Classical Mechanics and Borders of Applicability

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Energy

If a force F is applied to a particle that achieves a displacement Δs, the work done by the force is the scalar quantity

$\Delta W = \mathbf{F} \cdot \Delta \mathbf{s}$ .

If the mass of the particle is constant, and ΔW_total is the total work done on the particle, obtained by summing the work done by each applied force, from Newton's second law:

$\Delta W_{\rm total} = \Delta E_k \,\!$ ,

where E_k is called the kinetic energy. For a point particle, it is defined as

$E_k = \begin{matrix} \frac{1}{2} \end{matrix} mv^2$ .

For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.

A particular class of forces, known as conservative forces, can be expressed as the gradient of a scalar function, known as the potential energy and denoted E_p:

$\mathbf{F} = - \nabla E_p$ .

If all the forces acting on a particle are conservative, and E_p is the total potential energy, obtained by summing the potential energies corresponding to each force

$\mathbf{F} \cdot \Delta \mathbf{s} = - \nabla E_p \cdot \Delta \mathbf{s} = - \Delta E_p \Rightarrow - \Delta E_p = \Delta E_k \Rightarrow \Delta (E_k + E_p) = 0 \,\!$ .

This result is known as conservation of energy and states that the total energy,

$\sum E = E_k + E_p \,\!$

is constant in time. It is often useful, because many commonly encountered forces are conservative.