## 数学代写|辛几何代写symplectic geometry代考|MATH257

statistics-lab™ 为您的留学生涯保驾护航 在代写辛几何symplectic geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写辛几何symplectic geometry代写方面经验极为丰富，各种代写辛几何symplectic geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|辛几何代写symplectic geometry代考|The symplectic linear group

Recall that for every $n \in \mathbb{N}$ there exists only one symplectic vector space of dimension $2 n$ (up to symplectic linear isomorphism). So we need only consider $\mathbb{R}^n \times\left(\mathbb{R}^n\right)^$ with the canonical symplectic structure $\omega$ of Example 1.4. If $\langle$,$\rangle denotes the euclidean$ inner product on $\mathbb{R}^{2 n}$, then $\mathbb{R}^n \times\left(\mathbb{R}^n\right)^$ can be identified with $\mathbb{R}^{2 n} \cong \mathbb{R}^n \oplus \mathbb{R}^n$, where the symplectic form is given by the formula
$$\omega\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right)=\left\langle x, y^{\prime}\right\rangle-\left\langle x^{\prime}, y\right\rangle .$$
The complex structure $J: \mathbb{R}^{2 n} \rightarrow \mathbb{R}^{2 n}$ is the orthogonal transformation $J(x, y)=$ $(-y, x)$ for $x, y \in \mathbb{R}^n$. Then, $J^2=-i d$ and
$$\omega\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right)=\left\langle J(x, y),\left(x^{\prime}, y^{\prime}\right)\right\rangle .$$
A linear map $f: \mathbb{R}^{2 n} \rightarrow \mathbb{R}^{2 n}$ with matrix $A$ (with respect to the canonical basis) is symplectic if and only if
$$\langle J v, w\rangle=\omega(v, w)=\omega(A v, A w)=\langle J A v, A w\rangle=\left\langle A^t J A v, w\right\rangle,$$
for every $v, w \in \mathbb{R}^{2 n}$. So, $f$ is symplectic if and only if $A^t J A=J$. The set of symplectic linear maps
$$\operatorname{Sp}(n, \mathbb{R})=\left{A \in \mathbb{R}^{2 n \times 2 n}: A^t J A=J\right}$$
is a Lie group, as a closed subgroup of $G L(2 n, \mathbb{R})$, and is called the symplectic group. To see that $\operatorname{Sp}(n, \mathbb{R})$ is a Lie group in an elementary way which gives directly its Lie algebra, let $F: G L(2 n, \mathbb{R}) \rightarrow \mathfrak{s o}(2 n, \mathbb{R})$ be the smooth map
$$F(A)=A^t J A .$$
Then, $\operatorname{Sp}(n, \mathbb{R})=F^{-1}(J)$ and it suffices to show that $J$ is a regular value of $F$. The derivative of $F$ at $A$ is $F_{* A}(H)=H^t J A+A^t J H, H \in \mathbb{R}^{2 n \times 2 n}$. Let $A \in \operatorname{Sp}(n, \mathbb{R})$ and $B \in \mathfrak{s o}(2 n, \mathbb{R})$. If $H=-\frac{1}{2} A J B$, since $A^t J=J A^{-1}$, then
$$H^t J A+A^t J H–\frac{1}{2}(J B)^t J+J\left(-\frac{1}{2} J B\right)–\frac{1}{2} B^t J^t J-\frac{1}{2} J^2 B-B .$$

## 数学代写|辛几何代写symplectic geometry代考|Symplectic manifolds

A symplectic manifold is a pair $(M, \omega)$, where $M$ is a smooth manifold and $\omega$ is a closed 2-form on $M$ such that $\left(T_p M, \omega_p\right)$ is a symplectic vector space for every $p \in M$. Necessarily then $M$ is even dimensional and if $\operatorname{dim} M=2 n$, then $\frac{1}{n !} \omega^n$ is a volume $2 n$-form on $M$. So $M$ is orientable and $\omega$ determines in this way an orientation. However, not every orientable, even-dumensional, smooth manifold admits a symplectic structure. If $(M, \omega)$ is a compact, symplectic manifold of dimension $2 n$, then $\omega$ defines a real cohomology class $a=[\omega] \in H^2(M ; \mathbb{R})$ and the cohomology class $a^n=a \cup \cdots \cup a \in H^{2 n}(M ; \mathbb{R})$ is represented by $\omega^n=\omega \wedge \cdots \wedge \omega$. So, $a^n \neq 0$ and the symplectic form $\omega$ cannot be exact. It follows that if $M$ is an orientable, compact, smooth manifold such that $H^2(M ; \mathbb{R})={0}$, then $M$ admits no symplectic structure. For example, the $n$-sphere $S^n$ cannot be symplectic for $n>2$, as well as the 4-manifold $S^1 \times S^3$.

