Figure 1.
A rigid transformation with spatial and body frames
- Previous Article
- JGM Home
- This Issue
-
Next Article
On the relationship between the energy shaping and the Lyapunov constraint based methods
December
2017, 9(4): 487-574.
doi: 10.3934/jgm.2017019
The physical foundations of geometric mechanics
Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada |
The principles of geometric mechanics are extended to the physical elements of mechanics, including space and time, rigid bodies, constraints, forces, and dynamics. What is arrived at is a comprehensive and rigorous presentation of basic mechanics, starting with precise formulations of the physical axioms. A few components of the presentation are novel. One is a mathematical presentation of force and torque, providing certain well-known, but seldom clearly exposited, fundamental theorems about force and torque. The classical principles of Virtual Work and Lagrange-d'Alembert are also given clear mathematical statements in various guises and contexts. Another novel facet of the presentation is its derivation of the Euler-Lagrange equations. Standard derivations of the Euler-Lagrange equations from the equations of motion for Newtonian mechanics are typically done for interconnections of particles. Here this is carried out in a coordinate-free rmner for rigid bodies, giving for the first time a direct geometric path from the Newton-Euler equations to the Euler-Lagrange equations in the rigid body setting.
Keywords:
Rigid body dynamics,
force and torque,
Newton-Euler equations,
Euler-Lagrange equations,
nonholonomic constraints.
Mathematics Subject Classification:
Primary: 70E55; Secondary: 70G45, 70G65, 70G75, 70H03.
Citation:
Andrew D. Lewis. The physical foundations of geometric mechanics. Journal of Geometric Mechanics,
2017, 9
(4)
: 487-574.
doi: 10.3934/jgm.2017019
![]()
References:
| [1] |
R. Abraham and J. E. Marsden,
Foundations of Mechanics, 2nd edition, Addison Wesley, Reading, MA, 1978. |
| [2] |
R. Abraham, J. E. Marsden and T. S. Ratiu,
Manifolds, Tensor Analysis, and Applications, 2nd edition, no. 75 in Applied Mathematical Sciences, Springer-Verlag, New York/Heidelberg/Berlin, 1988.
doi: 10.1007/978-1-4612-1029-0. |
| [3] |
V. I. Arnol'ed,
Mathematical Methods of Classical Mechanics, 2nd edition, no. 60 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1978, New edition: [ |
| [4] |
V. I. Arnol'd,
Mathematical Methods of Classical Mechanics, 2nd edition, no. 60 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1989, First edition: [ |
| [5] |
R. E. Artz,
Classical mechanics in Galilean space-time, Foundations of Physics, 11 (1981), 679-697.
doi: 10.1007/BF00726944. |
| [6] |
M. Berger,
Geometry I, Universitext, Springer-Verlag, New York/Heidelberg/Berlin, 1987.
doi: 10.1007/978-3-540-93815-6. |
| [7] |
A. Bhand and A. D. Lewis, Rigid body mechanics in Galilean spacetimes Journal of Mathematical Physics, 46 (2005), 102902, 29 pp.
doi: 10.1063/1.2060547. |
| [8] |
A. M. Bloch,
Nonholonomic Mechanics and Control, no. 24 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 2003.
doi: 10.1007/b97376. |
| [9] |
B. Brogliato,
Nonsmooth Mechanics, Third edition. Communications and Control Engineering Series. Springer, [Cham], 2016.
doi: 10.1007/978-3-319-28664-8. |
| [10] |
F. Bullo and A. D. Lewis,
Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Systems, no. 49 in Texts in Applied Mathematics. Springer-Verlag, New York, 2005.
doi: 10.1007/978-1-4899-7276-7. |
| [11] |
H. Cendra, D. D. Holm, J. E. Marsden and T. S. Ratiu,
Lagrangian reduction, the Euler-Poincaré equations, and semidirect products, American Mathematical Society Translations, Series 2, 186 (1998), 1-25.
doi: 10.1090/trans2/186/01. |
| [12] |
D. L. Cohn,
Measure Theory, 2nd edition, Birkhäuser Advanced Texts, Birkhäuser, Boston/Basel/Stuttgart, 2013.
doi: 10.1007/978-1-4614-6956-8. |
| [13] |
J. Cortés, M. de León, D. M. de Diego and S. Martínez,
Mechanical systems subjected to generalized non-holonomic constraints, Proceedings of the Royal Society. London. Series A. Mathematical and Physical Sciences, 457 (2001), 651-670.
doi: 10.1098/rspa.2000.0686. |
| [14] |
M. Crampin,
On the concept of angular velocity, European Journal of Physics, 7 (1986), 287-293.
