Define the difference between the newtonian mechanics and a lagrangian mechanics

The difference between Newtonian and Lagrangian mechanics is summarized below:

In Newtonian mechanics the fundamental thing of conceren is force that acts on a object. According to newton's laws of motion when a force acts on a body it produces acceleration and the acceleration is then related to velocity and position of the object. The basic equation of Newtonian mechanics are

where

'F' is the vector sum of all the forces that act on the object.

'a' is the acceleration that is produced in the body as a result of the forces.

The acceleration is related to the position of the body as

Thus we can know the position of any object if we know the acceleration of the object and the boundary condition of the object.

However in Lagrangian mechanics the basic parameter upon which the motion of the object is defined is the  a mathematical definition of position and change in position, thus an object can take an arbitrary path while travelling between 2 positions but only that path is physically possible in which the change in potential energy is minimum or takes least amount of work to be done this is known as principle of least work.

Mathematically

and we try to minimize the work that needs to be done thus giving us the path taken by the particle.

The equations of motions can be derived from this basic premise of Lagrangian mechanics.

i'm just gonna leave this comment here, so i can get a challange done-

explanation: