### 数学代写|辛几何代写symplectic geometry代考|MAT9551

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## 数学代写|辛几何代写symplectic geometry代考|Newtonian mechanics

In newtonian mechanics the state of a mechanical system is described by a finite number of real parameters. The set of all possible positions, of a material point for example, is a finite dimensional smooth manifold $M$, called the configuration space. A motion of the system is a smooth curve $\gamma: I \rightarrow M$, where $I \subset \mathbb{R}$ is an open interval. The velocity field of $\gamma$ is smooth curve $\dot{\gamma}: I \rightarrow T M$. The (total space of the) tangent bundle $T M$ of $M$ is called the phase space.

According to Newton, the total force is a vector field $F$ that acts on the points of the configuration space. Locally, a motion is a solution of the second order differential equation $F=m \ddot{x}$, where $m$ is the mass. Equivalently, $\dot{\gamma}$ is locally a solution of the first order differential equation
$$\left(\begin{array}{l} \dot{x} \ \dot{v} \end{array}\right)=\left(\begin{array}{c} v \ \frac{1}{m} F(x, v, t) \end{array}\right) \text {. }$$
Consider a system of $N$ particles in $\mathbb{R}^3$ subject to some forces. If $x_i$ denotes the position of the i-th particle then the configuration space is $\left(\mathbb{R}^3\right)^N$ and Newton’s law of motion is
$$m_i \frac{d^2 x_i}{d t^2}=F_i\left(x_1, \ldots, x_N, \dot{x}_1, \ldots, \dot{x}_N, t\right), \quad 1 \leq i \leq N,$$
where $m_i$ is the mass and $F_i$ is the force on the i-th particle. Relabeling the variables setting $q^{3 i}, q^{3 i+1}$ and $q^{3 i+2}$ the coordinates of $x_i$ in this order, the configuration space becomes $\mathbb{R}^n, n=3 N$, and the equations of motion take the form
$$m_j \frac{d^2 q^j}{d t^2}=F_j\left(q^1, \ldots, q^n, \dot{q}^1, \ldots, \dot{q}^n, t\right), \quad 1 \leq j \leq n$$
Suppose that the forces do not depend on time and are conservative. This means that there is a smooth function $V: \mathbb{R}^n \rightarrow \mathbb{R}$ such that
$$F_j\left(q^1, \ldots, q^n, \dot{q}^1, \ldots, \dot{q}^n\right)=-\frac{\partial V}{\partial q^j}, \quad 1 \leq j \leq n$$

## 数学代写|辛几何代写symplectic geometry代考|Lagrangian mechanics

Let $(M, g, V)$ be a newtonian mechanical system with potential energy $V$ and let $L: T M \rightarrow \mathbb{R}$ be the smooth function $L=T-V \circ \pi$, where $T$ is the kinetic energy and $\pi: T M \rightarrow M$ is the tangent bundle projection.

Theorem 2.1. (d’Alembert-Lagrange) A smooth curve $\gamma: I \rightarrow M$ is a motion of the mechanical system $(M, g, V)$ if and only if
$$\frac{d}{d t}\left(\frac{\partial L}{\partial v^i}(\dot{\gamma}(t))\right)=\frac{\partial L}{\partial x^i}(\dot{\gamma}(t))$$
for every $t \in I$ and $i=1,2 \ldots, n$, where $n$ is the dimension of $M$.
Proof. Suppose that in local coordinates we have $\gamma=\left(x^1, x^2, \ldots, x^n\right)$. Recall that $\gamma$ is a motion of $(M, g, V)$ if and only if
$$\ddot{x}^k=-\sum_{i, j=1}^n \Gamma_{i j}^k \dot{x}^i \dot{x}^j-\sum_{l=1}^n \frac{\partial V}{\partial x^l} g^{l k}$$
Since
$$L(\dot{\gamma})=\frac{1}{2} \sum_{i, j=1}^n g_{i j} \dot{x}^i \dot{x}^j-V(\gamma)$$
for every $i=1,2, \ldots, n$ we have
$$\begin{gathered} \frac{d}{d t}\left(\frac{\partial L}{\partial v^i}(\dot{\gamma}(t))\right)-\frac{\partial L}{\partial x^i}(\dot{\gamma}(t))=\frac{d}{d t}\left(\sum_{j=1}^n g_{i j} \dot{x}^j\right)-\frac{1}{2} \sum_{m, l=1}^n \frac{\partial g_{m l}}{\partial x^i} \dot{x}^m \dot{x}^l+\frac{\partial V}{\partial x^i}(\gamma(t))= \ \sum_{j=1}^n \sum_{l=1}^n \frac{\partial g_{i j}}{\partial x^l} \dot{x}^l \dot{x}^j+\sum_{j=1}^n g_{i j} \ddot{x}^j-\frac{1}{2} \sum_{m, l=1}^n \frac{\partial g_{m l}}{\partial x^i} \dot{x}^m \dot{x}^l+\frac{\partial V}{\partial x^i}(\gamma(t))= \ \sum_{m, l=1}^n\left(\frac{\partial g_{i m}}{\partial x^l}-\frac{1}{2} \frac{\partial g_{m l}}{\partial x^i}\right) \dot{x}^m \dot{x}^l+\sum_{j=1}^n g_{i j} \ddot{x}^j+\frac{\partial V}{\partial x^i}(\gamma(t)) \end{gathered}$$

# 辛几何代写

## 数学代写|辛几何代写symplectic geometry代考|Newtonian mechanics

$$(\dot{x} \dot{v})=\left(v \frac{1}{m} F(x, v, t)\right) .$$

$$m_i \frac{d^2 x_i}{d t^2}=F_i\left(x_1, \ldots, x_N, \dot{x}_1, \ldots, \dot{x}_N, t\right), \quad 1 \leq i \leq N$$

$$m_j \frac{d^2 q^j}{d t^2}=F_j\left(q^1, \ldots, q^n, \dot{q}^1, \ldots, \dot{q}^n, t\right), \quad 1 \leq j \leq n$$

$$F_j\left(q^1, \ldots, q^n, \dot{q}^1, \ldots, \dot{q}^n\right)=-\frac{\partial V}{\partial q^j}, \quad 1 \leq j \leq n$$

## 数学代写|辛几何代写symplectic geometry代考|Lagrangian mechanics

$$\frac{d}{d t}\left(\frac{\partial L}{\partial v^i}(\dot{\gamma}(t))\right)=\frac{\partial L}{\partial x^i}(\dot{\gamma}(t))$$

$$\ddot{x}^k=-\sum_{i, j=1}^n \Gamma_{i j}^k \dot{x}^i \dot{x}^j-\sum_{l=1}^n \frac{\partial V}{\partial x^l} g^{l k}$$

$$L(\dot{\gamma})=\frac{1}{2} \sum_{i, j=1}^n g_{i j} \dot{x}^i \dot{x}^j-V(\gamma)$$

$$\frac{d}{d t}\left(\frac{\partial L}{\partial v^i}(\dot{\gamma}(t))\right)-\frac{\partial L}{\partial x^i}(\dot{\gamma}(t))=\frac{d}{d t}\left(\sum_{j=1}^n g_{i j} \dot{x}^j\right)-\frac{1}{2} \sum_{m, l=1}^n \frac{\partial g_{m l}}{\partial x^i} \dot{x}^m \dot{x}^l+\frac{\partial V}{\partial x^i}(\gamma(t))=\sum_{j=1}^n \sum_{l=1}^n$$

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