Conservative Forces and Conservation of Energy (Proof)

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Conservative Forces and Conservation of Energy (Proof)

Intro

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In this short section we will

a) define conservative forces and
b) prove that such forces conserve a quantity we call energy for constant mass systems.

A conservative force \vec{F} is defined as:

(1)   \begin{equation*} \vec{F} = - \nabla U(\vec{r}) \end{equation*}

Where U is a function of the coordinates.Now we multiply both sides by \vec{v} = d \vec{x} / dt. We get:

(2)   \begin{equation*} \vec{v}\cdot \vec{F} = m \vec{v} \cdot \frac{d \vec{v}}{dt} = - \vec{v} \cdot \nabla U. \end{equation*}

Where we used Newton’s second law in the first equal sign. It is left to the reader to prove:

(3)   \begin{equation*} \vec{v} \cdot \frac{d \vec{v}}{dt} = \frac{1}{2} \frac{d v^2}{dt} \,\,\, \text{and} \,\,\, \vec{v}\cdot \nabla U = \frac{d U}{dt} \end{equation*}

Where v is the velocity. Using the equations above we have:

(4)   \begin{equation*} \frac{d}{dt} \left( \frac{1}{2} m v^2 + U \right) = 0. \end{equation*}

Defining energy to be the sum of the kinetic energy m v^2/2 and the potential energy U we have:

(5)   \begin{equation*} \frac{dE}{dt} = 0. \end{equation*}

In other words E_{final} = E_{initial}, the standard conservation of energy.

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Sample Problem

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Prove

(1)   \begin{equation*} \vec{v} \cdot \frac{d \vec{v}}{dt} = \frac{1}{2} \frac{d v^2}{dt} \,\,\, \text{and} \,\,\, \vec{v}\cdot \nabla U = \frac{d U}{dt} \end{equation*}

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Solution

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For the first equality we have:

    \begin{align*} \frac{1}{2} \frac{d v^2}{dt} &= \frac{1}{2} \frac{ d( v_x^2 + v_y^2 +v_z^2)}{dt} \\ &= v_x \frac{dv_x}{dt} + v_y \frac{dv_y}{dt}+ v_z \frac{dv_z}{dt} \\ &= \vec{v} \cdot \frac{d \vec{v}}{dt}. \end{align*}

For the second equality we have:

(1)   \begin{equation*} \vec{v} \cdot U(x,y,z) = \frac{d x}{dt} \frac{\partial U}{\partial x} + \frac{d y}{dt} \frac{\partial U}{\partial y} + \frac{d z}{dt} \frac{\partial U}{\partial z} = \frac{d U }{dt}. \end{equation*}

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