Chapter 13 - Hamiltonian Mechanics

Section 13.1 - The Basic Variables

Definition. Consider a Laplacian defined as \(\mathcal{L} = \mathcal{L}(q_1, \ldots, q_n, \dot{q}_1, \ldots, \dot{q}_n, t)\). Then, the set of coordinates \(q_1, \ldots, q_n\) are the configuration space while the set of coordinates \(q_1, \ldots, q_n, \dot{q}_1, \ldots, \dot{q}_n\) are known as the state space.

Recall that the generalized momenta \(p_i\) is also defined such that

\[p_i = \frac{\partial \mathcal{L}}{\partial \dot{q}_i}\]

Definition. The generalized momenta is also called the canonical momentum or the momentum conjugate to \(q_i\).

Definition. The Hamiltonian is defined as

\[\mathcal{H} = \sum_{i = 1}^n p_i \dot{q_i} - \mathcal{L}\]

Section 13.2 - Hamilton's Equations for One-Dimensional Systems

We see that for a pendulum, \(\mathcal{L} = \frac{1}{2} m L^2 \dot{\phi}^2 - mgL(1 - \cos \phi)\). For a bead sliding on a frictionless wire of height \(y = f(x)\), we see \(\mathcal{L} = \frac{1}{2}m[1 + f'(x)^2] - mgf(x)\).

Notably, using natural coordinates, \(\mathcal{L} = \frac{1}{2}A(q)\dot{q}^2 - U(q)\). Then, we can define \(\mathcal{H} = p\dot{q} - \mathcal{L}\).

We know that \(p = \frac{\partial \mathcal{L}}{\partial \dot{q}} = A(q)\dot{q}\). Then, \(\mathcal{H} = p\dot{q} - \mathcal{L} = A(q)\dot{q}^2 - \frac{1}{2} A(q) \dot{q}^2 + U(q) = 2T - T + U = T + U\)

Similarly, we can solve for \(\dot{q}\) from the definition of the generalized momentum to see that \(\dot{q} = \frac{q}{A(q)}\).

Deriving Hamilton's Equations is thus simple. We see that \(\frac{\partial \mathcal{H}}{\partial q} = p \frac{\partial \dot{q}}{\partial q} - [\frac{\partial \mathcal{L}}{\partial q} + \frac{\mathcal{L}}{\partial \dot{q}} \frac{\partial \dot{q}}{\partial q}] = p \frac{\partial \dot{q}}{\partial q} - [\frac{\partial \mathcal{L}}{\partial q} + q\frac{\partial \dot{q}}{\partial q}] = -\frac{\partial \mathcal{L}}{\partial q} = -\dot{p}\)

Differentiating instead with respect to \(p\), we see that \(\frac{\partial \mathcal{H}}{\partial p} = [\dot{q} + p \frac{\partial \dot{q}}{\partial p}] - \frac{\partial \mathcal{L}}{\partial \dot{q}} \frac{\partial \dot{q}}{\partial p} = [\dot{q} + p \frac{\partial \dot{q}}{\partial p}] - p \frac{\partial \dot{q}}{\partial p} = \dot{q}\)

Section 13.3 - Hamilton's Equations in Several Dimensions

We know that

\[\mathcal{H} = \sum_{i = 1}^N p_i \dot{q}_i - \mathcal{L}\]

Here, the generalized momenta are defined as

\[p_i = \frac{\partial \mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}, t)},{\partial \dot{q}_i}\]

This tells us that \(\dot{\mathbf{q}} = \dot{\mathbf{q}}(\mathbf{q}, \mathbf{p}, t)\). Then, we can define the Hamiltonian as

\[\mathcal{H} = \mathcal{H}(\mathbf{q}, \mathbf{p}, t) = \sum_{i = 1}^N p_i \dot{q}_i(\mathbf{q}, \mathbf{p}, t) - \mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}(\mathbf{q}, \mathbf{p}, t), t)\]

We can differentiate with respect to \(p_i\) to see that

\[\dot{q}_i = \frac{\partial \mathcal{H}}{\partial p_i}\]

We can differentiate with respect to \(q_i\) to see that

\[\dot{p}_i = - \frac{\partial \mathcal{H}}{\partial q_i}\]

For a system with \(n\) coordinates, this gives us \(2n\) first-order differential equations rather than \(n\) second-order differential equations as seen in the Lagrange equations.

We then can calculate

\[\frac{d \mathcal{H}}{dt} = \sum_{i=1}^N (\frac{\partial \mathcal{H}}{\partial q_i} \dot{q}_i + \frac{\partial \mathcal{H}}{\partial p_i} \dot{p}_i) + \frac{\partial \mathcal{H}}{\partial t}\]

We can then substitute Hamilton's equations to see that

\[\frac{d \mathcal{H}}{dt} = \frac{\partial \mathcal{H}}{\partial t}\]

From section 7.8, we know that if the relation from the generalized coordinates to rectangular coordinates is independent of \(t\) (that is, our generalized coordinates are natural), than \(\mathcal{H} = T + U\).

Section 13.4 - Ignorable Coordinates

Definition. If \(\mathcal{H}\) is independent of a coordinate \(q_i\), it immediately follows that \(\dot{p}_i = 0\) and thus \(p_i\) is a constant. Note that this definition immediately follows from the Lagrangian definition.

Section 13.5 - Lagrange's Equations vs. Hamilton's Equations

Skipped.

Section 13.6 - Phase-Space Orbits

Skipped.

Section 13.7 - Liouville's Theorem

Skipped.