The principle of least action

The notion of least action is a key concept in the Hamiltonian formulation. This is a more general kind of path variation where time and coordinates for position are both flexible. So Position coordinates are kept constant at the path's endpoints, but changes in time are allowed. Such a variation is shown in fig and is denoted by a letter △ instead of δ.

In mechanics the quantity

Like the δ variation process, we can also designate here the varied paths by a parameter α. But as the △ variation includes the time associated with each point on the path, t will be also function of α and t, i.e. qj = qj (α,t). So that

Referring to the fig. above point p on the actual path now goes over to p' on the varied path with the correspondence:

Advantage of Hamiltonian formulation

The two variables are not treated equally in the Lagrangian approach since the latter, which is the time derivative of the first, is ultimately thought to be a dependent variable. However, in the Hamiltonian scheme, coordinates and momenta are treated equally. This gives frequent freedom in selecting coordinates and momenta, which successfully aids in the formulation of mechanics, whose natural starting point is based on the Hamiltonian technique, in a more abstract and profound manner. In the study of potential energy charges in atoms and nuclei, the relationship between the numerical values of the Hamiltonian and the energy of a conservative system is crucial.

Although the lagrangian formulation of the mechanical problem admits no novel approaches, the equality of status of coordinates and momenta offers a useful provides a convenient basis for the development of quantum mechanics and statistical mechanics.

The knowledge of Hamiltonian of a system is extremely important particular if we are interested in quantising a dynamical system. For setting up a schrodinger equation, one replaces generalized momenta by the corresponding differential operators.

This note is taken from Classical mechanics, MSC physics, Nepal.

This note is a part of the Physics Repository.