Newtons equations for the pendulum in Cartesian coordinates

In the absence of friction, there are two forces acting on the mass m whose position vector at time t is denoted by the vector r[r] measured relative to the support point, which is the origin of a set of Cartesian axes with three-component k in the upward vertical direction. The first is the force of gravity on the mass, which acts downwards with a value —mgk. The second is the centripetal force, unknown for the moment, which is directed along the support towards the universal point. We denote this force by — Tr t, where Tis a scalar function of time to be found. The Newtonian equations of motion can then be written as 

Another way of thinking of the centripetal force is that it is a force arising hecause the mass is constrained to move on the surface of a sphere, expressed by the constraint 

This equation can be differentiated with respect to time to obtain the equations 

The first of these expresses the condition that the centripetal constraint force does no work, because the velocity is perpendicular to the radius. The second states that the radial component of the acceleration is directed inwards and equal to the square of the speed. This relation can be used to calculate the constraint force by taking the scalar product of the equation of motion (2) with the vector function r[t], and using the constraint and its time derivatives to obtain 

Although this equation is derived in the classic works of Webster [2] and Sommerfeld [5], only Webster notes that the equation of conservation of energy 

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