Conservation of Energy in Mechanical Systems

In a system of mass points 1... k on an individual mass point i the force should act according to Eq. (3.2)  

We use here a Cartesian coordinate system. X,-, T,-, and Z, are the forces in redirection, y-direction, and z-direction, respectively, that are acting on the ith particle. 

The right-hand side of Eq. (3.2) is a partial derivative of a function y, z). Note that this function is only dependent on the position of the particle i. The function (h is called the potential energy of the particle. 

According to d Alembert we introduce now the forces of inertness. A particle that is under stress will try to lower the stresses and will follow the stresses. The forces of inertness are 

The forces of inertness and the forces arising from the potential energy sum to zero  

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