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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.