-
Previous Article
The projective symplectic geometry of higher order variational problems: Minimality conditions
- JGM Home
- This Issue
-
Next Article
An approximation theorem in classical mechanics
Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics
1. | Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany |
2. | Head of Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany |
References:
[1] |
E. Andruchow, G. Larotonda, L. Recht and A. Varela, The left invariant metric in the general linear group,, Journal of Geometry and Physics, 86 (2014), 241.
doi: 10.1016/j.geomphys.2014.08.009. |
[2] |
D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas (Second Edition),, Princeton reference, (2009).
doi: 10.1515/9781400833344. |
[3] |
A. M. Bloch, P. E. Crouch, N. Nordkvist and A. K. Sanyal, Embedded geodesic problems and optimal control for matrix Lie groups,, Journal of Geometric Mechanics, 3 (2011), 197.
doi: 10.3934/jgm.2011.3.197. |
[4] |
P. G. Ciarlet, Three-Dimensional Elasticity,, Number 1 in Studies in Mathematics and its Applications, (1988).
|
[5] |
C. De Boor, A naive proof of the representation theorem for isotropic, linear asymmetric stress-strain relations,, Journal of Elasticity, 15 (1985), 225.
doi: 10.1007/BF00041995. |
[6] |
M. P. do Carmo, Riemannian Geometry,, Birkhäuser Basel, (1992).
doi: 10.1007/978-1-4757-2201-7. |
[7] |
J.-H. Eschenburg and J. Jost, Differentialgeometrie und Minimalflächen,, Springer, (2007).
|
[8] |
S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry,, Springer, (1990).
doi: 10.1007/978-3-642-97242-3. |
[9] |
H. Hencky, Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern?,, Zeitschrift für Physik, 55 (1929), 145. |
[10] |
N. J. Higham, Functions of Matrices: Theory and Computation,, Society for Industrial and Applied Mathematics, (2008).
doi: 10.1137/1.9780898717778. |
[11] |
J. Jost, Riemannian Geometry and Geometric Analysis (2nd ed.),, Springer, (1998).
doi: 10.1007/978-3-662-22385-7. |
[12] |
K. Königsberger, Analysis 1,, Analysis, (2004). |
[13] |
J. Lankeit, P. Neff and Y. Nakatsukasa, The minimization of matrix logarithms: On a fundamental property of the unitary polar factor,, Linear Algebra and its Applications, 449 (2014), 28.
doi: 10.1016/j.laa.2014.02.012. |
[14] |
S. Lee, M. Choi, H. Kim and F. C. Park, Geometric direct search algorithms for image registration,, IEEE Transactions on Image Processing, 16 (2007), 2215.
doi: 10.1109/TIP.2007.901809. |
[15] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, volume 17., Springer, (1999).
doi: 10.1007/978-0-387-21792-5. |
[16] |
R. J. Martin and P. Neff, The GL(n)-geodesic distance on $SO(n)$,, in preparation, (2016). |
[17] |
A. Mielke, Finite elastoplasticity, Lie groups and geodesics on $SL(d)$,, in Geometry, (2002), 61.
doi: 10.1007/0-387-21791-6_2. |
[18] |
M. Moakher, Means and averaging in the group of rotations,, SIAM Journal on Matrix Analysis and Applications, 24 (2002), 1.
doi: 10.1137/S0895479801383877. |
[19] |
P. Neff, Convexity and coercivity in nonlinear, anisotropic elasticity and some useful relations,, Technical report, (2008). |
[20] |
P. Neff, B. Eidel and R. J. Martin, Geometry of logarithmic strain measures in solid mechanics,, Archive for Rational Mechanics and Analysis, 222 (2016), 507.
doi: 10.1007/s00205-016-1007-x. |
[21] |
P. Neff, B. Eidel, F. Osterbrink and R. Martin, A Riemannian approach to strain measures in nonlinear elasticity,, Comptes Rendus Mécanique, 342 (2014), 254.
doi: 10.1016/j.crme.2013.12.005. |
[22] |
P. Neff, Y. Nakatsukasa and A. Fischle, A logarithmic minimization property of the unitary polar factor in the spectral and frobenius norms,, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 1132.
doi: 10.1137/130909949. |
[23] |
C. H. Taubes, Differential Geometry: Bundles, Connections, Metrics and Curvature,, Oxford Graduate Texts in Mathematics, (2011).
doi: 10.1093/acprof:oso/9780199605880.001.0001. |
[24] |
B. Vandereycken, P.-A. Absil and S. Vandewalle, A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank,, IMA Journal of Numerical Analysis, 33 (2013), 481.
doi: 10.1093/imanum/drs006. |
show all references
References:
[1] |
E. Andruchow, G. Larotonda, L. Recht and A. Varela, The left invariant metric in the general linear group,, Journal of Geometry and Physics, 86 (2014), 241.
doi: 10.1016/j.geomphys.2014.08.009. |
[2] |
D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas (Second Edition),, Princeton reference, (2009).
doi: 10.1515/9781400833344. |
[3] |
A. M. Bloch, P. E. Crouch, N. Nordkvist and A. K. Sanyal, Embedded geodesic problems and optimal control for matrix Lie groups,, Journal of Geometric Mechanics, 3 (2011), 197.
doi: 10.3934/jgm.2011.3.197. |
[4] |
P. G. Ciarlet, Three-Dimensional Elasticity,, Number 1 in Studies in Mathematics and its Applications, (1988).
|
[5] |
C. De Boor, A naive proof of the representation theorem for isotropic, linear asymmetric stress-strain relations,, Journal of Elasticity, 15 (1985), 225.
