14.3: Hamilton's Equations of Motion

In classical mechanics we can describe the state of a system by specifying its Lagrangian as a function of the coordinates and their time rates of change:

\[ L=L(q_{i},\dot{q}) \label{14.3.2} \]

If the coordinates and the velocities increase, the corresponding increment in the Lagrangian is

\[ dL=\sum_{i}\dfrac{\partial L}{\partial q_{i}}dq_{i}+\sum_{i}\dfrac{\partial L}{\partial \dot{q_{i}}}d\dot{q_{i}}. \label{14.3.3} \]

Definition: generalized momenta

The generalized momentum p_i associated with the generalized coordinate q_i is defined as

\[ p_{i}=\dfrac{\partial L}{\partial \dot{q_{i}}}. \label{14.3.4} \]

[You have seen this before, in Section 13.4. Remember “ignorable coordinate”?]

It follows from the Lagrangian equation of motion (Equation 13.4.14)

\[ \dfrac{d}{dt}\dfrac{\partial L}{\partial \dot{q_{i}}}=\dfrac{\partial L}{\partial q_{i}} \nonumber \]

that

\[ \dot{p}_{i}=\dfrac{\partial L}{\partial q_{i}}. \label{14.3.5} \]

Thus

\[ dL=\sum_{i}\dot{p}_{i}dq_{i}+\sum_{i}p_{i}d\dot{q}_{i}. \label{14.3.1} \]

(I am deliberately numbering this Equation \( \ref{14.3.1}\), to maintain an analogy between this section and Section 14.2.)

However, it is sometimes convenient to change the basis of the description of the state of a system from \( q_{i}\) and \( \dot{q_{i}}\) to \( q_{i}\) and \( \dot{p_{i}}\) by defining a quantity called the hamiltonian \( H\) defined by

\[ H=\sum_{i}p_{i}\dot{q_{i}}-L. \label{14.3.6} \]

Definition: hamiltonian

In that case, if the state of the system changes, then

\[ \begin{align*} dH&=\sum_{i}p_{i}d\dot{q_{i}}+\sum_{i}\dot{q_{i}}dp_{i}-dL \label{14.3.7} \\[5pt] &=\sum_{i}p_{i}d\dot{q_{i}}+\sum_{i}\dot{q_{i}}dp_{i}-\sum_{i}\dot{p_{i}}dq_{i}-\sum_{i}p_{i}d\dot{q_{i}} \label{14.3.8} \end{align*} \]

That is

\[ dH=\sum_{i}\dot{q_{i}}dp_{i}-\sum_{i}\dot{p_{i}}dq_{i}. \label{14.3.9} \]

We are regarding the hamiltonian as a function of the generalized coordinates and generalized momenta:

\[ H=H(q_{i},p_{i}) \label{14.3.10} \]

so that

\[ dH=\sum_{i}\dfrac{\partial H}{\partial q_{i}}dq_{i}+\sum_{i}\dfrac{\partial H}{\partial p_{i}}dp_{i}, \label{14.3.11} \]

from which we see that

\[ -\dot{p}_{i}=\dfrac{\partial H}{\partial q_{i}} \label{14.3.12} \]

and

\[ \dot{q}_{i}=\dfrac{\partial H}{\partial p_{i}} \label{14.3.13} \]

In summary, then, Equations \( \ref{14.3.4}\), \( \ref{14.3.5}\), \( \ref{14.3.12}\) and \( \ref{14.3.13}\):

\[ p_{i}=\dfrac{\partial L}{\partial\dot{q_{i}}} \label{A} \]

\[ \dot{p_{i}}=\dfrac{\partial L}{\partial q_{i}} \label{B} \]

\[ - \dot{p_{i}}=\dfrac{\partial H}{\partial q_{i}} \label{C} \]

\[ \dot{q_{i}}=\dfrac{\partial H}{\partial p_{i}} \label{D} \]

which I personally find impossible to commit accurately to memory (although note that there is one dot in each equation) except when using them frequently, may be regarded as Hamilton’s equations of motion. I’ll refer to these equations as A, B, C and D.

Note that, in Equation \ref{B}, if the Lagrangian is independent of the coordinate \( q_{i}\) the coordinate \( q_{i}\) is referred to as an “ignorable coordinate”. I suppose it is called “ignorable” because you can ignore it when calculating the lagrangian, but in fact a so-called “ignorable” coordinate is usually a very interesting coordinate indeed, because it means (look at the second equation) that the corresponding generalized momentum is conserved.

Now the kinetic energy of a system is given by \( T=\dfrac{1}{2}\sum_{i}p_{i}\dot{q_{i}}\) (for example, \( \dfrac{1}{2}m\nu\nu\)), and the hamiltonian (Equation \( \ref{14.3.6}\)) is defined as \( H=\sum_{i}p_{i}\dot{q_{i}}-L\). For a conservative system, \( L=T-V\), and hence, for a conservative system, \( H=T+V\). If you are asked in an examination to explain what is meant by the hamiltonian, by all means say it is \( T+V\). That’s fine for a conservative system, and you’ll probably get half marks. That’s 50% - a D grade, and you’ve passed. If you want an A+, however, I recommend Equation \( \ref{14.3.6}\).