American Institute of Mathematical Sciences

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June  2020, 12(2): 165-308. doi: 10.3934/jgm.2020013

Nonholonomic and constrained variational mechanics

Andrew D. Lewis

Department of Mathematics and Statistics, Queeen's University, Kingston, ON K7L 3N6, Canada

Received  December 2018 Revised  January 2020 Published  June 2020 Early access  June 2020

Fund Project: Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada

Equations governing mechanical systems with nonholonomic constraints can be developed in two ways: (1) using the physical principles of Newtonian mechanics; (2) using a constrained variational principle. Generally, the two sets of resulting equations are not equivalent. While mechanics arises from the first of these methods, sub-Riemannian geometry is a special case of the second. Thus both sets of equations are of independent interest.

The equations in both cases are carefully derived using a novel Sobolev analysis where infinite-dimensional Hilbert manifolds are replaced with infinite-dimensional Hilbert spaces for the purposes of analysis. A useful representation of these equations is given using the so-called constrained connection derived from the system's Riemannian metric, and the constraint distribution and its orthogonal complement. In the special case of sub-Riemannian geometry, some observations are made about the affine connection formulation of the equations for extremals.

Using the affine connection formulation of the equations, the physical and variational equations are compared and conditions are given that characterise when all physical solutions arise as extremals in the variational formulation. The characterisation is complete in the real analytic case, while in the smooth case a locally constant rank assumption must be made. The main construction is that of the largest affine subbundle variety of a subbundle that is invariant under the flow of an affine vector field on the total space of a vector bundle.

Keywords: Nonholonomic mechanics, calculus of variations, affine differential geometry, Sobolev spaces of mappings, linear and affine vector fields on vector bundles, invariant subbundles, sub-Riemannian geometry.
Mathematics Subject Classification: Primary: 70F25; Secondary: 34C45, 46E35, 49K05, 53B05, 53C05, 53C17, 58A30, 70G45.
Citation: Andrew D. Lewis. Nonholonomic and constrained variational mechanics. Journal of Geometric Mechanics, 2020, 12 (2) : 165-308. doi: 10.3934/jgm.2020013
References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences, 75, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.  Google Scholar

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A. A. Agrachev and R. V. Gamkrelidze, Exponential representation of flows and a chronological enumeration, Mat. Sb. (N.S.), 107 (1978), 467–532,639.  Google Scholar

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A. A. Agrachev, D. Barilari and U. Boscain, A Comprehensive Introduction to SubRiemannian Geometry, Cambridge Studies in Advanced Mathematics, 181, Cambridge University Press, Cambridge, 2020. doi: 10.1017/9781108677325.  Google Scholar

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A. V. Borisov, I. S. Mamaev and I. A. Bizyaev, Dynamical systems with non-integrable constraints: Vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics, Russian Math. Surveys, 72 (2017), 783-840.  doi: 10.4213/rm9783.  Google Scholar

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F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

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show all references

References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences, 75, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.  Google Scholar

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[3]

A. A. Agrachev and R. V. Gamkrelidze, Exponential representation of flows and a chronological enumeration, Mat. Sb. (N.S.), 107 (1978), 467–532,639.  Google Scholar

[4]

A. A. Agrachev, D. Barilari and U. Boscain, A Comprehensive Introduction to SubRiemannian Geometry, Cambridge Studies in Advanced Mathematics, 181, Cambridge University Press, Cambridge, 2020. doi: 10.1017/9781108677325.  Google Scholar

[5]

C. D. Aliprantis and K. C. Border, Infinite-Dimensional Analysis. A Hitchhiker's Guide, Springer, Berlin, 2006. doi: 10.1007/3-540-29587-9.  Google Scholar

[6]

M. Berger, Geometry I, Universitext, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-540-93815-6.  Google Scholar

[7]

A. V. Borisov, I. S. Mamaev and I. A. Bizyaev, Dynamical systems with non-integrable constraints: Vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics, Russian Math. Surveys, 72 (2017), 783-840.  doi: 10.4213/rm9783.  Google Scholar

