5 pages
There are mainly three ways to describe class mechanics, i.e. Newtonian, Lagrangian and Hamiltonian mechanics. They are all equivalent, but the equations of motion for a given system may appear simpler in one of the approaches than in the others. In some particular fields, such as quantum physics and accelerator physics, Hamiltonian equations have some advantages. Given a function \(H(x,p;t)\) (called Hamiltonian), the equations of motion for a dynamical system are given by Hamilton’s equations:
\[\begin{aligned} %\label{eq:Hamiltonian} \frac{\partial H}{\partial x_i}=- \frac{d p_i(t)}{d t}\\ \frac{\partial H}{\partial p_i}= \frac{d x_i(t)}{d t}\end{aligned}\]
, for all \(i=1,\ldots,d\), where \(x\in \mathbf{R}^n\) is the coordinates vector (hence \(\dot x\) is the velocity vector) and \(p\in \mathbf{R}^n\) is the momentum vector. In simple cases, \(p_i\) is equivalent to the mechanical momentum \(m \frac{d x_i(t)}{d t}\), but this is not always the case. The Hamiltonian plays as the same role in Hamiltonian mechanics as force plays in Newtonian mechanics. It defines a \(2\times{}n\) first oder partial derivative equations instead of second order partial derivative equations defined in Lagrangian mechanics (ref). One import advantage is that Hamiltonian systems are “conservative systems” (ref, Liouville’s Theorem), which keeps conservation of area in phase space. In simple example, Hamiltonian can be written as
\[%label{eq:HamilTotal} H(x,p)=U(x)+K(p)\]
for kinetic energy \(K(p)\) and potential energy \(U(x)\). From this equation, we could find Hamiltonian is very similar to the “total” energy of the system, expressed in terms of the coordinates and conjugate momentum. In fact, this phenomenon ensures Hamiltonian could be extended to more complicated field (ref), which we will not discuss here. In this manuscript, we mainly concentrated on the form (xxx). Back to statistics, we treat
\[\begin{array}{c@{ as }l} U(x)&-loglik(x\mbox{that we wish to sample})+Constant\\ K(p)&-loglik(N(p;0,M))+Constant \end{array}\]
or more straightforward: \[H(x,p)=-logLik(x)-logLik(p), \mbox{assuming }p\sim N_n(0,M)\]
Here \(M>0\) is a “mass matrix” is often chosen as a diagonal matrix. Hence the Hamiltonian we used in this manuscript is reduced to \[\begin{aligned} -\frac{\partial H}{\partial x_i}=\frac{d p_i(t)}{d t}=-\frac {d U(x_i)}{d x_i}\\ \frac{\partial H}{\partial p_i}=\frac{d x_i(t)}{d t}=\frac{d K(p_i)}{d p_i}=[M^{-1}p]_i\end{aligned}\]
Since it choose the direction adaptively, we must verify the ergodecity very carefully. Following theorem ensures this properties. We described the theorem and proved some of them.
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