# American Institute of Mathematical Sciences

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

## Nonholonomic and constrained variational mechanics

 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.

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. [2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, 140, Elsevier/Academic Press, Amsterdam, 2003. [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. [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. [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. [6] M. Berger, Geometry I, Universitext, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-540-93815-6. [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. [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. [9] 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. [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. [11] H. Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France, 85 (1957), 77-99. [12] W.-L. Chow, Über Systemen von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117 (1939), 98-105.  doi: 10.1007/BF01450011. [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. [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. [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. [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. [17] M. Crampin and T. Mestdag, Anholonomic frames in constrained dynamics, Dyn. Syst., 25 (2010), 159-187.  doi: 10.1080/14689360903360888. [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. [19] H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band, 153, Springer-Verlag New York Inc., New York, 1969. [20] H. Federer, Geometric Measure Theory, Classics in Mathematics, Springer-Verlag Berlin Heidelberg, 1996. doi: 10.1007/978-3-642-62010-2. [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. [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. [23] H. L. Goldschmidt, Existence theorems for analytic linear partial differential equations, Ann. of Math. (2), 86 (1967), 246-270.  doi: 10.2307/1970689. [24] H. Goldschmidt, Integrability criteria for systems of nonlinear partial differential equations, J. Differential Geometry, 1 (1967), 269-307.  doi: 10.4310/jdg/1214428094. [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. [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. [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. [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. [29] J. Jost, Postmodern Analysis, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05306-5. [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. [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. [32] W. Klingenberg, Riemannian Geometry, de Gruyter Studies in Mathematics, 1, Walter de Gruyter & Co., Berlin/New York, 1982. [33] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. [34] J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148.  doi: 10.1007/BF00375092. [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. [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. [37] I. Kupka and W. M. Oliva, The non-holonomic mechanics, J. Differential Equations, 169 (2001), 169-189.  doi: 10.1006/jdeq.2000.3897. [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. [39] J. M. Lee, Riemannian Manifolds, Graduate Texts in Mathematics, 176, Springer-Verlag, New York, 1997. doi: 10.1007/b98852. [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. [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. [42] A. D. Lewis, The physical foundations of geometric mechanics, J. Geom. Mech., 9 (2017), 487-574.  doi: 10.3934/jgm.2017019. [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. [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. [45] P. W. Michor, Manifolds of Differentiable Mappings, Shiva Mathematics Series, 3, Shiva Publishing Ltd., Nantwich, 1980. [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. [47] J. Nestruev, Smooth Manifolds and Observables, Graduate Texts in Mathematics, 220, Springer-Verlag, New York, 2003. doi: 10.1007/b98871. [48] G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1600-1. [49] P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, New York, 2006. doi: 10.1007/978-0-387-29403-2. [50] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, Matematicheskaya teoriya optimal' nykh protsessov, Gosudarstvennoe izdatelstvo fiziko-matematicheskoi literatury, Moscow, 1961. [51] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962. [52] S. Ramanan, Global Calculus, Graduate Studies in Mathematics, 65, American Mathematical Society, Providence, RI, 2005. [53] V. V. Rumiantsev, On Hamilton's principle for nonholonomic systems, Prikl. Mat. Mekh., 42 (1978), 387-399. [54] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. (2), 10 (1958), 338-354.  doi: 10.2748/tmj/1178244668. [55] E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7. [56] H. J. Sussmann, Smooth distributions are globally finitely spanned, in Analysis and Design of Nonlinear Control Systems, Springer, Berlin, 2008, 3–8. doi: 10.1007/978-3-540-74358-3_1. [57] J. L. Synge, Geodesics in non-holonomic geometry, Math. Ann., 99 (1928), 738-751.  doi: 10.1007/BF01459122. [58] G. Terra, Vakonomic versus nonholonomic mechanics revisited, São Paulo J. Math. Sci., 12 (2018), 136-145.  doi: 10.1007/s40863-017-0062-z. [59] G. Terra and M. H. Kobayashi, On the classical mechanical systems with non-linear constraints, J. Geom. Phys., 49 (2004), 385-417.  doi: 10.1016/j.geomphys.2003.08.005. [60] G. Terra and M. H. Kobayashi, On the variational mechanics with non-linear constraints, J. Math. Pures Appl. (9), 83 (2004), 629-671.  doi: 10.1016/S0021-7824(03)00069-2. [61] A. M. Vershik and V. Y. Gershkovich, Nonholonomic dynamical systems. Geometry of distributions and variational problems, in Current problems in mathematics. Fundamental directions, Vol. 16 (Russian), Itogi Nauki i Tekhniki, 307, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987, 5–85. [62] R. O. Wells Jr, Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics, 65, Springer, New York, 2008. doi: 10.1007/978-0-387-73892-5. [63] H. Whitney, Differentiable manifolds, Ann. of Math. (2), 37 (1936), 645-680.  doi: 10.2307/1968482. [64] G. Zampieri, Nonholonomic versus vakonomic dynamics, J. Differential Equations, 163 (2000), 335-347.  