[8]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

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F. Cantrijn, J. Cortés, M. de León and D. Martín de Diego, On the geometry of generalised Chaplygin systems, Math. Proc. Cambridge Philos. Soc., 132 (2002), 323-351.  doi: 10.1017/S0305004101005679.  Google Scholar

[10]

F. Cardin and M. Favretti, On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys., 18 (1996), 295-325.  doi: 10.1016/0393-0440(95)00016-X.  Google Scholar

[11]

H. Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France, 85 (1957), 77-99.   Google Scholar

[12]

W.-L. Chow, Über Systemen von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117 (1939), 98-105.  doi: 10.1007/BF01450011.  Google Scholar

[13]

A. Convent and J. Van Schaftingen, Geometric partial differentiability on manifolds: The tangential derivative and the chain rule, J. Math. Anal. Appl., 435 (2016), 1672-1681.  doi: 10.1016/j.jmaa.2015.11.036.  Google Scholar

[14]

A. Convent and J. Van Schaftingen, Intrinsic co-local weak derivatives and Sobolev spaces between manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 16 (2016), 97-128.  doi: 10.2422/2036-2145.201312_005.  Google Scholar

[15]

A. Convent and J. Van Schaftingen, Higher order intrinsic weak differentiability and Sobolev spaces between manifolds, Adv. Calc. Var., 12 (2019), 303-332.  doi: 10.1515/acv-2017-0008.  Google Scholar

[16]

J. Cortés, M. de León, D. Martín de Diego and S. Martínez, Geometric description of vakonomic and nonholonomic dynamics. Comparison of solutions, SIAM J. Control Optim., 41 (2002), 1389-1412.  doi: 10.1137/S036301290036817X.  Google Scholar

[17]

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[18]

M. Favretti, Equivalence of dynamics for nonholonomic systems with transverse constraints, J. Dynam. Differential Equations, 10 (1998), 511-536.  doi: 10.1023/A:1022667307485.  Google Scholar

[19]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band, 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[20]

H. Federer, Geometric Measure Theory, Classics in Mathematics, Springer-Verlag Berlin Heidelberg, 1996. doi: 10.1007/978-3-642-62010-2.  Google Scholar

[21]

C. Fefferman and J. Kollár, Continuous solutions of linear equations, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, Springer, New York, 2013,233–282. doi: 10.1007/978-1-4614-4075-8_10.  Google Scholar

[22]

O. E. Fernandez and A. M. Bloch, Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data, J. Phys. A, 41 (2008), 20pp. doi: 10.1088/1751-8113/41/34/344005.  Google Scholar

[23]

H. L. Goldschmidt, Existence theorems for analytic linear partial differential equations, Ann. of Math. (2), 86 (1967), 246-270.  doi: 10.2307/1970689.  Google Scholar

[24]

H. Goldschmidt, Integrability criteria for systems of nonlinear partial differential equations, J. Differential Geometry, 1 (1967), 269-307.  doi: 10.4310/jdg/1214428094.  Google Scholar

[25]

X. Grácia, J. Marin-Solano and M.-C. Muñoz-Lecanda, Some geometric aspects of variational calculus in constrained systems, Rep. Math. Phys., 51 (2003), 127-148.  doi: 10.1016/S0034-4877(03)80006-X.  Google Scholar

[26]

H. Grauert, On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. (2), 68 (1958), 460-472.  doi: 10.2307/1970257.  Google Scholar

[27]

H. Grauert and R. Remmert, Coherent Analytic Sheaves, Grundlehren der Mathematischen Wissenschaften, 265, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69582-7.  Google Scholar

[28]

S. Jafarpour and A. D. Lewis, Time-Varying Vector Fields and Their Flows, SpringerBriefs in Mathematics, Springer, New York, 2014. doi: 10.1007/978-3-319-10139-2.  Google Scholar

[29]

J. Jost, Postmodern Analysis, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05306-5.  Google Scholar

[30]

