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\section{\label{Hamiltonian mechanics}Hamiltonian Mechanics}

The \textit{Lagrangian} is given by 
\[
L=\sum T(\dot{q}_{i})-V(q_{i}) 
\]
The \textit{Hamiltonian} is given by 
\[
H=\sum \dot{q}_{i}p_{i}-L 
\]
In 2 dimensions 
\[
L=T-U=\frac{1}{2}m(\dot{r}^{2}+r^{2}\dot{\phi}^{2})-U(r,\phi ) 
\]

If the coordinates used are \textit{canonical} the \label{Hamilton equations}%
Hamilton equations are the equations of motion for the system 
\[
\frac{dq_{i}}{dt}=\frac{\partial H}{\partial p_{i}}\qquad \frac{dp_{i}}{dt}=-%
\frac{\partial H}{\partial q_{i}} 
\]
Coordinates are canonical if 
\[
\{q_{i},q_{j}\}=0\qquad \{p_{i},p_{j}\}=0\qquad \{q_{i},p_{j}\}=\delta _{ij} 
\]
where $\{,\}$ is the \textit{Poisson bracket} 
\[
\{A,B\}=\sum\limits_{i}\left[ \frac{\partial A}{\partial q_{i}}\frac{%
\partial B}{\partial p_{i}}-\frac{\partial A}{\partial p_{i}}\frac{\partial B%
}{\partial q_{i}}\right] 
\]
The Hamiltonian of a Harmonic oscillator is given by 
\[
H(x,p)=p^{2}/2m+\frac{1}{2}m\omega ^{2}x^{2} 
\]
With new coordinates $(\theta ,I)$, obtained by the canonical transformation 
$x=\sqrt{\dfrac{2I}{m\omega }}\cos (\theta )$ and $p=-\sqrt{2Im\omega }\sin
(\theta )$, with inverse $\theta =\arctan (-\dfrac{p}{m\omega x})$ and $I=%
\frac{1}{2}\dfrac{p^{2}}{m\omega }+\frac{1}{2}m\omega x^{2}$ it follows that 
\[
H(\theta ,I)=\omega I 
\]

The Hamiltonian of a charged particle with charge $q$ in an external
electromagnetic field is given by 
\[
H=\frac{1}{2m}\left( \vec{p}-q\vec{A}\right) ^{2}+qV 
\]
This Hamiltonian can be derived from the Hamiltonian of a free particle $%
H=p^{2}/2m$ with the transformations $\vec{p}\rightarrow \vec{p}-q\vec{A}$
and $H\rightarrow H-qV$. This is elegant from a relativistic point of
view---this is equivalent to the transformation of the momentum 4-vector $%
p^{\alpha }\rightarrow p^{\alpha }-qA^{\alpha }$. A gauge transformation on
the potentials $A^{\alpha }$ corresponds to a canonical transformation,
which makes the Hamilton equations the equations of motion for the system.

\subsection{\ Related Reference Topics}

{\small \hyperref{Differential Equations}{}{}{../Mathematics/ODE_contents.tex%
}}

{\small \hyperref{Table of Integrals}{}{}{../Mathematics/TAB_integrals.tex}}

{\small \hyperref{Table of Derivatives}{}{}{%
../Mathematics/TAB_derivatives.tex}}

{\small \hyperref{Table of Physical Constants}{}{}{PhysicalConstants.tex}}

{\small \hyperref{Table of SI Units}{}{}{SIUnits.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Numbers, Functions, and Units}{}{}{../../Help/DM2.tex}}

{\small \hyperref{Trigonometry}{}{}{../../Help/DM4.tex}}

{\small \hyperref{Calculus}{}{}{../../Help/DM7.tex}}

{\small \hyperref{Matrix Algebra}{}{}{../../Help/DM8.tex}}

{\small \hyperref{Vector Calculus}{}{}{../../Help/DM9.tex}}

{\small \hyperref{Differential Equations}{}{}{../../Help/DM10.tex}}\hfill

\noindent \hrulefill

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