7.3 The Canonical equations

Again, consider the relations for transforming back and forth between the Lagrangian and the Hamiltonian:

\begin{align} \dot q_i&=\die{H}{p_i} \\ p_i &= \die{L}{\dot q_i} \tag{7.17}\end{align}

The second equation can be re-arranged using the Euler-Lagrange equation:

\begin{align} \frac{d}{dt}\die{L}{\dot q_i}&=\die{L}{q_i}\\ \therefore \dot p&=\die{L}{q_i}= -\die{H}{q_i} \tag{7.18}\end{align}

We can write the transformation equations as:

\begin{align} \dot q_i&=\die{H}{p_i} \\ \dot p_i &= -\die{H}{q_i} \tag{7.19}\end{align}

These are called the “canonical equations of Hamilton”. They are \(2n\) first order differential equations that specify the location, \(p_i\), \(q_i\), of the system in phase space, and have the property that the total time derivatives are isolated on one side of the equation. Note that if a coordinate is cyclic in the Hamiltonian, conservation of the associated generalized momentum follows immediately. In order to solve these equations in configuration space, one can substitute the first equation into the second one and re-obtain the second-order equation for the coordinate as a function of time that one obtains from the Euler-Lagrange equations.

Consider now the total time derivative of the Hamiltonian, \(H(p_i,q_i,t)\):

\begin{align} \frac{dH}{dt}&= \sum_i\left(\die{H}{p_i}\dot p_i +\die{H}{q_i}\dot q_i \right)+\die{H}{t}\\ &= \sum_i\left(\dot q_i\dot p_i -\dot p_i\dot q_i \right)+\die{H}{t}\\ &= \die{H}{t} \left(=-\die{L}{t}\right) \tag{7.20}\end{align}

where we used the canonical equations. We see that if the Lagrangian (or Hamiltonian) does not explicitly depend on time, then the total time derivative of \(H\) is zero (i.e. \(H\) is a constant). Often, the Hamiltonian is equal to the total energy, in particular, if it does not depend on time explicitly and if the potential does not depend on velocity. This is the result that we obtained for the Jacobian integral. In the case that the Hamiltonian is the total energy, it can be written as:

\begin{align} H=T+V \tag{7.21}\end{align}

One should however be careful and keep in mind that this form is not the general definition of the Hamiltonian, which is given by the Legendre transformation.