doi: 10.1088/0143-0807/7/4/014. |
| [15] |
M. R. Flannery,
The enigma of nonholonomic constraints, American Journal of Physics, 73 (2005), 265-272.
doi: 10.1119/1.1830501. |
| [16] |
C. Glocker,
Set Valued Force Laws, no. 1 in Lecture Notes in Applied Mechanics, Springer-Verlag, New York/Heidelberg/Berlin, 2001.
doi: 10.1007/978-3-540-44479-4. |
| [17] |
H. Goldstein,
Classical Mechanics, Addison Wesley, Reading, MA, 1951, New edition: [ |
| [18] |
H. Goldstein, C. P. Poole, Jr and J. L. Safko,
Classical Mechanics, 3rd edition, Addison Wesley, Reading, MA, 2001, Original edition: [ |
| [19] |
X. Grácia, J. Marin-Solano and M.-C. Muñoz-Lecanda,
Some geometric aspects of variational calculus in constrained systems, Reports on Mathematical Physics, 51 (2003), 127-148.
doi: 10.1016/S0034-4877(03)80006-X. |
| [20] |
A. Hatcher,
Algebraic Topology, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2002. |
| [21] |
D. Husemoller,
Fibre Bundles, 3rd edition, no. 20 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1994.
doi: 10.1007/978-1-4757-2261-1. |
| [22] |
S. Jafarpour and A. D. Lewis,
Time-Varying Vector Fields and Their Flows, Springer Briefs in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 2014.
doi: 10.1007/978-3-319-10139-2. |
| [23] |
Y. Kanno,
Nonsmooth Mechanics and Convex Optimization, CRC Press, Boca Raton, FL, 2011.
doi: 10.1201/b10839. |
| [24] |
P. V. Kharlomov,
A critique of some mathematical models of mechanical systems with differential constraints, Journal of Applied Mathematics and Mechanics. Translation of the Soviet journal Prikladnaya Matematika i Mekhanika, 56 (1992), 584-594.
doi: 10.1016/0021-8928(92)90016-2. |
| [25] |
S. Kobayashi and K. Nomizu,
Foundations of Differential Geometry, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. |
| [26] |
V. V. Kozlov,
The problem of realizing constraints in dynamics, Journal of Applied Mathematics and Mechanics. Translation of the Soviet journal Prikladnaya Matematika i Mekhanika, 56 (1992), 594-600.
doi: 10.1016/0021-8928(92)90017-3. |
| [27] |
J. L. Lagrange,
Méchanique Analitique, Chez la Veuve Desaint, Paris, 1788, Translation: [ |
| [28] |
J. L. Lagrange,
Analytical Mechanics, No. 191 in Boston Studies in the Philosophy of Science, Kluwer
Academic Publishers, Dordrecht, 1997, Original edition: [ |
| [29] |
A. D. Lewis,
The geometry of the Gibbs-Appell equations and Gauss's Principle of Least Constraint, Reports on Mathematical Physics, 38 (1996), 11-28.
doi: 10.1016/0034-4877(96)87675-0. |
| [30] |
A. D. Lewis,
Affine connections and distributions with applications to nonholonomic mechanics, Reports on Mathematical Physics, 42 (1998), 135-164.
doi: 10.1016/S0034-4877(98)80008-6. |
| [31] |
A. D. Lewis,
Is it worth learning differential geometric methods for modelling and control of mechanical systems?, Robotica, 26 (2007), 765-777.
|
| [32] |
A. D. Lewis and R. M. Murray,
Variational principles for constrained systems: Theory and experiment, International Journal of Non-Linear Mechanics, 30 (1995), 793-815.
doi: 10.1016/0020-7462(95)00024-0. |
| [33] |
P. Liberrmn and C. -M. Marle,
Symplectic Geometry and Analytical Mechanics, no. 35 in Mathematics and its Applications, D. Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo, 1987.
doi: 10.1007/978-94-009-3807-6. |
| [34] |
J. E. Marsden and T. R. Hughes,
Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983, Reprint: [ |
| [35] |
J. E. Marsden and T. R. Hughes,
Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Original
edition: [ |
| [36] |
J. E. Marsden and J. Scheurle,
The reduced Euler-Lagrange equations, Fields Institute Communications, Fields Institute, 1 (1993), 139-164.
|
| [37] |
M. D. P. Monteiro Marques,
Differential Inclusions in Nonsmooth Mechanical Problems, No. 9 in Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston/Basel/Stuttgart, 1993.
