30 lines
1.0 KiB
Markdown
30 lines
1.0 KiB
Markdown
# Chapter 13 - Hamiltonian Mechanics
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## Section 13.1 - The Basic Variables
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**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*.
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Recall that the generalized momenta $p_i$ is also defined such that
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$$p_i = \frac{\partial \mathcal{L}}{\partial \dot{q}_i}$$
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**Definition**. The generalized momenta is also called the *canonical momentum* or the *momentum conjugate to $q_i$*.
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**Definition**. The *Hamiltonian* is defined as
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$$\mathcal{H} = \sum_{i = 1}^n p_i \dot{q_i} - \mathcal{L}$$
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## Section 13.2 - Hamilton's Equations for One-Dimensional Systems
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## Section 13.3 - Hamilton's Equations in Several Dimensions
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## Section 13.4 - Ignorable Coordinates
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## Section 13.5 - Lagrange's Equations vs. Hamilton's Equations
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## Section 13.6 - Phase-Space Orbits
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## Section 13.7 - Lioville's Theorem
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