| Version 4 |
Version 3 |
| The Hamilton equations are a formulation of the equations of motion in classical mechanics. |
The Hamilton equations are a formulation of the equations of motion in classical mechanics. |
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| \subsubsection*{Local formulation} |
\subsubsection*{Local formulation} |
| Suppose $U\subseteq \R^n$ is an open set, suppose $I$ is an interval |
Suppose $U\subseteq \R^n$ is an open set, suppose $I$ is an interval |
| (representing time), and $H\colon U\times \R^n\times I\to \R$ |
(representing time), and $H\colon U\times \R^n\times I\to \R$ |
| is a smooth function. Then the equations |
is a smooth function. Then the equations |
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\begin{align}
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\begin{eqnarray}
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| \label{HE1} |
\label{HE1} |
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\dot{q}_j &= \frac{\partial H}{\partial p_j}(q(t),p(t),t), \\
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\dot{q}_j &=& \frac{\partial H}{\partial p_j}(q(t),p(t),t), \\
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| \label{HE2} |
\label{HE2} |
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\dot{p}_j &= -\frac{\partial H}{\partial q_j}(q(t),p(t),t),
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\dot{p}_j &=& -\frac{\partial H}{\partial q_j}(q(t),p(t),t),
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\end{align}
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\end{eqnarray}
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| are the \emph{Hamilton equations} for the curve |
are the \emph{Hamilton equations} for the curve |
| $$ |
$$ |
| (q, p)=(q_1,\ldots, q_n, p_1,\ldots, p_n) \colon I\to U\times \R^n. |
(q, p)=(q_1,\ldots, q_n, p_1,\ldots, p_n) \colon I\to U\times \R^n. |
| $$ |
$$ |
| Such a solution is called a \emph{bicharacteristic}, and $H$ is |
Such a solution is called a \emph{bicharacteristic}, and $H$ is |
| called a \emph{Hamiltonian function}. Here we use classical notation; |
called a \emph{Hamiltonian function}. Here we use classical notation; |
| the $q_i$'s represent the location of the particles, |
the $q_i$'s represent the location of the particles, |
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the $p_i$'s represent the momenta of the particles.
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the $p_i$'s represent the momentum of the particles.
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| \subsubsection*{Global formulation} |
\subsubsection*{Global formulation} |
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Suppose $P$ is a symplectic manifold with symplectic form $\omega$ and that $H\colon P\to \R$
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Suppose $P$ is a symplectic manifold with symplectic form $\omega$ and that $H\colon T^\ast Q\to \R$
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| is a smooth function. Then $X_H$, the Hamiltonian |
is a smooth function. Then $X_H$, the Hamiltonian |
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vector field corresponding to $H$ is determined by
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vector field corresponding to $H$ determined by
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| $$ |
$$ |
| dH=\omega(X_H,\cdot). |
dH=\omega(X_H,\cdot). |
| $$ |
$$ |
| The most common case is when $P$ is the cotangent bundle of a manifold $Q$ |
The most common case is when $P$ is the cotangent bundle of a manifold $Q$ |
| equipped with the canonical symplectic form $\omega=-d\alpha$, |
equipped with the canonical symplectic form $\omega=-d\alpha$, |
| where $\alpha$ is the \PMlinkname{Poincar\'e $1$-form}{Poincare1Form}. (Note that other authors may have different sign convention.) Then Hamilton's equations are the equations for the flow of the vector field $X_H$. Given a system of coordinates $x^1, \ldots x^{2n}$ on the manifold $P$, they can be written as follows: |
where $\alpha$ is the \PMlinkname{Poincar\'e $1$-form}{Poincare1Form}. |
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Then Hamilton's equations are the equations for the flow of the vector field $X_H$. Given a system of coordinates $x^1, \ldots x^{2n}$ on the manifold $P$, they can be written as follows: |
| $$ |
$$ |
| \dot x^i = (X_H)^i (x_1, \ldots x_{2n}, t) |
\dot x^i = (X_H)^i (x_1, \ldots x_{2n}, t) |
| $$ |
$$ |
| The relation with the former definition is that in canonical |
The relation with the former definition is that in canonical |
| local coordinates $(q_i,p_j)$ for $T^\ast Q$, the flow of $X_H$ |
local coordinates $(q_i,p_j)$ for $T^\ast Q$, the flow of $X_H$ |
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is determined by equations \eqref{HE1}-\eqref{HE2}.
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is determined by equations \ref{HE1}-\ref{HE2}.
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Also, the following terminology is frequently encountered --- the manifold $P$ is known as the phase space, the manifold $Q$ is known as the configuration space, and the product $P \times\R$ is known as state space.
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Also, the following terminology is frequently encountered --- the manifold $P$ is known as the phase space, the manifold $Q$ is known as the cnfiguration space, and the product $P \times \mathbb{R}$ is known as state space.
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