Hamilton Equations

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

 

$$\dot{x} \frac{\partial E}{\partial p_x}$$ =

$$\dot{p_x} - \frac{\partial E}{\partial x}$$ =

$$\dot{y} \frac{\partial E}{\partial p_y}$$ =

$$\dot{p_y} - \frac{\partial E}{\partial y}$$ =

$$\dot{z} \frac{\partial E}{\partial p_z}$$ =

$$\dot{p_z} - \frac{\partial E}{\partial z}$$ =

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

$$\frac{m v^2}{2} = \frac{p^2}{2m}$$

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

$$E = \frac{p^2}{2m} + q V$$

Now you can easily see that:

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

so the equation

$$\dot{x} = \frac{\partial E}{\partial p_x}$$

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

$$\dot{p}_x = m\dot{v}_x = m a_x = - \frac{\partial qV}{\partial x}$$

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:

 

$$\dot{q}_i \frac{\partial H}{\partial p_i}$$ =

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

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