| 31 |
|
$F_{ji}$ be the force that particle $j$ exerts on particle $i$. |
| 32 |
|
Newton's third law states that |
| 33 |
|
\begin{equation} |
| 34 |
< |
F_{ij} = -F_{ji} |
| 34 |
> |
F_{ij} = -F_{ji}. |
| 35 |
|
\label{introEquation:newtonThirdLaw} |
| 36 |
|
\end{equation} |
| 37 |
– |
|
| 37 |
|
Conservation laws of Newtonian Mechanics play very important roles |
| 38 |
|
in solving mechanics problems. The linear momentum of a particle is |
| 39 |
|
conserved if it is free or it experiences no force. The second |
| 72 |
|
\subsection{\label{introSection:lagrangian}Lagrangian Mechanics} |
| 73 |
|
|
| 74 |
|
Newtonian Mechanics suffers from two important limitations: motions |
| 75 |
< |
can only be described in cartesian coordinate systems. Moreover, It |
| 76 |
< |
become impossible to predict analytically the properties of the |
| 75 |
> |
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 |
| 84 |
|
|
| 85 |
|
Hamilton introduced the dynamical principle upon which it is |
| 86 |
|
possible to base all of mechanics and most of classical physics. |
| 87 |
< |
Hamilton's Principle may be stated as follows, |
| 88 |
< |
|
| 89 |
< |
The actual trajectory, along which a dynamical system may move from |
| 90 |
< |
one point to another within a specified time, is derived by finding |
| 91 |
< |
the path which minimizes the time integral of the difference between |
| 93 |
< |
the kinetic, $K$, and potential energies, $U$. |
| 87 |
> |
Hamilton's Principle may be stated as follows: the actual |
| 88 |
> |
trajectory, along which a dynamical system may move from one point |
| 89 |
> |
to another within a specified time, is derived by finding the path |
| 90 |
> |
which minimizes the time integral of the difference between the |
| 91 |
> |
kinetic, $K$, and potential energies, $U$. |
| 92 |
|
\begin{equation} |
| 93 |
|
\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} , |
| 94 |
|
\label{introEquation:halmitonianPrinciple1} |
| 95 |
|
\end{equation} |
| 98 |
– |
|
| 96 |
|
For simple mechanical systems, where the forces acting on the |
| 97 |
|
different parts are derivable from a potential, the Lagrangian |
| 98 |
|
function $L$ can be defined as the difference between the kinetic |
| 135 |
|
p_i = \frac{{\partial L}}{{\partial q_i }} |
| 136 |
|
\label{introEquation:generalizedMomentaDot} |
| 137 |
|
\end{equation} |
| 141 |
– |
|
| 138 |
|
With the help of the generalized momenta, we may now define a new |
| 139 |
|
quantity $H$ by the equation |
| 140 |
|
\begin{equation} |
| 142 |
|
\label{introEquation:hamiltonianDefByLagrangian} |
| 143 |
|
\end{equation} |
| 144 |
|
where $ \dot q_1 \ldots \dot q_f $ are generalized velocities and |
| 145 |
< |
$L$ is the Lagrangian function for the system. |
| 146 |
< |
|
| 151 |
< |
Differentiating Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, |
| 152 |
< |
one can obtain |
| 145 |
> |
$L$ is the Lagrangian function for the system. Differentiating |
| 146 |
> |
Eq.~\ref{introEquation:hamiltonianDefByLagrangian}, one can obtain |
| 147 |
|
\begin{equation} |
| 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 |
| 174 |
|
t}} |
| 175 |
|
\label{introEquation:motionHamiltonianTime} |
| 176 |
|
\end{equation} |
| 183 |
– |
|
| 177 |
|
Eq.~\ref{introEquation:motionHamiltonianCoordinate} and |
| 178 |
|
Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's |
| 179 |
|
equation of motion. Due to their symmetrical formula, they are also |
| 281 |
|
statistical characteristics. As a function of macroscopic |
| 282 |
|
parameters, such as temperature \textit{etc}, the partition function |
| 283 |
|
can be used to describe the statistical properties of a system in |
| 284 |
< |
thermodynamic equilibrium. |
| 285 |
< |
|
| 286 |
< |
As an ensemble of systems, each of which is known to be thermally |
| 294 |
< |
isolated and conserve energy, the Microcanonical ensemble (NVE) has |
| 295 |
< |
a partition function like, |
| 284 |
> |
thermodynamic equilibrium. As an ensemble of systems, each of which |
