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|
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\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
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|
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Newtonian Mechanics suffers from two important limitations: motions |
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can only be described in cartesian coordinate systems. Moreover, it |
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becomes impossible to predict analytically the properties of the |
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system even if we know all of the details of the interaction. In |
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order to overcome some of the practical difficulties which arise in |
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attempts to apply Newton's equation to complex system, approximate |
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numerical procedures may be developed. |
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Newtonian Mechanics suffers from a important limitation: motions can |
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only be described in cartesian coordinate systems which make it |
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impossible to predict analytically the properties of the system even |
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> |
if we know all of the details of the interaction. In order to |
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overcome some of the practical difficulties which arise in attempts |
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to apply Newton's equation to complex system, approximate numerical |
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procedures may be developed. |
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|
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\subsubsection{\label{introSection:halmiltonPrinciple}\textbf{Hamilton's |
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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} |
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then Eq.~\ref{introEquation:halmitonianPrinciple1} becomes |
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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} |
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|
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dH = \sum\limits_k {\left( {p_k d\dot q_k + \dot q_k dp_k - |
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\frac{{\partial L}}{{\partial q_k }}dq_k - \frac{{\partial |
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L}}{{\partial \dot q_k }}d\dot q_k } \right)} - \frac{{\partial |
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L}}{{\partial t}}dt \label{introEquation:diffHamiltonian1} |
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L}}{{\partial t}}dt . \label{introEquation:diffHamiltonian1} |
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\end{equation} |
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Making use of Eq.~\ref{introEquation:generalizedMomenta}, the second |
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and fourth terms in the parentheses cancel. Therefore, |
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Eq.~\ref{introEquation:diffHamiltonian1} can be rewritten as |
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\begin{equation} |
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dH = \sum\limits_k {\left( {\dot q_k dp_k - \dot p_k dq_k } |
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\right)} - \frac{{\partial L}}{{\partial t}}dt |
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\right)} - \frac{{\partial L}}{{\partial t}}dt . |
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\label{introEquation:diffHamiltonian2} |
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\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 |
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conserves the total energy |
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(Eq.~\ref{introEquation:energyConservation}). It follows that |
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Hamilton's equations of motion conserve the total Hamiltonian. |
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Hamilton's equations of motion conserve the total Hamiltonian |
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\begin{equation} |
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\frac{{dH}}{{dt}} = \sum\limits_i {\left( {\frac{{\partial |
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H}}{{\partial q_i }}\dot q_i + \frac{{\partial H}}{{\partial p_i |
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}}\dot p_i } \right)} = \sum\limits_i {\left( {\frac{{\partial |
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H}}{{\partial q_i }}\frac{{\partial H}}{{\partial p_i }} - |
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\frac{{\partial H}}{{\partial p_i }}\frac{{\partial H}}{{\partial |
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q_i }}} \right) = 0} \label{introEquation:conserveHalmitonian} |
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q_i }}} \right) = 0}. \label{introEquation:conserveHalmitonian} |
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\end{equation} |
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|
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\section{\label{introSection:statisticalMechanics}Statistical |
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momentum variables. Consider a dynamic system of $f$ particles in a |
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cartesian space, where each of the $6f$ coordinates and momenta is |
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assigned to one of $6f$ mutually orthogonal axes, the phase space of |
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this system is a $6f$ dimensional space. A point, $x = (\rightarrow |
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q_1 , \ldots ,\rightarrow q_f ,\rightarrow p_1 , \ldots ,\rightarrow |
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p_f )$, with a unique set of values of $6f$ coordinates and momenta |
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is a phase space vector. |
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this system is a $6f$ dimensional space. A point, $x = |
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(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
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\over q} _1 , \ldots |
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,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
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> |
\over q} _f |
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,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
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\over p} _1 \ldots |
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> |
,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} |
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\over p} _f )$ , with a unique set of values of $6f$ coordinates and |
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momenta is a phase space vector. |
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%%%fix me |
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|
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In statistical mechanics, the condition of an ensemble at any time |
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is known to be thermally isolated and conserve energy, the |
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Microcanonical ensemble (NVE) has a partition function like, |
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\begin{equation} |
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\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition}. |
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\Omega (N,V,E) = e^{\beta TS}. \label{introEquation:NVEPartition} |
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|
\end{equation} |
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A canonical ensemble (NVT)is an ensemble of systems, each of which |
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can share its energy with a large heat reservoir. The distribution |