Geometric Action Principles in Classical Dynamics (original) (raw)

General principles of classical dynamics are usually developed in the framework of phase spaces, that is tangent or cotangent bundles over the control manifold. A more effective approach is proposed here by applying POINCARÉ-CARTAN theory of differential forms directly to the control manifold, so that lifting operations are completely avoided. The basic distinction between action principles and stationarity of functionals is pointed out. The EULER-LAGRANGE-HAMILTON variational theory is formulated without end constraints on the trajectory variations. A careful treatment of natural and essential conditions for the variational problem leads to a proper formulation of MAUPERTUIS action principle and to assess its equivalence to HAMILTON principle. POINCARÉ-CARTAN and HAMILTON-PONTRYAGIN hybrid principles, involving vertical variations of vector and covector fields, are addressed with an appropriate geometric approach. Riassunto I principi della dinamica classica sono usualmente svilupp...