Abstract
This paper considers dynamical systems described by Hamilton’s equations. It deals with the development of the explicit equations of motion for such systems when constraints are imposed on them. Such explicit equations do not appear to have been obtained hereto. The holonomic and/or nonholonomic constraints imposed can be nonlinear functions of the canonical momenta, the coordinates, and time, and they can be functionally dependent. These explicit equations of motion for constrained systems are obtained through the development of the connection between the Lagrangian concept of virtual displacements and Hamiltonian dynamics. A simple three-step approach for obtaining the explicit equations of motion of constrained Hamiltonian systems is presented. Four examples are provided illustrating the ease and simplicity with which these equations can be obtained by using the proposed three-step approach.
Similar content being viewed by others
References
Udwadia, F.E., Kalaba, R.E.: A new perspective on constrained motion. Proc. R. Soc. Lond. Ser. A 439, 407–410 (1992)
Udwadia, F.E., Kalaba, R.E.: On the foundations of analytical dynamics. Int. J. Nonlinear Mech. 37, 1079–1090 (2002)
Udwadia, F.E., Schutte, A.D.: Equations of Motion for general constrained systems in lagrangian mechanics. Acta Mech. 213, 111–129 (2010)
Udwadia, F.E., Wanichanon, T.: On general nonlinear constrained mechanical systems. Numer. Algebra Control Optim. 3(3), 425–443 (2013)
Pars, L.A.: A Treatise on Analytical Dynamics. Oxbow Press, Oxford (1972)
Udwadia, F.E., Kalaba, R.E.: Analytical Dynamics: A New Approach. Cambridge University Press, Cambridge (1996)
Penrose, R.: A generalized inverse of matrices. Proc. Camb. Philos. Soc. 51, 406–413 (1955)
Moore, E.H.: On the reciprocal of the general algebraic matrix. Abstr. Bull. Am. Soc. 26, 394–395 (1910)
Graybill, F.: Matrices with Application to Statistics, Duxbury Classics Series, Pacific Grove. Duxbury Publishing Company, CA (2001)
Goldstein, H.: Classical Mechanics. Addison-Wesley, New York, NY (1976)
Udwadia, F.E., Phohomsiri, P.: Explicit poincare equations of motion for general constrained systems. Part I. analytical results. Proc. R. Soc. Lond. Ser. A 463, 1421–1434 (2007)
Udwadia, F.E., Phohomsiri, P.: Explicit poincare equations of motion for general constrained systems. Part II. Applications to multi-body dynamics and nonlinear control. Proc. R. Soc. Lond. Ser. A 463, 1435–1446 (2007)
Udwadia, F.E., Mylapilli, H.: Constrained motion of mechanical systems and tracking control of nonlinear systems: connections and closed-form results. Nonlinear Dyn. Syst. Theory 15(1), 73–89 (2015)
Udwadia, F.E.: Optimal tracking control of nonlinear dynamical systems. Proc. R. Soc. Lond. Ser. A 464, 2341–2363 (2008)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The author declares that he has no conflict of interest.
Research involving Human Participants and/or Animals
The research carried out in this article does not involve any human participants or animals.
Rights and permissions
About this article
Cite this article
Udwadia, F.E. Constrained motion of Hamiltonian systems. Nonlinear Dyn 84, 1135–1145 (2016). https://doi.org/10.1007/s11071-015-2558-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-015-2558-3