Singular Lagrangians. Classical dynamics and quantization. Lectures for young scientists
Abstract
The lectures are devoted to the classical and quantum dynamics of the systems described by singular (or degenerate) Lagrangians. The complete set of the Hamiltonian constraints is constructed in the framework of the Lagrangian formalism. The equations of motion in the phase space are derived by taking into account all the constraints in the theory. It is proved that the dynamic on the physical submanifold of the phase space has the Hamiltonian form. On lectures the second Noether theorem is widely used. On its basis the properties of the Poisson brackets of the primary constraints are investigated and the invariance of the Lagrangian constraints during evolution is proved. The setting of the Cauchy problem in the theories with singular Lagrangians is discussed. The quantization of the systems with constraints is carried out by the functional integration in the phase space. There is considered the most general case of the first class and the second class constraints with an explicit time dependence. The gauge conditions may be noninvoluntary and time dependent. The material is illustrated by some examples (relativistic point particle, relativistic string, electromagnetic field, and YangMills fields).
 Publication:

Unknown
 Pub Date:
 1986
 Bibcode:
 1986slcd.rept.....N
 Keywords:

 Classical Mechanics;
 Equations Of Motion;
 Lagrange Coordinates;
 Quantum Mechanics;
 Singularity (Mathematics);
 Electromagnetic Fields;
 Feynman Diagrams;
 Hamiltonian Functions;
 PhaseSpace Integral;
 Time Dependence;
 YangMills Fields;
 YangMills Theory;
 Thermodynamics and Statistical Physics