Preface


Hamiltonian mechanics is a mathematically sophisticated formulation of classical mechanics.

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Introduction to Linear Algebra with Mathematica

Hamilton Principle


Hamiltonian mechanics is another reformulation of classical mechanics that is naturally extended to statistical mechanics and quantum mechanics. Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting from Lagrangian mechanics. The Hamiltonian is defined in terms of Lagrangian \( L({\bf q}, \dot{\bf q}, t) \) by
\[ H({\bf p}, {\bf q}, t) = \sum_{i=1}^n p_i \,\frac{{\text d} q_i}{{\text d}t} - L({\bf q}, \dot{\bf q}, t) , \]
where pi are generalized momentum and are related to the generalized coordinates q by \( p_i = \frac{{\text d}L({\bf q}, \dot{\bf q}, t)}{{\text d} \dot{q}_i} . \) The equations of motion follow from
\[ \dot{p}_i = - \frac{\partial H({\bf p}, {\bf q}, t)}{\partial q_i} , \qquad \dot{q}_i = \frac{\partial H({\bf p}, {\bf q}, t)}{\partial p_i} . \]
The Poisson bracket is defined by
\begin{equation} \label{EqH.1} \left\{ f , G \right\} = \sum_{i=1}^n \left( \frac{\partial F}{\partial q_i} \,\frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i} \,\frac{\partial G}{\partial q_i} \right) \end{equation}
for any two functions F and G of canonical coordinates and momenta. It is linear for both F and G. It is anti symmetric: { F, G } = −{ G, F }. It is an easy exercise to prove that the Poisson bracket also satisfies the Jacobi identity:
\[ \left\{ \left\{ F, G \right\} , H \right\} + \left\{ \left\{ G, H \right\} , F \right\} + \left\{ \left\{ H, F \right\} , G \right\} = 0 . \]
The canonical coordinates and momenta themselves have Poisson brqackets
\[ \begin{split} \left\{ q_k , q_i \right\} &= 0, \qquad \left\{ p_k , p_i \right\} = 0, \\ \left\{ q_k , p_i \right\} &= \delta_{k,i} , \end{split} \]
for k, i = 1, 2, … , n. Hamilton's equations can be written as
\begin{equation} \label{EqH.2} \frac{{\text d}q_k}{{\text d}t} = \left\{ q_k ,H \right\} , \qquad \frac{{\text d}p_k}{{\text d}t} = \left\{ p_k ,H \right\} . \end{equation}

 

Canonical transformations


The 2n-dimensional space of points specified by the canonical coordinates and momenta is called phase space.

 

Hamiltonian evolution


When the dynamics is described by Hamilton's equations, the evolution in time is made by canonical transformations.

 

  1. Jordan, T., Steppingstones in Hamiltonian dynamics, The American Journal of Physics, 2004, 72, No. 8, pp. 1095--1099. doi: 10.1119/1.1737394

 

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