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Hamilton Equations

Another way in which Newton equations are often rewritten is due to Hamilton:

 
$\displaystyle \dot{x}$ = $\displaystyle \frac{\partial E}{\partial p_x}$  
$\displaystyle \dot{p_x}$ = $\displaystyle - \frac{\partial E}{\partial x}$  
$\displaystyle \dot{y}$ = $\displaystyle \frac{\partial E}{\partial p_y}$  
$\displaystyle \dot{p_y}$ = $\displaystyle - \frac{\partial E}{\partial y}$  
$\displaystyle \dot{z}$ = $\displaystyle \frac{\partial E}{\partial p_z}$  
$\displaystyle \dot{p_z}$ = $\displaystyle - \frac{\partial E}{\partial z}$  

where p = m v is the momentum of a material point.

It is easy to see that these equations are equivalent to the Newton's equation too. In order to see that first observe that

\begin{displaymath}\frac{m v^2}{2} = \frac{p^2}{2m}
\end{displaymath}

Hence the total energy of a material point can be rewritten as:

\begin{displaymath}E = \frac{p^2}{2m} + q V
\end{displaymath}

Now you can easily see that:

\begin{displaymath}\frac{\partial E}{\partial p_x} =
\frac{\partial}{\partial ...
..._z^2}{2m}
= \frac{p_x}{m} = \frac{m v_x}{m} = v_x = \dot{x}
\end{displaymath}

so the equation

\begin{displaymath}\dot{x} = \frac{\partial E}{\partial p_x}
\end{displaymath}

simply states that $\dot{x} = v_x = p_x / m$. The second Hamilton equation states that:

\begin{displaymath}\dot{p}_x = m\dot{v}_x = m a_x = - \frac{\partial qV}{\partial x}
\end{displaymath}

so it is equivalent to the Newton equation.

Hamilton equations are often expressed in terms of generalized coordinates too and written in the following way:

 
$\displaystyle \dot{q}_i$ = $\displaystyle \frac{\partial H}{\partial p_i}$  
$\displaystyle \dot{p}_i$ = $\displaystyle - \frac{\partial H}{\partial q_i}, \quad
i = 1, 2, 3$  

where H is the Hamilton function, or the Hamiltonian. It is, as you have seen, simply a total energy of the material point.


next up previous index
Next: Lagrange and Hamilton Equations Up: Kinematics and Dynamics of Previous: Lagrange Equations
Zdzislaw Meglicki
2001-02-26