| 285 |
> |
is known to be thermally isolated and conserve energy, the |
| 286 |
> |
Microcanonical ensemble (NVE) has a partition function like, |
| 287 |
|
\begin{equation} |
| 288 |
|
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
| 289 |
|
\end{equation} |
| 578 |
|
The hidden geometric properties\cite{Budd1999, Marsden1998} of an |
| 579 |
|
ODE and its flow play important roles in numerical studies. Many of |
| 580 |
|
them can be found in systems which occur naturally in applications. |
| 590 |
– |
|
| 581 |
|
Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is |
| 582 |
|
a \emph{symplectic} flow if it satisfies, |
| 583 |
|
\begin{equation} |
| 591 |
|
\begin{equation} |
| 592 |
|
{\varphi '}^T J \varphi ' = J \circ \varphi |
| 593 |
|
\end{equation} |
| 594 |
< |
is the property that must be preserved by the integrator. |
| 595 |
< |
|
| 596 |
< |
It is possible to construct a \emph{volume-preserving} flow for a |
| 597 |
< |
source free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ |
| 598 |
< |
\det d\varphi = 1$. One can show easily that a symplectic flow will |
| 609 |
< |
be volume-preserving. |
| 610 |
< |
|
| 611 |
< |
Changing the variables $y = h(x)$ in an ODE |
| 594 |
> |
is the property that must be preserved by the integrator. It is |
| 595 |
> |
possible to construct a \emph{volume-preserving} flow for a source |
| 596 |
> |
free ODE ($ \nabla \cdot f = 0 $), if the flow satisfies $ \det |
| 597 |
> |
d\varphi = 1$. One can show easily that a symplectic flow will be |
| 598 |
> |
volume-preserving. Changing the variables $y = h(x)$ in an ODE |
| 599 |
|
(Eq.~\ref{introEquation:ODE}) will result in a new system, |
| 600 |
|
\[ |
| 601 |
|
\dot y = \tilde f(y) = ((dh \cdot f)h^{ - 1} )(y). |
| 602 |
|
\] |
| 603 |
|
The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$. |
| 604 |
|
In other words, the flow of this vector field is reversible if and |
| 605 |
< |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. |
| 606 |
< |
|
| 620 |
< |
A \emph{first integral}, or conserved quantity of a general |
| 605 |
> |
only if $ h \circ \varphi ^{ - 1} = \varphi \circ h $. A |
| 606 |
> |
\emph{first integral}, or conserved quantity of a general |
| 607 |
|
differential function is a function $ G:R^{2d} \to R^d $ which is |
| 608 |
|
constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ , |
| 609 |
|
\[ |
| 616 |
|
which is the condition for conserving \emph{first integral}. For a |
| 617 |
|
canonical Hamiltonian system, the time evolution of an arbitrary |
| 618 |
|
smooth function $G$ is given by, |
| 633 |
– |
|
| 619 |
|
\begin{eqnarray} |
| 620 |
|
\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\ |
| 621 |
|
& = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\ |
| 622 |
|
\label{introEquation:firstIntegral1} |
| 623 |
|
\end{eqnarray} |
| 639 |
– |
|
| 640 |
– |
|
| 624 |
|
Using poisson bracket notion, Equation |
| 625 |
|
\ref{introEquation:firstIntegral1} can be rewritten as |
| 626 |
|
\[ |
| 633 |
|
\] |
| 634 |
|
As well known, the Hamiltonian (or energy) H of a Hamiltonian system |
| 635 |
|
is a \emph{first integral}, which is due to the fact $\{ H,H\} = |
| 636 |
< |
0$. |
| 654 |
< |
|
| 655 |
< |
When designing any numerical methods, one should always try to |
| 636 |
> |
0$. When designing any numerical methods, one should always try to |
| 637 |
|
preserve the structural properties of the original ODE and its flow. |
| 638 |
|
|
| 639 |
|
\subsection{\label{introSection:constructionSymplectic}Construction of Symplectic Methods} |
| 674 |
|
\label{introEquation:FlowDecomposition} |
| 675 |
|
\end{equation} |
| 676 |
|
where each of the sub-flow is chosen such that each represent a |
| 677 |
< |
simpler integration of the system. |
| 678 |
< |
|
| 698 |
< |
Suppose that a Hamiltonian system takes the form, |
| 677 |
> |
simpler integration of the system. Suppose that a Hamiltonian system |
| 678 |
> |
takes the form, |
| 679 |
|
\[ |
| 680 |
|
H = H_1 + H_2. |
| 681 |
|
\] |