doi: 10.1007/978-3-0348-7614-8. |
| [38] |
R. M. Murray, Z. X. Li and S. S. Sastry,
A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994. |
| [39] |
O. M. O'Reilly,
Intermediate Dynamics for Engineers, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2008.
doi: 10.1017/CBO9780511791352. |
| [40] |
J. G. Papastavridis,
Tensor Calculus and Analytical Dynamics, Library of Engineering Mathematics, CRC Press, Boca Raton, FL, 1999. |
| [41] |
L. A. Pars,
A Treatise on Analytical Dynamics, John Wiley and Sons, New York, 1965. |
| [42] |
M. Spivak,
Physics for Mathematicians. Mechanics I, Publish or Perish, Inc., Houston, 2010. |
| [43] |
C. Truesdell and W. Noll,
The Non-Linear Field Theories of Mechanics, no. 3 in Handbuch der Physik, Springer-Verlag, New York/Heidelberg/Berlin, 1965, New edition: [ |
| [44] |
C. Truesdell and W. Noll,
The Non-Linear Field Theories of Mechanics, Springer-Verlag, New York/Heidelberg/Berlin, 2004,
Original edition: [ |
| [45] |
E. T. Whittaker,
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. Cambridge University Press, New York, 1959, New
edition: [ |
| [46] |
E. T. Whittaker,
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge Mathematical Library, Cambridge University Press, New
York/Port Chester/Melbourne/Sydney, 1988, Original
edition: [ |
show all references
Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.
References:
| [1] |
R. Abraham and J. E. Marsden,
Foundations of Mechanics, 2nd edition, Addison Wesley, Reading, MA, 1978. |
| [2] |
R. Abraham, J. E. Marsden and T. S. Ratiu,
Manifolds, Tensor Analysis, and Applications, 2nd edition, no. 75 in Applied Mathematical Sciences, Springer-Verlag, New York/Heidelberg/Berlin, 1988.
doi: 10.1007/978-1-4612-1029-0. |
| [3] |
V. I. Arnol'ed,
Mathematical Methods of Classical Mechanics, 2nd edition, no. 60 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1978, New edition: [ |
| [4] |
V. I. Arnol'd,
Mathematical Methods of Classical Mechanics, 2nd edition, no. 60 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1989, First edition: [ |
| [5] |
R. E. Artz,
Classical mechanics in Galilean space-time, Foundations of Physics, 11 (1981), 679-697.
doi: 10.1007/BF00726944. |
| [6] |
M. Berger,
Geometry I, Universitext, Springer-Verlag, New York/Heidelberg/Berlin, 1987.
doi: 10.1007/978-3-540-93815-6. |
| [7] |
A. Bhand and A. D. Lewis, Rigid body mechanics in Galilean spacetimes Journal of Mathematical Physics, 46 (2005), 102902, 29 pp.
doi: 10.1063/1.2060547. |
| [8] |
A. M. Bloch,
Nonholonomic Mechanics and Control, no. 24 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 2003.
doi: 10.1007/b97376. |
| [9] |
B. Brogliato,
Nonsmooth Mechanics, Third edition. Communications and Control Engineering Series. Springer, [Cham], 2016.
doi: 10.1007/978-3-319-28664-8. |
| [10] |
F. Bullo and A. D. Lewis,
Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Systems, no. 49 in Texts in Applied Mathematics. Springer-Verlag, New York, 2005.
doi: 10.1007/978-1-4899-7276-7. |
| [11] |
H. Cendra, D. D. Holm, J. E. Marsden and T. S. Ratiu,
Lagrangian reduction, the Euler-Poincaré equations, and semidirect products, American Mathematical Society Translations, Series 2, 186 (1998), 1-25.
doi: 10.1090/trans2/186/01. |
| [12] |
D. L. Cohn,
Measure Theory, 2nd edition, Birkhäuser Advanced Texts, Birkhäuser, Boston/Basel/Stuttgart, 2013.
doi: 10.1007/978-1-4614-6956-8. |
| [13] |
J. Cortés, M. de León, D. M. de Diego and S. Martínez,
Mechanical systems subjected to generalized non-holonomic constraints, Proceedings of the Royal Society. London. Series A. Mathematical and Physical Sciences, 457 (2001), 651-670.
doi: 10.1098/rspa.2000.0686. |
| [14] |
M. Crampin,
On the concept of angular velocity, European Journal of Physics, 7 (1986), 287-293.
doi: 10.1088/0143-0807/7/4/014. |
| [15] |
M. R. Flannery,
The enigma of nonholonomic constraints, American Journal of Physics, 73 (2005), 265-272.
