, 102 (1995), 818-820.  doi: 10.1080/00029890.1995.12004668.

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##### References:
 [1] S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Physics Reports, 470 (2009), 151-238.  doi: 10.1016/j.physrep.2008.11.001. [2] S. Blanes, F. Casas and J. Ros, Improved high order integrators based on the Magnus expansion, BIT Numerical Mathematics, 40 (2000), 434-450.  doi: 10.1023/A:1022311628317. [3] F. Casas and B. Owren, Cost efficient Lie group integrators in the RKMK class, BIT Numerical Mathematics, 43 (2003), 723-742.  doi: 10.1023/B:BITN.0000009959.29287.d4. [4] E. Celledoni, H. Marthinsen and B. Owren, An introduction to Lie group integrators-basics, new developments and applications, Journal of Computational Physics, 257 (2014), 1040-1061.  doi: 10.1016/j.jcp.2012.12.031. [5] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds., NIST Digital Library of Mathematical Functions, Release 1.0.24 of 2019-09-15, URL http://dlmf.nist.gov/. [6] E. B. Dynkin, Calculation of the coefficients in the Campbell-Hausdorff formula, Doklady Akademii Nauk SSSR (N.S.), 57 (1947), 323–326, English translation available at: http://people.math.umass.edu/ gunnells/S14/lie/dynkin-BCHfmla.pdf. [7] K. Engø, A. Marthinsen and H. Z. Munthe-Kaas, Diffman: An object-oriented MATLAB toolbox for solving differential equations on manifolds, Applied Numerical Mathematics, 39 (2001), 323-347.  doi: 10.1016/S0168-9274(00)00042-8. [8] T. M. N. Gonçalves and E. L. Mansfield, Moving frames and conservation laws for Euclidean invariant Lagrangians, Studies in Applied Mathematics, 130 (2012), 134-166.  doi: 10.1111/j.1467-9590.2012.00566.x. [9] T. M. N. Gonçalves and E. L. Mansfield, On moving frames and Noether's conservation laws, Studies in Applied Mathematics, 128 (2011), 1-29.  doi: 10.1111/j.1467-9590.2011.00522.x. [10] T. M. N. Gonçalves and E. L. Mansfield, Moving frames and Noether's conservation laws-the general case, Forum Math. Sigma, 4 (2016), e29, 55 pp. doi: 10.1017/fms.2016.24. [11] J. K. Hale, Ordinary Differential Equations, Second edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. [12] B. C. Hall, Lie Groups, Lie Algebras and Representations: An Elementary Introduction, Second edition, Graduate Texts in Mathematics, 222. Springer, Cham, 2015. doi: 10.1007/978-3-319-13467-3. [13] A. Iserles, H. Z. Munthe-Kaas, S. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, Acta Numer., Cambridge Univ. Press, Cambridge, 9 (2000), 215-365.  doi: 10.1017/S0962492900002154. [14] A. Iserles and S. Nørsett, On the solution of linear differential equations in Lie groups, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 983-1020.  doi: 10.1098/rsta.1999.0362. [15] A. Iserles, S. P. Nørsett and A. F. Rasmussen, Time symmetry and high-order Magnus methods, Applied Numerical Mathematics, 39 (2001), 379-401.  doi: 10.1016/S0168-9274(01)00088-5. [16] I. A. Kogan and P. J. Olver, Invariant Euler-Lagrange equations and the invariant variational bicomplex, Acta Applicandae Mathematicae, 76 (2003), 137-193.  doi: 10.1023/A:1022993616247. [17] E. L. Mansfield, A Practical Guide to the Invariant Calculus, Cambridge Monographs on Applied and Computational Mathematics, 26. Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511844621. [18] P. C. Moan and J. Niesen, Convergence of the Magnus series, J. Found Comput Math, 8 (2008), 291-301.  doi: 10.1007/s10208-007-9010-0. [19] H. Munthe-Kaas, Lie-Butcher theory for Runge-Kutta methods, BIT Numerical Mathematics, 35 (1995), 572-587.  doi: 10.1007/BF01739828. [20] H. Munthe-Kaas, Runge-Kutta methods on Lie groups, BIT Numerical Mathematics, 38 (1998), 92-111.  doi: 10.1007/BF02510919. [21] H. Munthe-Kaas, Higher order Runge-Kutta methods on manifolds, Appl. Numer. Math., 29 (1999), 115-127.  doi: 10.1016/S0168-9274(98)00030-0. [22] H. Munthe-Kaas and A. Zanna, Numerical integration of ordinary differential equations on homogenous manifolds, In: Cucker F., Shub M. (eds) Foundations of Computational Mathematics. Springer, Berlin, Heidelberg. (1997), 305-315. doi: 10.1007/978-3-642-60539-0_24. [23] M. Suzuki, On the convergence of exponential operators-the Zassenhaus formula, BCH formula and systematic approximants, Commun.Math. Phys., 57 (1977), 193-200.  doi: 10.1007/BF01614161. [24] G. M. Tuynman, The derivation of the exponential map of matrices, Amer. Math. Monthly, 102 (1995), 818-820.  doi: 10.1080/00029890.1995.12004668.
The two different paths $\gamma_1$, $\gamma_2$
2-Norm of the difference between the two moving frames
The imaginary component of $u$
Plots of solutions to the variational problem defined by (49), computed integrating the two different ways; the plots look identical to the naked eye
Absolute value of the difference between the two surfaces in Figure 4
Plots of solutions to the variational problem defined by (53), computed integrating the two different ways; the plots look identical to the naked eye
Absolute value of the difference between the two surfaces in Figure 6
2–norm of the difference between the two moving frames
A plot of the minimiser as an evolving curve, $(t, x(s,t), u(s,t))$
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 $h^4k$ $\frac{1}{24} \partial_x^3 R - \frac{1}{24} {\rm ad}_{ \mathcal{Q}^x}( \partial_x^2 R) + \frac{1}{24} {\rm ad}_{{ \partial_x^2 \mathcal{Q}^x}}(R)$ $h^3k^2$ $\frac{1}{12} \partial_x^2 \partial_y R - \frac{1}{24} {\rm ad}_{ \mathcal{Q}^x} ( \partial_x \partial_y R) -\frac{1}{24} {\rm ad}_{ \partial_x \mathcal{Q}^x} ( \partial_y R)$ $- \frac{1}{24} {\rm ad}_{ \mathcal{Q}^x} ({\rm ad}_{ \mathcal{Q}^y} ( \partial_x R)) -\frac{1}{12} {\rm ad}_R( \partial_x R) + \frac{1}{2} {\rm ad}_{ \partial_x \mathcal{Q}^y} ( \partial_x R)$ $+ \frac{1}{6} {\rm ad}_{ \partial_x^2 \mathcal{Q}^y} (R) - \frac{1}{24} {\rm ad}_{ \mathcal{Q}^y}( {\rm ad}_{ \partial_y \mathcal{Q}^x}(R)) -\frac{1}{24}{\rm ad}_{ \partial_x \mathcal{Q}^y}({\rm ad}_{ \mathcal{Q}^x}(R))$ $+ \frac{1}{8}{\rm ad}_{\left[ \partial_x \mathcal{Q}^y, \mathcal{Q}^x\right]} (R)+\frac{1}{8}{\rm ad}_{\left[ \mathcal{Q}^y, \partial_x \mathcal{Q}^x\right]}(R)$
 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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