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|
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\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
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|
| 74 |
< |
Newtonian Mechanics suffers from two important limitations: motions |
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< |
can only be described in cartesian coordinate systems. Moreover, it |
| 76 |
< |
becomes impossible to predict analytically the properties of the |
| 77 |
< |
system even if we know all of the details of the interaction. In |
| 78 |
< |
order to overcome some of the practical difficulties which arise in |
| 79 |
< |
attempts to apply Newton's equation to complex system, approximate |
| 80 |
< |
numerical procedures may be developed. |
| 74 |
> |
Newtonian Mechanics suffers from a important limitation: motions can |
| 75 |
> |
only be described in cartesian coordinate systems which make it |
| 76 |
> |
impossible to predict analytically the properties of the system even |
| 77 |
> |
if we know all of the details of the interaction. In order to |
| 78 |
> |
overcome some of the practical difficulties which arise in attempts |
| 79 |
> |
to apply Newton's equation to complex system, approximate numerical |
| 80 |
> |
procedures may be developed. |
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|
|
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\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's |
| 83 |
|
Principle}} |
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|
function $L$ can be defined as the difference between the kinetic |
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|
energy of the system and its potential energy, |
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|
\begin{equation} |
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L \equiv K - U = L(q_i ,\dot q_i ) , |
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> |
L \equiv K - U = L(q_i ,\dot q_i ). |
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|
\label{introEquation:lagrangianDef} |
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|
\end{equation} |
| 104 |
< |
then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
| 104 |
> |
Thus, Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
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|
\begin{equation} |
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< |
\delta \int_{t_1 }^{t_2 } {L dt = 0} , |
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> |
\delta \int_{t_1 }^{t_2 } {L dt = 0} . |
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|
\label{introEquation:halmitonianPrinciple2} |
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|
\end{equation} |
| 109 |
|
|
| 148 |
|
dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k - |
| 149 |
|
\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial |
| 150 |
|
L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial |
| 151 |
< |
L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1} |
| 151 |
> |
L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1} |
| 152 |
|
\end{equation} |
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|
Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second |
| 154 |
|
and fourth terms in the parentheses cancel. Therefore, |
| 155 |
|
Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as |
| 156 |
|
\begin{equation} |
| 157 |
|
dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k } |
| 158 |
< |
\right)} - \frac{{\partial L}}{{\partial t}}dt |
| 158 |
> |
\right)} - \frac{{\partial L}}{{\partial t}}dt . |
| 159 |
|
\label{introEquation:diffHamiltonian2} |
| 160 |
|
\end{equation} |
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|
By identifying the coefficients of $dq_k$, $dp_k$ and dt, we can |
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|
In Newtonian Mechanics, a system described by conservative forces |
| 192 |
|
conserves the total energy |
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|
(Eq.~\ref{introEquation:energyConservation}). It follows that |
| 194 |
< |
Hamilton's equations of motion conserve the total Hamiltonian. |
| 194 |
> |
Hamilton's equations of motion conserve the total Hamiltonian |
| 195 |
|
\begin{equation} |
| 196 |
|
\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial |
| 197 |
|
H}}{{\partial q_i }}\dot q_i + \frac{{\partial H}}{{\partial p_i |
| 198 |
|
}}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial |
| 199 |
|
H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - |
| 200 |
|
\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial |
| 201 |
< |
q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
| 201 |
> |
q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian} |
| 202 |
|
\end{equation} |
| 203 |
|
|
| 204 |
|
\section{\label{introSection:statisticalMechanics}Statistical |
| 219 |
|
momentum variables. Consider a dynamic system of $f$ particles in a |
| 220 |
|
cartesian space, where each of the $6f$ coordinates and momenta is |
| 221 |
|
assigned to one of $6f$ mutually orthogonal axes, the phase space of |
| 222 |
< |
this system is a $6f$ dimensional space. A point, $x = (\rightarrow |
| 223 |
< |
q_1 , \ldots ,\rightarrow q_f ,\rightarrow p_1 , \ldots ,\rightarrow |
| 224 |
< |
p_f )$, with a unique set of values of $6f$ coordinates and momenta |
| 225 |
< |
is a phase space vector. |
| 222 |
> |
this system is a $6f$ dimensional space. A point, $x = |
| 223 |
> |
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 224 |
> |
\over q} _1 , \ldots |
| 225 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 226 |
> |
\over q} _f |
| 227 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 228 |
> |
\over p} _1 \ldots |
| 229 |
> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
| 230 |
> |
\over p} _f )$ , with a unique set of values of $6f$ coordinates and |
| 231 |
> |
momenta is a phase space vector. |
| 232 |
|
%%%fix me |
| 233 |
|
|
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|
In statistical mechanics, the condition of an ensemble at any time |
| 290 |
|
is known to be thermally isolated and conserve energy, the |
| 291 |
|
Microcanonical ensemble (NVE) has a partition function like, |
| 292 |
|
\begin{equation} |
| 293 |
< |
\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition}. |
| 293 |
> |
\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition} |
| 294 |
|
\end{equation} |
| 295 |
|
A canonical ensemble (NVT)is an ensemble of systems, each of which |
| 296 |
|
can share its energy with a large heat reservoir. The distribution |