doi: 10.1119/1.1830501. |
| [16] |
C. Glocker,
Set Valued Force Laws, no. 1 in Lecture Notes in Applied Mechanics, Springer-Verlag, New York/Heidelberg/Berlin, 2001.
doi: 10.1007/978-3-540-44479-4. |
| [17] |
H. Goldstein,
Classical Mechanics, Addison Wesley, Reading, MA, 1951, New edition: [ |
| [18] |
H. Goldstein, C. P. Poole, Jr and J. L. Safko,
Classical Mechanics, 3rd edition, Addison Wesley, Reading, MA, 2001, Original edition: [ |
| [19] |
X. Grácia, J. Marin-Solano and M.-C. Muñoz-Lecanda,
Some geometric aspects of variational calculus in constrained systems, Reports on Mathematical Physics, 51 (2003), 127-148.
doi: 10.1016/S0034-4877(03)80006-X. |
| [20] |
A. Hatcher,
Algebraic Topology, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2002. |
| [21] |
D. Husemoller,
Fibre Bundles, 3rd edition, no. 20 in Graduate Texts in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 1994.
doi: 10.1007/978-1-4757-2261-1. |
| [22] |
S. Jafarpour and A. D. Lewis,
Time-Varying Vector Fields and Their Flows, Springer Briefs in Mathematics, Springer-Verlag, New York/Heidelberg/Berlin, 2014.
doi: 10.1007/978-3-319-10139-2. |
| [23] |
Y. Kanno,
Nonsmooth Mechanics and Convex Optimization, CRC Press, Boca Raton, FL, 2011.
doi: 10.1201/b10839. |
| [24] |
P. V. Kharlomov,
A critique of some mathematical models of mechanical systems with differential constraints, Journal of Applied Mathematics and Mechanics. Translation of the Soviet journal Prikladnaya Matematika i Mekhanika, 56 (1992), 584-594.
doi: 10.1016/0021-8928(92)90016-2. |
| [25] |
S. Kobayashi and K. Nomizu,
Foundations of Differential Geometry, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. |
| [26] |
V. V. Kozlov,
The problem of realizing constraints in dynamics, Journal of Applied Mathematics and Mechanics. Translation of the Soviet journal Prikladnaya Matematika i Mekhanika, 56 (1992), 594-600.
doi: 10.1016/0021-8928(92)90017-3. |
| [27] |
J. L. Lagrange,
Méchanique Analitique, Chez la Veuve Desaint, Paris, 1788, Translation: [ |
| [28] |
J. L. Lagrange,
Analytical Mechanics, No. 191 in Boston Studies in the Philosophy of Science, Kluwer
Academic Publishers, Dordrecht, 1997, Original edition: [ |
| [29] |
A. D. Lewis,
The geometry of the Gibbs-Appell equations and Gauss's Principle of Least Constraint, Reports on Mathematical Physics, 38 (1996), 11-28.
doi: 10.1016/0034-4877(96)87675-0. |
| [30] |
A. D. Lewis,
Affine connections and distributions with applications to nonholonomic mechanics, Reports on Mathematical Physics, 42 (1998), 135-164.
doi: 10.1016/S0034-4877(98)80008-6. |
| [31] |
A. D. Lewis,
Is it worth learning differential geometric methods for modelling and control of mechanical systems?, Robotica, 26 (2007), 765-777.
|
| [32] |
A. D. Lewis and R. M. Murray,
Variational principles for constrained systems: Theory and experiment, International Journal of Non-Linear Mechanics, 30 (1995), 793-815.
doi: 10.1016/0020-7462(95)00024-0. |
| [33] |
P. Liberrmn and C. -M. Marle,
Symplectic Geometry and Analytical Mechanics, no. 35 in Mathematics and its Applications, D. Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo, 1987.
doi: 10.1007/978-94-009-3807-6. |
| [34] |
J. E. Marsden and T. R. Hughes,
Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983, Reprint: [ |
| [35] |
J. E. Marsden and T. R. Hughes,
Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1994, Original
edition: [ |
| [36] |
J. E. Marsden and J. Scheurle,
The reduced Euler-Lagrange equations, Fields Institute Communications, Fields Institute, 1 (1993), 139-164.
|
| [37] |
M. D. P. Monteiro Marques,
Differential Inclusions in Nonsmooth Mechanical Problems, No. 9 in Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston/Basel/Stuttgart, 1993.
doi: 10.1007/978-3-0348-7614-8. |
| [38] |
R. M. Murray, Z. X. Li and S. S. Sastry,
A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994. |
| [39] |
O. M. O'Reilly,
Intermediate Dynamics for Engineers, Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 2008.
