Presentation by Maria Sorokina

Hamiltonian formalism on manifolds with singularities.

  1. Motivation
  2. Definitions
  3. Theory
  4. Examples
  5. Application

Motivation

The configuration spaces of many real mechanical systems appear to be manifolds with singularity. A singularity often indicates that geometry of motion might change at the point. We face conceptual difficulties even describing mechanics of the ideal models. Namely, since the configuration space is not a smooth manifold the whole set of techniques of Hamiltonian mechanics cannot be applied.

Considering a certain algebra whose real spectrum coincides with the above configuration space is the way to solve this problem. The structure of this algebra is completely determined by the geometry of the singularity. For a broad class of singularities, the desired algebra can be described directly as the pullback of the two already known algebras.

To develop Hamiltonian formalism on a manifold one needs to define the Poisson bracket. Algebra $A$ with the induced Poisson bracket is called Poisson manifold. Hamiltonian is the element of algebra $A$ . Hamiltonian system is the triple $(A,\{,\},H)$.

Correspondence between a hamiltonian and a vector field is given by the formula $X_H(f)=\{H,f\}$. It has the following physical meaning: a point of the phase space moves along the field $X_H$. Algebraic approach allows to develop a unified formalism for the classical and quantum mechanics.

Definitions

Let $A$ will always be a commutative, associative algebra with unit over $\R$. All $\R$-algebra homomorphisms $h\colon A_1\to A_2$ are assumed unit-preserving.

Let $A$ be a $\R$-algebra. Then

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a $\R$-homomorphism $\Delta\colon A\to A$ is called a linear differential operator of order $\leq k$ with values in $A$ if for any $a_0,…a_k \in A$ we have
$$(\delta_{a_0}\circ\dots\circ\delta_{a_k})(\Delta)=0,$$ where $\delta_{a_i}(\Delta)=[\Delta, a_i]$.

Let us denote the set of all differential operator of order $\leq k$ acting

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$A$ to $A$ by $\Diff_{k}$.

Embedding of $A$-modules $\Diff_{k-1}(A)\subset\Diff_k(A)$ allows one to define the quotient module $S_k(A)=\frac{\Diff_k(A)}{\Diff_{k-1}(A)},$ which is called the module of symbols of order $k$.

The coset of an operator $\Delta\in\Diff_k(A)$ modulo $\Diff_{k-1}(A)$ will be denoted by $smbl_k\Delta$ and called the symbol of $\Delta$.

Let us define the algebra of symbols for the algebra $A$

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by setting
$$Smbl(A)= \bigoplus_{n=0}^{\infty} S_n(A).$$

For two elements
$$smbl_l(\Delta_1)\in S_l(A), smbl_m(\Delta_2)\in S_m(A)$$
let us set by definition
$$smbl_l(\Delta_1)\cdot smbl_m(\Delta_2):=smbl_{l+m}(\Delta_1\circ\Delta_2)\in S_{l+m}(A).$$ $\Smbl(A)$ is a commutative graded $\R$-algebra.

One can assign to the pair $(smbl_m(\Delta),smbl_l(\nabla))$ the element
$$\{smbl_m(\Delta),smbl_l(\nabla)\} = smbl_{l+m-1}([\Delta,\nabla])\in S_{l+m-1}(A),$$
which is called Poisson bracket.

It satisfies the Jacobi identity, since the commutator of linear differential operators satisfies this identity. Thus, $Smbl(A)$ is a Lie algebra with respect to this operation.

Construction of the algebra

Let us consider algebras $A_1 = C^\infty(M_1),$ $A_2 = C^\infty(M_2)$ and homomorphisms $p_i\colon A_i\to C$, where $C$ is a certain $\R$-algebra. Then consider pullback:
$$\xymatrix{A\ar@{->>}[r]^{\pi_2}\ar@{->>}[d]_{\pi_1} & A_2\ar@{->>}[d]^{p_2}\\ A_1 \ar@{->>}[r]_{p_1}&C}$$
Here $A=A_1 \oplus_C A_2 =\{(a_1,a_2)\in A_1\oplus A_2: p_1(a_1) = p_2(a_2)\}.$

Examples

  1. $A_1=A_2=C^\infty(\R)$, $C=\R$, $p_1(f)=f(0),\,p_2(g)=g(0)$.
  2. $A_1=A_2=C^\infty(\R)$, $C=\R[x]/(x^{k+1}),$
    $$p_i(f)=\sum_{n=0}^k \frac{f^{(n)}(0) \epsilon^n}{n!}.$$

Lemma (Spectrum of the algebra)

Let us consider the following commutative diagram, $\R$-algebras homomorphisms $h_{A_i}$ are defined as compositions $h_{A_i}=h_C \circ p_i$:
$$
\xymatrix{ A\ar@{->>}[r]^{\pi_2}\ar@{->>}[d]_{\pi_1} & A_2\ar@{->>}[d]^{p_2}\\ A_1 \ar@{->>}[r]_{p_1}& C\\ & &R\ar@{<-}[ul]|-{h_C}\ar@{<-}[uul]_{h_{A_2}} \ar@{<-}[ull]^{h_{A_1}} \ar@{<.}[ulul]}
$$

