Hamiltonian

From Physics wiki

In the Lagrangian formulation of a system, one deals with $N\,$ generalized coordinates $q_i\,$ , their velocities $\dot{q}_i\,$ , and a Lagrangian $L(q_i, \dot{q}_i)\,$ , and one is tasked with solving the Euler-Lagrange equations which are generally second order in time:

$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0\,$ .

The canonical momentum conjugate to $q_i\,$ is

$p_i \equiv \frac{\partial L}{\partial \dot{q}_i}\,$ .

Let us rephrase the problem to consider $q_i\,$ and $p_i\,$ to be independent variables. Being $2N\,$ variables, we expect that this should be an equally valid description of the system. We perform a Legendre transformation of $L\,$ with respect to $\dot{q}_i\,$ . First, we formally invert the previous expression to obtain each $\dot{q}_i\,$ in terms of $q_j\,$ and $p_j\,$ . Then we define the Hamiltonian to be

$H(p,q) \equiv \sum_i p_i \dot{q}_i - L(q_j, \dot{q}_j)\,$ .

Note that

$\frac{\partial H(p,q)}{\partial p_j} = \dot{q}_j + \sum_i p_i\frac{\partial \dot{q}_i}{\partial p_j} - \sum_i \frac{\partial L}{\partial \dot{q}_i} \frac{\partial \dot{q}_i}{\partial p_j} = \dot{q}_j + \sum_i \left(p_i - \frac{\partial L}{\partial \dot{q}_i} \right)\frac{\partial \dot{q}_i}{\partial p_j}\,$ ,

which, by virtue of the definition of $p_i\,$ becomes

$\dot{q}_i = \frac{\partial H(p,q)}{\partial p_i}\,$ .

Furthermore,

$\frac{\partial H(p,q)}{\partial q_j}\,$	$= \sum_i p_i \frac{\partial \dot{q}_i}{\partial q_j} - \frac{\partial L}{\partial q_j} - \sum_i \frac{\partial L}{\partial \dot{q}_i} \frac{\partial \dot{q}_j}{\partial q_i}\,$ ,
	$= -\frac{d}{dt} \frac{\partial L}{\partial \dot{q}_j}\,$ ,
	$= -\frac{dp_j}{dt}\,$ .