# 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

## Section 13.3 - Hamilton's Equations in Several Dimensions

## Section 13.4 - Ignorable Coordinates

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

## Section 13.6 - Phase-Space Orbits

## Section 13.7 - Lioville's Theorem