M. Jóźwikowski and W. Respondek, A comparison of vakonomic and nonholonomic dynamics with applications to non-invariant Chaplygin systems, J. Geom. Mech., 11 (2019), 77-122.  doi: 10.3934/jgm.2019005.  Google Scholar

[31]

P. V. Kharlomov, A critique of some mathematical models of mechanical systems with differential constraints, J. Appl. Math. Mech., 56 (1992), 584-594.  doi: 10.1016/0021-8928(92)90016-2.  Google Scholar

[32]

W. Klingenberg, Riemannian Geometry, de Gruyter Studies in Mathematics, 1, Walter de Gruyter & Co., Berlin/New York, 1982.  Google Scholar

[33]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963.  Google Scholar

[34]

J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148.  doi: 10.1007/BF00375092.  Google Scholar

[35]

I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.  Google Scholar

[36]

V. V. Kozlov, The problem of realizing constraints in dynamics, J. Appl. Math. Mech., 56 (1992), 594-600.  doi: 10.1016/0021-8928(92)90017-3.  Google Scholar

[37]

I. Kupka and W. M. Oliva, The non-holonomic mechanics, J. Differential Equations, 169 (2001), 169-189.  doi: 10.1006/jdeq.2000.3897.  Google Scholar

[38]

B. Langerok, A connection theoretic approach to sub-Riemannian geometry, J. Geom. Phys., 46 (2003), 203-230.  doi: 10.1016/S0393-0440(02)00026-8.  Google Scholar

[39]

J. M. Lee, Riemannian Manifolds, Graduate Texts in Mathematics, 176, Springer-Verlag, New York, 1997. doi: 10.1007/b98852.  Google Scholar

[40]

A. D. Lewis, Affine connections and distributions with applications to nonholonomic mechanics, Rep. Math. Phys., 42 (1998), 135-164.  doi: 10.1016/S0034-4877(98)80008-6.  Google Scholar

[41]

A. D. Lewis, Generalised Subbundles, Distributions, and Families of Vector Fields. A Comprehensive Review, Lecture Notes from ICMAT Summer School, Madrid, 2012. Available from: http://www.mast.queensu.ca/ andrew/notes/abstracts/2011a.html. Google Scholar

[42]

A. D. Lewis, The physical foundations of geometric mechanics, J. Geom. Mech., 9 (2017), 487-574.  doi: 10.3934/jgm.2017019.  Google Scholar

[43]

A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment, Internat. J. Non-Linear Mech., 30 (1995), 793-815.  doi: 10.1016/0020-7462(95)00024-0.  Google Scholar

[44]

W. Liu and H. J. Sussmann, Abnormal sub-Riemannian minimizers, in Differential Equations, Dynamical Systems, and Control Science, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, 1994,705–716.  Google Scholar

[45]

P. W. Michor, Manifolds of Differentiable Mappings, Shiva Mathematics Series, 3, Shiva Publishing Ltd., Nantwich, 1980.  Google Scholar

[46]

E. Minguzzi, The equality of mixed partial derivatives under weak differentiability conditions, Real Anal. Exchange, 40 (2014/15), 81-97.  doi: 10.14321/realanalexch.40.1.0081.  Google Scholar

[47]

J. Nestruev, Smooth Manifolds and Observables, Graduate Texts in Mathematics, 220, Springer-Verlag, New York, 2003. doi: 10.1007/b98871.  Google Scholar

[48]

G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1600-1.  Google Scholar

[49]

P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, New York, 2006. doi: 10.1007/978-0-387-29403-2.  Google Scholar

[50]

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Figure 1.  A depiction of $ \nu\hat{{\sigma}} $ and $ \delta\hat{{\sigma}} $. Note that $ \nu\hat{{\sigma}}_s $ is the tangent vector field for $ \hat{{\sigma}}_s $ and $ \delta\hat{{\sigma}}^t $ is the tangent vector field for $ \hat{{\sigma}}^t $
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