## Relativistic Lagrangian and equation of motion

Here I have tried to solve problem in Classical mechanics which is about relativistic Lagrangian and equation of motion.

**Question**

Our problem is to show that relativistic Lagrangian give the equation of motion

**Solution**

From non relativistic Lagrangian

where is the kinetic energy and is the potential energy and is the momentum of particle.

Thus,

, ,

If we assume similar equations of motion in relativistic mechanics then,

(1)

(2)

(3)

where,

and

,

and is the speed of particle in the inertial frame under consideration.

From equation (1)

On integrating it we get

where,

, and is a constant of integration and may be taken as the potential energy of the particle and .

Now we use above definition of Lagrangian for obtaining relativistic equation of motion.

Lagrangian equation of motion for x co-ordinate is

and

or,

Thus

or,

Same way we can find the equation of motion in and direction as

and

Thus in generalized form we can write,

where is the generalized co-ordinates.

This is the required relativistic equation of motion.