A geometric analysis of dynamical systems with singular Lagrangians. (English) Zbl 1251.37052

The author investigates a singular mechanical system given by the Lagrangian \[ L= \dot q_1\dot q_3- q_2\dot q_3+ q_1 q_3. \] This system has been studied previously using the constraint method of Dirac, but the results were incomplete and conclusions from different authors were not in agreement. The author rectifies this situation by providing a rigorous solution using the Hamiltonian exterior differential systems method. Given the Lagrangian, one obtains the corresponding dynamical distribution in the first jet bundle. She shows that this distribution is not completely integrable and has a non-constant rank. To obtain the dynamics a general integration method (developed by O. Krupková) called the “geometric constraint algorithm” is applied.
The author calculates the Euler-Lagrange and Hamiltonian equations in terms of the corresponding distributions and finds the complete structure of the solutions. It is worth noting that the Hamiltonian and Euler-Lagrange equations are not equivalent and that the dynamics are not representable by a vector field.


37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H03 Lagrange’s equations
70H45 Constrained dynamics, Dirac’s theory of constraints
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