Mechanics of Constrained Systems within Lagrangian and Hamiltonian Formalisms

Classical Mechanics of Constrained Systems within Lagrangian and Hamiltonian Formalisms 

We consider an A -particle mechanical system. A set of K constraints applied to it is holonomic whenever all the relationships connecting the natural coordinates Qi, i = 1,2. .. 3 N of the particles in the system plus the time t — and which are a mathematical counterpart to the existence of constraints inside the system — are of the form  

The elimination of K dependent coordinates results in the introduction of a set of generalized coordinates qj,j =, 2. .. n] where n = 3 N - K, m terms of which the natural coordinates are expressed parametrically  

Any conservative mechanical system which is either free or subject to holonomic constraints and whose potential does not depend on the generalized velocities is described by standard equations of motion (either Lagrangian or Hamiltonian). The kinetic energy of the iV-particIe system is  

On the scale of molecules, all the constraints to be taken into account are mathematically idealized and of the holonomic type. Moreover, the defining transformation Eq. (31) do not depend on time explicitly. 

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