Mechanics Chapter 13

docs/physics/mechanics/13-hamiltonian-mechanics.md

@@ -14,16 +14,66 @@ $$p_i = \frac{\partial \mathcal{L}}{\partial \dot{q}_i}$$
 
 $$\mathcal{H} = \sum_{i = 1}^n p_i \dot{q_i} - \mathcal{L}$$
 
-
-
 ## 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
 
-## Section 13.7 - Lioville's Theorem
+Skipped.
+
+## Section 13.7 - Liouville's Theorem
+
+Skipped.