A smooth map $f:(M, \omega) \rightarrow\left(M^{\prime}, \omega^{\prime}\right)$ between symplectic manifolds is called symplectic if $f^* \omega^{\prime}=\omega$. If $f$ is also a diffeomorphism, it is called symplectomorphism. In this way symplectic manifolds form a category. The product of two symplectic manifolds $\left(M_1, \omega_1\right)$ and $\left(M_2, \omega_2\right)$ is the symplectic manifold
$$\left(M_1 \times M_2, \pi_1^* \omega_1+\pi_2^* \omega\right),$$
where $\pi_j: M_1 \times M_2 \rightarrow M_j, j=1,2$, are the projections.

# 辛几何代写

## 数学代写|辛几何代写symplectic geometry代考|The symplectic linear group

jdenotestheeuclidean 上的内积 $\mathbb{R}^{2 n}$ ，然峝 $\backslash m$ mathbb {}$^{\wedge} n \backslash$ times $\backslash$ eft $\left(\backslash m a t h b b{R}^{\wedge} n \backslash \text { ight }\right)^{\wedge}$ 可以识别为 $\mathbb{R}^{2 n} \cong \mathbb{R}^n \oplus \mathbb{R}^n$ ，其中辛形式由公式给出
$$\omega\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right)=\left\langle x, y^{\prime}\right\rangle-\left\langle x^{\prime}, y\right\rangle .$$

$$\omega\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right)=\left\langle J(x, y),\left(x^{\prime}, y^{\prime}\right)\right\rangle .$$

$$\langle J v, w\rangle=\omega(v, w)=\omega(A v, A w)=\langle J A v, A w\rangle=\left\langle A^t J A v, w\right\rangle,$$

loperatorname ${S p}(n, \backslash m a t h b b{R})=\backslash l$ left ${A \backslash i n \backslash m a t h b b{R} \wedge{2 n \backslash$ Itimes $2 n}: A \wedge t ~ J A=\backslash \backslash r i g h t}$

$$F(A)=A^t J A .$$

$$H^t J A+A^t J H-\frac{1}{2}(J B)^t J+J\left(-\frac{1}{2} J B\right)-\frac{1}{2} B^t J^t J-\frac{1}{2} J^2 B-B$$

## 数学代写|辛几何代写symplectic geometry代考|Symplectic manifolds

$$\left(M_1 \times M_2, \pi_1^* \omega_1+\pi_2^* \omega\right),$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|辛几何代写symplectic geometry代考|MAT4551

statistics-lab™ 为您的留学生涯保驾护航 在代写辛几何symplectic geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写辛几何symplectic geometry代写方面经验极为丰富，各种代写辛几何symplectic geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|辛几何代写symplectic geometry代考|The equations of Hamilton

A Lagrangian system $(M, L)$ is called hyperregular if the Legendre transformation $\mathcal{L}: T M \rightarrow T^* M$ is a diffeomorphism. For example a newtonian mechanical system with potential energy and the system of example $2.4$ are hyperregular.

Definition 3.1. In a hyperregular Lagrangian system as above, the smooth function $H=E \circ \mathcal{L}^{-1}: T^* M \rightarrow \mathbb{R}$, where $E$ is the energy, is called the Hamiltonian function of the system.

Example 3.2. Let $(M, g, V)$ is a newtonian mechanical system with potential energy. The Legendre transformation gives
$$q^i=x^i, \quad p_i=\frac{\partial L}{\partial v^i}=\sum_{j=1}^n g_{i j} v^j .$$
The inverse Legendre transformation is given by
$$x^i=q^i, \quad v^i=\sum_{j=1}^n g^{i j} p_j .$$
So we have
\begin{aligned} E & =\frac{1}{2} \sum_{i, j=1}^n g_{i j} v^i v^j+V\left(x^1, x^2, \ldots, x^n\right) \ L & =\frac{1}{2} \sum_{i, j=1}^n g_{i j} v^i v^j-V\left(x^1, x^2, \ldots, x^n\right) \end{aligned}
and therefore
$$H=\frac{1}{2} \sum_{i, j=1}^n g^{i j} p_i p_j+V\left(q^1, q^2, \ldots, q^n\right)$$

