1.0 KiB
1.0 KiB
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}