doi: 10.1006/jdeq.1999.3727. [65] W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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. [2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, 140, Elsevier/Academic Press, Amsterdam, 2003. [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. [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. [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. [6] M. Berger, Geometry I, Universitext, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-540-93815-6. [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. [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. [9] 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. [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. [11] H. Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France, 85 (1957), 77-99. [12] W.-L. Chow, Über Systemen von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117 (1939), 98-105.  doi: 10.1007/BF01450011. [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. [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. [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. [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. [17] M. Crampin and T. Mestdag, Anholonomic frames in constrained dynamics, Dyn. Syst., 25 (2010), 159-187.  doi: 10.1080/14689360903360888. [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. [19] H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band, 153, Springer-Verlag New York Inc., New York, 1969. [20] H. Federer, Geometric Measure Theory, Classics in Mathematics, Springer-Verlag Berlin Heidelberg, 1996. doi: 10.1007/978-3-642-62010-2. [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. [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. [23] H. L. Goldschmidt, Existence theorems for analytic linear partial differential equations, Ann. of Math. (2), 86 (1967), 246-270.  doi: 10.2307/1970689. [24] H. Goldschmidt, Integrability criteria for systems of nonlinear partial differential equations, J. Differential Geometry, 1 (1967), 269-307.  doi: 10.4310/jdg/1214428094. [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. [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. [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. [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. [29] J. Jost, Postmodern Analysis, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05306-5. [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. [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. [32] W. Klingenberg, Riemannian Geometry, de Gruyter Studies in Mathematics, 1, Walter de Gruyter & Co., Berlin/New York, 1982. [33] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. [34] J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148.  doi: 10.1007/BF00375092. [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. [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. [37] I. Kupka and W. M. Oliva, The non-holonomic mechanics, J. Differential Equations, 169 (2001), 169-189.  doi: 10.1006/jdeq.2000.3897. [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. [39] J. M. Lee, Riemannian Manifolds, Graduate Texts in Mathematics, 176, Springer-Verlag, New York, 1997. doi: 10.1007/b98852. [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. [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. [42] A. D. Lewis, The physical foundations of geometric mechanics, J. Geom. Mech., 9 (2017), 487-574.  doi: 10.3934/jgm.2017019. [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. [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. [45] P. W. Michor, Manifolds of Differentiable Mappings, Shiva Mathematics Series, 3, Shiva Publishing Ltd., Nantwich, 1980. [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. [47] J. Nestruev, Smooth Manifolds and Observables, Graduate Texts in Mathematics, 220, Springer-Verlag, New York, 2003. doi: 10.1007/b98871. [48] G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1600-1. [49] P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, New York, 2006. doi: 10.1007/978-0-387-29403-2. [50] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, Matematicheskaya teoriya optimal' nykh protsessov, Gosudarstvennoe izdatelstvo fiziko-matematicheskoi literatury, Moscow, 1961. [51] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962. [52] S. Ramanan, Global Calculus, Graduate Studies in Mathematics, 65, American Mathematical Society, Providence, RI, 2005. [53] V. V. Rumiantsev, On Hamilton's principle for nonholonomic systems, Prikl. Mat. Mekh., 42 (1978), 387-399. [54] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. (2), 10 (1958), 338-354.  doi: 10.2748/tmj/1178244668. [55] E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7. [56] H. J. Sussmann, Smooth distributions are globally finitely spanned, in Analysis and Design of Nonlinear Control Systems, Springer, Berlin, 2008, 3–8. doi: 10.1007/978-3-540-74358-3_1. [57] J. L. Synge, Geodesics in non-holonomic geometry, Math. Ann., 99 (1928), 738-751.  doi: 10.1007/BF01459122. [58] G. Terra, Vakonomic versus nonholonomic mechanics revisited, São Paulo J. Math. Sci., 12 (2018), 136-145.  doi: 10.1007/s40863-017-0062-z. [59] G. Terra and M. H. Kobayashi, On the classical mechanical systems with non-linear constraints, J. Geom. Phys., 49 (2004), 385-417.  doi: 10.1016/j.geomphys.2003.08.005. [60] G. Terra and M. H. Kobayashi, On the variational mechanics with non-linear constraints, J. Math. Pures Appl. (9), 83 (2004), 629-671.  doi: 10.1016/S0021-7824(03)00069-2. [61] A. M. Vershik and V. Y. Gershkovich, Nonholonomic dynamical systems. Geometry of distributions and variational problems, in Current problems in mathematics. Fundamental directions, Vol. 16 (Russian), Itogi Nauki i Tekhniki, 307, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987, 5–85. [62] R. O. Wells Jr, Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics, 65, Springer, New York, 2008. doi: 10.1007/978-0-387-73892-5. [63] H. Whitney, Differentiable manifolds, Ann. of Math. (2), 37 (1936), 645-680.  doi: 10.2307/1968482. [64] G. Zampieri, Nonholonomic versus vakonomic dynamics, J. Differential Equations, 163 (2000), 335-347.  doi: 10.1006/jdeq.1999.3727. [65] W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
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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