Theorem 3.3. (Hamilton) Let $(M, L)$ be a hyperregular Lagrangian system on the n-dimensional manifold $M$. A smooth curve $\gamma: I \rightarrow M$ is a Lagrangian motion if and only if the smooth curve $\mathcal{L} \circ \dot{\gamma}: I \rightarrow T^* M$ locally solves the system of differential equations
$$\dot{q}^i=\frac{\partial H}{\partial p_i}, \quad \dot{p}_i=-\frac{\partial H}{\partial q^i}, \quad i=1,2, \ldots, n$$

## 数学代写|辛几何代写symplectic geometry代考|Symplectic linear algebra

A synplectic form on a (real) vector space $V$ of finite dimension is a non-degenerate, skew-symmetric, bilinear form $\omega: V \times V \rightarrow \mathbb{R}$. This means that the map $\tilde{\omega}: V \rightarrow V^*$ defined by $\tilde{\omega}(v)(w)=\omega(v, w)$, for $v, w \in V$, is a linear isomorphism. The pair $(V, \omega)$ is then called a symplectic vector space.

Lemma 1.1. (Cartan) Let $V$ be a vector space of dimension $n$ and $\omega$ be a skewsymmetric, bilinear form on $V$. If $\omega \neq 0$, then the rank of $\tilde{\omega}$ is even. If $\operatorname{dim} \tilde{\omega}(V)=$ $2 k$, there exists a basis $l^1, l^2, \ldots, l^{2 k}$ of $\tilde{\omega}(V)$ such that
$$\omega=\sum_{j=1}^k l^{2 j-1} \wedge l^{2 j}$$
Proof. Let $\left{v_1, v_2, \ldots, v_n\right}$ be a basis of $V$ and $\left{v_1^, v_2^, \ldots, v_n^\right}$ be the corresponding dual basis of $V^$. If $a_{i j}=\omega\left(v_i, v_j\right), i<j$, then
$$\omega=\sum_{i<j} a_{i j} v_i^* \wedge v_j^* .$$
Since $\omega \neq 0$, there are some $1 \leq i<j \leq n$ such that $a_{i j} \neq 0$. We may assume that $a_{12} \neq 0$, changing the numbering if necessary. Let
$$\begin{gathered} l^1=\frac{1}{a_{12}} \tilde{\omega}\left(v_1\right)=v_2^+\frac{1}{a_{12}} \sum_{j=3}^n a_{1 j} v_j^, \ l^2=\tilde{\omega}\left(v_2\right)=-a_{12} v_1^+\sum_{j=3}^n a_{2 j} v_j^ . \end{gathered}$$
The set $\left{l^1, l^2, v_3^, \ldots, v_n^\right}$ is now a new basis of $V^*$. If $\omega_1=\omega-l^1 \wedge l^2$, then
$$\tilde{\omega}1\left(v_1\right)=a{12} l^1-l^1\left(v_1\right) l^2+l^2\left(v_1\right) l^1=a_{12} l^1-0-a_{12} l^1=0,$$

$$\tilde{\omega}_1\left(v_2\right)=l^2-l^1\left(v_2\right) l^2+l^2\left(v_2\right) l^1=l^2-l^2+0=0 .$$
Thus, $\omega_1$ is an element of the subalgebra of the exterior algebra of $V$ generated by $v_3^, \ldots, v_n^$. If $\omega_1=0$, then $\omega=l^1 \wedge l^2$. If $\omega_1 \neq 0$, we repeat the above taking $\omega_1$ in the place of $\omega$. So, inductively, we arrive at the conclusion, since $V$ has finite dimension.

# 辛几何代写

## 数学代写|辛几何代写symplectic geometry代考|The equations of Hamilton

$$q^i=x^i, \quad p_i=\frac{\partial L}{\partial v^i}=\sum_{j=1}^n g_{i j} v^j .$$

$$x^i=q^i, \quad v^i=\sum_{j=1}^n g^{i j} p_j .$$

$$E=\frac{1}{2} \sum_{i, j=1}^n g_{i j} v^i v^j+V\left(x^1, x^2, \ldots, x^n\right) L \quad=\frac{1}{2} \sum_{i, j=1}^n g_{i j} v^i v^j-V\left(x^1, x^2, \ldots, x^n\right)$$

$$H=\frac{1}{2} \sum_{i, j=1}^n g^{i j} p_i p_j+V\left(q^1, q^2, \ldots, q^n\right)$$

$$\dot{q}^i=\frac{\partial H}{\partial p_i}, \quad \dot{p}_i=-\frac{\partial H}{\partial q^i}, \quad i=1,2, \ldots, n$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。