Lemma. Spectrum of algebra $A$ is equal to splice of spectrums of algebras $A_1$ and $A_2$ by spectrum of algebra $C$:
$$\Spec_\R{A}=\frac{\Spec_\R{A_1}\coprod \Spec_\R{A_2}}{\Spec_\R{C}} \label{lem1} \tag{L1} .$$

Construction of differential operator

Let us consider algebra $A$ as a pullback of $A_1$ and $A_2$ over algebra $C$. Assume that for $i=1,2$
$$Closure\Int |\pi_i|(|A_i|) = |\pi_i|(|A_i|).$$
Let us consider $\Delta_i\in \Diff_k(A_i)$, $i=1,2$ and $\Delta\in \Diff(A)$

Theorem. Differential operator over algebra $A$ may be defined as a pair of differential operators over $A_1$ and $A_2$:
$$\begin{equation}
\Diff_{k}(A)=\{(\Delta_1, \Delta_2) \mid p_1\circ\Delta_1\circ\pi_1 = p_2\circ\Delta_2\circ\pi_2 \}. \label{thm1}
\end{equation}
$$

Singularity of zero order in $\R^2$

Let us consider coordinate cross on the plane:
$$\mathbb{K}_{0} = \{(x,y)\subset {\mathbb{R}}^2\mid xy=0\}.$$
Algebra of smooth functions on the cross is given by the formula:
$$ A=C^\infty(\mathbb{K}_{0})={C^{\infty}({\mathbb{R}}^2)|}_{\mathbb{K}_{0}}= \{(f(x),g(y))| f(0)=g(0)\}$$
Applying the algorithm (lemma $\ref{lem1}$ and theorem $\ref{thm1}$, see $\eqref{thm1}$) we get in the above notations
$$A_1=A_2=C^\infty(\R), C=\R , p_1(f)=f(0),\,p_2(g)=g(0).$$
In that case…

Differentional operators:
$$\Diff_{k}(A)=\{(\Delta_1, \Delta_2)\mid a_0(0)=b_0(0),a_i(0)=0=b_i(0)\}.$$
$$\Delta_1 =\sum_{i=0}^k a_i(x){\left(\frac{\partial}{\partial x}\right)}^{i},$$
$$\Delta_2 =\sum_{i=0}^k b_i(y){\left(\frac{\partial}{\partial y}\right)}^{i}.$$

Symbol of order $k$:
$$[\Delta]_{k}=([\Delta_1],[\Delta_2]),$$
$a_0(0)=b_0(0)$ $a_k(0)=0=b_k(0)$ if $k\geq 1$

Poisson bracket:
$$\{[\Delta],[\nabla]\}=(\{[\Delta_1],[\nabla_1]\},\{[\Delta_2],[\nabla_2]\}),$$
$\{[\Delta_1],[\nabla_1]\}=(la_{l_{\Delta}}(x)a’_{n_{\nabla}}(x)-na_{n_{\nabla}}(x)a’_{l_{\Delta}}(x)){\left(\frac{\partial}{\partial x}\right)}^{l+n-1}$
$$\{[\Delta_2],[\nabla_2]\}=(lb_{l_{\Delta}}(y)b’_{n_{\nabla}}(y)-nb_{n_{\nabla}}(y)b’_{l_{\Delta}}(y)){\left(\frac{\partial}{\partial
y}\right)}^{l+n-1}$$
$a_0(0)=b_0(0),$ $a_i(0)=0=b_i(0),$ if $i\geq 1$ $i =l_{\Delta}, n_{\nabla}$

Lemma. Dimension of $|\Smbl(A)|$ is equal to 0 at the singular point and is equal to 1 at other points.

Thank you!

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3 thoughts on “Presentation by Maria Sorokina

  1. Masha Sorokina Post author

    все ок)))

    The polynomial $P$ has integer coefficients and $P(x)=5$ for five different integers $x$.
    Show that there is no integer $x$ such that $-6le P(x)le4$ or $6le P(x)le16$.

  2. Nikita Zinoviev

    Ссылки на формулы тоже работают!
    - см Applying the algorithm (lemma L1 and theorem 8, see (8)) we get in the above notations.
    Только надо либо самим давать визульные названия формулам с помощью tag{L1} (это в дополнение к label{lem1}) – так сделана лемма

    либо включать настройку про автонумерацию (cм http://docs.mathjax.org/en/latest/tex.html)
    MathJax.Hub.Config({
    TeX: { equationNumbers: { autoNumber: “AMS” } }
    });
    и использовать begin{equation} так сделано в теореме, либо включать нумерацию вообще всех формул в двойных долларах c помощью
    equationNumbers: { autoNumber: “all” }
    - я думаю это в итоге выключить, но не знаю, может это удобно.

  3. Masha Sorokina Post author

    $$xymatrix{ Bar@{->>}[r]^{pi_2}ar@{->>}[d]_{pi_1} & A_2ar@{->>}[d]^{p_2}\ C ar@{->>}[r]_{p_1}& C\ & &Rar@{<-}[ul]|-{h_C}ar@{<-}[uul]_{h_{A}} ar@{<-}[ull]^{h_{A}} ar@{<.}[ulul]}$$
    Nikita: Tex работает только в тексте, который заключен в доллары – одинарные или двойные (если это несколько строк, то wordpress может их перемешать с html тегами, это можно исправить, посмотрев в text view, можно и прямо набирать в нем).

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