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These images have been taken from Goldstein, Classical Mechanics.
Why do we need Lagrangian formulation of mechanics when we already have Newtonian formulation of mechanics?
Newtonian formulation of mechanics demands us to solve the equation of motion given by equation 1. 19. for this we need to know all the forces acting on the particle. But in general there exists forces which are unknown to us but their effects on the system is known to us. These forces are known as constraint forces. The effects of these forces are known as constraints and and these effects are expressed in terms of equations.
Thus constraint forces introduce two types of difficulties : 1) these forces are an mom so we cannot solve the equations of motions directly. Hence we need a formulation in which we do not need to know the constraint forces.
2) the position coordinates are connected by equations of constraint and hence, these are no more linearly independent.
But, how can this be a difficulty? We can express linearly dependent coordinates in terms of linearly independent coordinates and then solve equations for linearly independent coordinates and then substitute the values of linearly independent coordinates to find the linearly dependent coordinates. Hence, how can this be a difficulty and why do we need generalized coordinates to remove it?
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