doi: 10.1007/BF00041995. |
[6] |
M. P. do Carmo, Riemannian Geometry,, Birkhäuser Basel, (1992).
doi: 10.1007/978-1-4757-2201-7. |
[7] |
J.-H. Eschenburg and J. Jost, Differentialgeometrie und Minimalflächen,, Springer, (2007).
|
[8] |
S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry,, Springer, (1990).
doi: 10.1007/978-3-642-97242-3. |
[9] |
H. Hencky, Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern?,, Zeitschrift für Physik, 55 (1929), 145. |
[10] |
N. J. Higham, Functions of Matrices: Theory and Computation,, Society for Industrial and Applied Mathematics, (2008).
doi: 10.1137/1.9780898717778. |
[11] |
J. Jost, Riemannian Geometry and Geometric Analysis (2nd ed.),, Springer, (1998).
doi: 10.1007/978-3-662-22385-7. |
[12] |
K. Königsberger, Analysis 1,, Analysis, (2004). |
[13] |
J. Lankeit, P. Neff and Y. Nakatsukasa, The minimization of matrix logarithms: On a fundamental property of the unitary polar factor,, Linear Algebra and its Applications, 449 (2014), 28.
doi: 10.1016/j.laa.2014.02.012. |
[14] |
S. Lee, M. Choi, H. Kim and F. C. Park, Geometric direct search algorithms for image registration,, IEEE Transactions on Image Processing, 16 (2007), 2215.
doi: 10.1109/TIP.2007.901809. |
[15] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, volume 17., Springer, (1999).
doi: 10.1007/978-0-387-21792-5. |
[16] |
R. J. Martin and P. Neff, The GL(n)-geodesic distance on $SO(n)$,, in preparation, (2016). |
[17] |
A. Mielke, Finite elastoplasticity, Lie groups and geodesics on $SL(d)$,, in Geometry, (2002), 61.
doi: 10.1007/0-387-21791-6_2. |
[18] |
M. Moakher, Means and averaging in the group of rotations,, SIAM Journal on Matrix Analysis and Applications, 24 (2002), 1.
doi: 10.1137/S0895479801383877. |
[19] |
P. Neff, Convexity and coercivity in nonlinear, anisotropic elasticity and some useful relations,, Technical report, (2008). |
[20] |
P. Neff, B. Eidel and R. J. Martin, Geometry of logarithmic strain measures in solid mechanics,, Archive for Rational Mechanics and Analysis, 222 (2016), 507.
doi: 10.1007/s00205-016-1007-x. |
[21] |
P. Neff, B. Eidel, F. Osterbrink and R. Martin, A Riemannian approach to strain measures in nonlinear elasticity,, Comptes Rendus Mécanique, 342 (2014), 254.
doi: 10.1016/j.crme.2013.12.005. |
[22] |
P. Neff, Y. Nakatsukasa and A. Fischle, A logarithmic minimization property of the unitary polar factor in the spectral and frobenius norms,, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 1132.
doi: 10.1137/130909949. |
[23] |
C. H. Taubes, Differential Geometry: Bundles, Connections, Metrics and Curvature,, Oxford Graduate Texts in Mathematics, (2011).
doi: 10.1093/acprof:oso/9780199605880.001.0001. |
[24] |
B. Vandereycken, P.-A. Absil and S. Vandewalle, A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank,, IMA Journal of Numerical Analysis, 33 (2013), 481.
doi: 10.1093/imanum/drs006. |
[1] |
Raz Kupferman, Asaf Shachar. On strain measures and the geodesic distance to $SO_n$ in the general linear group. Journal of Geometric Mechanics, 2016, 8 (4) : 437-460. doi: 10.3934/jgm.2016015 |
[2] |
Vladimir S. Matveev and Petar J. Topalov. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. Electronic Research Announcements, 2000, 6: 98-104. |
[3] |
Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365 |
[4] |
Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028 |
[5] |
John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019 |
[6] |
David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477 |
[7] |
Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699 |
[8] |
François Gay-Balmaz, Cesare Tronci, Cornelia Vizman. Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups. Journal of Geometric Mechanics, 2013, 5 (1) : 39-84. doi: 10.3934/jgm.2013.5.39 |
[9] |
Mahesh Nerurkar. Forced linear oscillators and the dynamics of Euclidean group extensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1201-1234. doi: 10.3934/dcdss.2016049 |
[10] |
Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684 |
[11] |
Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275 |
[12] |
Jesus Idelfonso Díaz, Jean Michel Rakotoson. On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1037-1058. doi: 10.3934/dcds.2010.27.1037 |
[13] |
Kamil Otal, Ferruh Özbudak. Explicit constructions of some non-Gabidulin linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 589-600. doi: 10.3934/amc.2016028 |
[14] |
Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81 |
[15] |
Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841 |
[16] |
Axel Kohnert, Johannes Zwanzger. New linear codes with prescribed group of automorphisms found by heuristic search. Advances in Mathematics of Communications, 2009, 3 (2) : 157-166. doi: 10.3934/amc.2009.3.157 |
[17] |
Xiaomei Feng, Zhidong Teng, Fengqin Zhang. Global dynamics of a general class of multi-group epidemic models with latency and relapse. Mathematical Biosciences & Engineering, 2015, 12 (1) : 99-115. doi: 10.3934/mbe.2015.12.99 |
[18] |
Song-Mei Huan, Xiao-Song Yang. On the number of limit cycles in general planar piecewise linear systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2147-2164. doi: 10.3934/dcds.2012.32.2147 |
[19] |
Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493 |
[20] |
Ingrid Daubechies, Gerd Teschke, Luminita Vese. Iteratively solving linear inverse problems under general convex constraints. Inverse Problems & Imaging, 2007, 1 (1) : 29-46. doi: 10.3934/ipi.2007.1.29 |
2016 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]