Overview
Formulation
Many forces in classical mechanics can be represented as a vector field. That is, for every point in space, there is a vector which represents the force present at that point. (think gravity for instance.) That is, you have a function such as
{% \vec{F}(x,y,z) %}
Often times, this vector field can be encapsulated in a single variable function, {% V(x,y,z) %} such that
{% \vec{F}(x,y,z) = \nabla V(x,y,z) %}
(see gradient)
When the force can be expressed this way, it is called a conservative force.
{% du = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy + \frac{\partial u}{\partial z}dz %}
{% d \vec{s} = dx \vec{i} + dy \vec{j} + dz \vec{k} %}
{% \nabla u \cdot d\vec{s} = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy + \frac{\partial u}{\partial z}dz = du %}
{% \displaystyle \int_{x_0}^{x_1} \nabla u \cdot d\vec{s} = \int_{x_0}^{x_1} du = u_1 - u_0 %}
If the force is given as above, then this implies that change in kinetic energy is given by
{% \Delta KE = V_2 - V_1 %}
Conservation of Energy
{% Energy = Kinetic + Potential %}