doi: 10.1017/CBO9780511791352. |
| [40] |
J. G. Papastavridis,
Tensor Calculus and Analytical Dynamics, Library of Engineering Mathematics, CRC Press, Boca Raton, FL, 1999. |
| [41] |
L. A. Pars,
A Treatise on Analytical Dynamics, John Wiley and Sons, New York, 1965. |
| [42] |
M. Spivak,
Physics for Mathematicians. Mechanics I, Publish or Perish, Inc., Houston, 2010. |
| [43] |
C. Truesdell and W. Noll,
The Non-Linear Field Theories of Mechanics, no. 3 in Handbuch der Physik, Springer-Verlag, New York/Heidelberg/Berlin, 1965, New edition: [ |
| [44] |
C. Truesdell and W. Noll,
The Non-Linear Field Theories of Mechanics, Springer-Verlag, New York/Heidelberg/Berlin, 2004,
Original edition: [ |
| [45] |
E. T. Whittaker,
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. Cambridge University Press, New York, 1959, New
edition: [ |
| [46] |
E. T. Whittaker,
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge Mathematical Library, Cambridge University Press, New
York/Port Chester/Melbourne/Sydney, 1988, Original
edition: [ |
Figure 2.
Rod with tip constrained to move in a plane
Figure 3.
Central torque-force on a rigid body in a configuration
| [1] |
Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577 |
| [2] |
Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511 |
| [3] |
Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure and Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51 |
| [4] |
Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449 |
| [5] |
Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure and Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549 |
| [6] |
Dmitry V. Zenkov, Anthony M. Bloch. Dynamics of generalized Euler tops with constraints. Conference Publications, 2001, 2001 (Special) : 398-405. doi: 10.3934/proc.2001.2001.398 |
| [7] |
Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987 |
| [8] |
Yuan Xu, Xin Jin, Saiwei Wang, Yang Tang. Optimal synchronization control of multiple euler-lagrange systems via event-triggered reinforcement learning. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1495-1518. doi: 10.3934/dcdss.2020377 |
| [9] |
Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic and Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687 |
| [10] |
Bernard Bonnard, Olivier Cots, Nataliya Shcherbakova. The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion. Mathematical Control and Related Fields, 2013, 3 (3) : 287-302. doi: 10.3934/mcrf.2013.3.287 |
| [11] |
Paola Goatin, Philippe G. LeFloch. $L^1$ continuous dependence for the Euler equations of compressible fluids dynamics. Communications on Pure and Applied Analysis, 2003, 2 (1) : 107-137. doi: 10.3934/cpaa.2003.2.107 |
| [12] |
Okihiro Sawada. Analytic rates of solutions to the Euler equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1409-1415. doi: 10.3934/dcdss.2013.6.1409 |
| [13] |
Luigi Ambrosio. Variational models for incompressible Euler equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 1-10. doi: 10.3934/dcdsb.2009.11.1 |
| [14] |
Franco Flandoli, Dejun Luo. Euler-Lagrangian approach to 3D stochastic Euler equations. Journal of Geometric Mechanics, 2019, 11 (2) : 153-165. doi: 10.3934/jgm.2019008 |
| [15] |
Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure and Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503 |
| [16] |
Andrey Tsiganov. Integrable Euler top and nonholonomic Chaplygin ball. Journal of Geometric Mechanics, 2011, 3 (3) : 337-362. doi: 10.3934/jgm.2011.3.337 |
| [17] |
Roderick V.N. Melnik, Ningning Song, Per Sandholdt. Dynamics of torque-speed profiles for electric vehicles and nonlinear models based on differential-algebraic equations. Conference Publications, 2003, 2003 (Special) : 610-617. doi: 10.3934/proc.2003.2003.610 |
| [18] |
Philippe G. LeFloch, Seiji Ukai. A symmetrization of the relativistic Euler equations with several spatial variables. Kinetic and Related Models, 2009, 2 (2) : 275-292. doi: 10.3934/krm.2009.2.275 |
| [19] |
Young-Pil Choi. Compressible Euler equations interacting with incompressible flow. Kinetic and Related Models, 2015, 8 (2) : 335-358. doi: 10.3934/krm.2015.8.335 |
| [20] |
Gianluca Crippa, Elizaveta Semenova, Stefano Spirito. Strong continuity for the 2D Euler equations. Kinetic and Related Models, 2015, 8 (4) : 685-689. doi: 10.3934/krm.2015.8.685 |
2020 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]