Browse & Filter

$\newcommand{\Label}[1]{\label{#1}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$

The Hamiltonian of a charged particle in electromagnetic field is derived starting from the Lagrangian.

$\newcommand{\Prime}{{^\prime}} \newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}\newcommand[\Pime][{^\prime}]\newcommand{\qbf}{{\mathbf q}}\newcommand{\pbf}{\mathbf p}$

In the canonical formulation of mechanics, the  state of a system is represented by a point in phase space. As time evolves, the system moves along a path in phase space. Principle of least action is  formulated in phase space. The Hamilton's equations motion follow if we demand  that, for infinitesimal variations, with coordinates at the end point fixed, the action be extremum. No restrictions on variations in momentum are imposed.

The Hamiltonian  for a charged particle in electromagnetic field is invariant under the gauge transformation \[\vec A \longrightarrow \vec A^\prime = \vec A -\nabla \Lambda\]if the canonical momentum is taken to transform as \[\vec p \longrightarrow \vec p^\prime -(e/c) \nabla \Lambda.\]
$\newcommand{\Prime}{{^\prime}}\newcommand{\Label}[1]{\label{#1}}$

$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}
\newcommand{\Label}[1]{\label{#1}}\newcommand{\qbf}{{\mathbf q}}\newcommand{\pbf}{{\mathbf p}}$

Transition from the Lagrangian to Hamiltonian formalism is described; Hamiltonian equations of motion are obtained.

Symmetries play an important role in many areas of Physics, Chemistry and Particle Physics.

The equations of motion in Newtonian mechanics are always written in Cartesian coordinates. If there are constraints among the coordinates, these have to be taken into account separately. A special type of constraints, called holonomic constraints and the number of degrees of freedom is defined.

For a periodic system two types of motion are possible. In the first type both coordinates and momenta are periodic functions of time. An example of this type is motion of a simple pendulum. In the second type of motion only the coordinates are periodic functions of time. An example of the second type of motion is the conical pendulum where the angle keeps increasing, but the momentum  is a periodic function of time.

$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$

Jacobi's complete integral is defined as the action integral expressed in terms of non additive constants of motion and initial and final times. Knowledge of the complete integral is equivalent to the knowledge of the solution of equations of motion. Its relation with the Hamilton's principal function is \begin{equation} \pp[S_J (q, \alpha, t)]{\alpha_k} -\pp[S_J (q_0, \alpha, t_0)]{\alpha_k} =0 \end{equation}