Lagrangian systems with constraints: a geometric approach to the method of Lagrange multipliers

A geometric approach to the method of Lagrange multipliers is presented using the framework of the tangent bundle geometry. The nonholonomic systems with constraint functions linear in the velocities are studied in the first place and then, and using this study of the nonholonomic mechanical systems as a previous result, the holonomic systems are considered. The Lagrangian inverse problem is also analysed and, finally, the theory is illustrated with several examples.

[1]  V. V. Rumyantsev.  The dynamics of rheonomic lagrangian systems with constraints , 1984 .

[2]  Mark J. Gotay,et al.  Presymplectic manifolds and the Dirac-Bergmann theory of constraints , 1978 .

[3]  R. Weber,et al.  Hamiltonian systems with constraints and their meaning in mechanics , 1986 .

[4]  A. Lichnerowicz,et al.  Proceedings of the IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, Torino, June 7-11, 1982 , 1983 .

[5]  G. Marmo,et al.  The inverse problem in the calculus of variations and the geometry of the tangent bundle , 1990 .

[6]  M. Crampin.  Tangent bundle geometry Lagrangian dynamics , 1983 .

[7]  J. Cariñena.  Theory of Singular Lagrangians , 1990 .

[8]  M. Crampin.  On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics , 1981 .

[9]  V. N. Brendelev.  On the realization of constraints in nonholonomic mechanics , 1981 .

[10]  F. Cantrijn.  Symplectic approach to nonconservative mechanics , 1984 .

[11]  Eugene J. Saletan,et al.  Dynamical Systems: A Differential Geometric Approach to Symmetry and Reduction , 1985 .

[12]  A. Hanson,et al.  CONSTRAINED HAMILTONIAN SYSTEMS , 2020, Quantization of Gauge Systems.

[13]  J. Koiller.  Reduction of some classical non-holonomic systems with symmetry , 1992 .

[14]  R. Huston,et al.  Nonholonomic systems with non-linear constraint equations , 1976 .

[15]  G. Giachetta,et al.  Jet Methods in Nonholonomic Mechanics , 1992 .

[16]  F. Cardin,et al.  On constrained mechanical systems: D’Alembert’s and Gauss’ principles , 1989 .

[17]  Eugene J. Saletan,et al.  A Variational Principle for Nonholonomic Systems , 1970 .

[18]  M. Crampin,et al.  Reduction of degenerate Lagrangian systems , 1986 .

[19]  Ralph Abraham,et al.  Foundations Of Mechanics , 2019 .

[20]  F. Cantrijn.  Vector fields generating analysis for classical dissipative systems , 1982 .