^{1}

^{2}

^{2}

Non-canonical Lagrangian (Lagrangian with non-quadratic kinetic term) has been studied in the context of cosmology. In this work, the non-canonical Lagrangian with potential energy term has been discussed. We have obtained the periodic and solitary wave solutions for certain types of potential. The solutions obtained here may provide some new direction in the theory of phase transition, quantum field theory and related phenomena.

Canonical Lagrangian has been studied in the context of classical mechanics, quantum field theory and other branches of physics. The standard form of canonical Lagrangian density is given by

Here

In this framework, Lagrangian density (1) can be written as

Here kinetic energy term

where

Also, we are using natural units in which

Thus the Lagrangian density is given by

From here one can see that the kinetic energy is non-quadratic and that’s why the Lagrangian is known as non-canonical Lagrangian. The corresponding equation of motion is given by

Here

Multiplying both sides by

where C is constant of integration. Let

For

In the next section we will solve Equation (11) for some specific potentials. Also, Equation (11) cannot be solved for every potential. We will consider only those potentials for which Equation (11) can be integrated.

In this section we are going to solve Equation (11) for the following potentials.

Case I: Let us first consider the potential

where

After integrating this equation, we obtain

where

For time dependent case, the corresponding solution can be obtained by Lorentz transformation

and hence

From this solution one can see that, for

where

where

where

For static case, we obtain

Using Equation (11), we get

Now

and hence

Note that

For time dependent case, energy density is given by

For

which is a kink solitary wave solution [

The energy density is localised near

Case II: Here we will consider the potential of the form

In this case, the solution of time independent field equation is given by

Note that this solution is a solitary wave solution for

The energy density of time independent case is given by

and the energy density of time dependent case is given by

Thus the energy density is localised.

In this work we have discussed the non-canonical lagrangian for different kinds of potential. We have obtained the periodic and solitary wave solutions. A comparison is also made between the solutions of canonical and non-canonical lagrangian. As one can see that for Equation (16), the solution exists only when

We would like to thank Sanil Unnikrishnan for helpful discussions.

Bhupendra Singh,Ranjit Kumar,Loukrakpam Kennedy Meitei, (2016) A Study of Non-Canonical Lagrangian. Natural Science,08,211-215. doi: 10.4236/ns.2016.85024