Introduction to Linear Algebra with Mathematica

# Preface

Hamiltonian mechanics is a mathematically sophisticated formulation of classical mechanics.

Degres of freedom

The important characteristi cs of a dynamical system is its number of degrees of freedom. It is customary to set this number equal to one half the number of independent variables that completely define the state of system, i. e., to one half the dimension of the phase space of the syst m. Such a definition of the number of degrees of freedom came from its first appearence in mechanics where the one-dimensional motion of a material point is completely defined by two variables: coordinate and velocity. According to this definition the number of degrees of freedom may not be an integer. For example, if a system is described by one differential equation of the third order or by three first order combined differential equations, then its number of degrees of freedom equals one and a half. We note that if a system is non-autonomous , i.e. the algorithm for its transition from one state to another is explicitly dependent on tim , then it may be considered as autonomous by means of the incorporation of time as one of the coordinates of its phase space. In so doing a system described by a second order differential equation with external action must be considered as a system with one and a half degrees of freedom.

All dynamical systems havin g a physical meaning can be separated into two main groups: systems with conservation of their phase volume and systems with decrease of their phase volume, or dissipative sustems. If the autonomous system is described by differential equations of the type $\dot{x}_j = f_j (x_1 , x_2 , \ldots , x_n ), \qquad j=1,2,\ldots , n.$ then it can be shown, based on the divergence theorem, that the variation of its phase volume dV in a time dt is ${\text d}V = {\text d}t \int \left( \frac{{\text d}\dot{x}_1}{{\text d}x_1} + \frac{{\text d}\dot{x}_2}{{\text d}x_2} + \cdots + \frac{{\text d}\dot{x}_n}{{\text d}x_n} \right) {\text d}x_1 {\text d} x_2 \cdots {\text d}x_n = {\text d}t \int \mbox{div}\, \dot{\bf x}\,{\text d}{\bf x} ,$ where x is a vector with com ponents x1, …, xn. It follows from this that the sufficient condition for the conservation of the phase volume is $\mbox{div}\, \dot{\bf x}\,{\text d}{\bf x} = 0.$

# 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
$$\label{EqH.1} \left\{ f , G \right\} = \sum_{i=1}^n \left( \frac{\partial F}{\partial q_i} \,\frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i} \,\frac{\partial G}{\partial q_i} \right)$$
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
$$\label{EqH.2} \frac{{\text d}q_k}{{\text d}t} = \left\{ q_k ,H \right\} , \qquad \frac{{\text d}p_k}{{\text d}t} = \left\{ p_k ,H \right\} .$$

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