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# 物理代写|理论力学代写Theoretical Mechanics代考|PHYS354 Motion in the Homogeneous Gravitational Field

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## 物理代写|理论力学代写Theoretical Mechanics代考|Motion in the Homogeneous Gravitational Field

According to the above-given program we have to at first formulate the equation of motion. Using (2.48) together with (2.43) and exploiting the equality of inertial and heavy mass we can write:
$$\ddot{\mathbf{r}}=\mathbf{g} ; \quad \mathbf{g}=(0,0,-g) .$$
The mass is eliminated; in the gravitational field all bodies are therefrom equally accelerated. It results in a
uniformly accelerated motion
as we have discussed it already in Sect. 2.1.2. We can directly take the former results $(2.30)$ and $(2.31)$ :
\begin{aligned} &\mathbf{v}(t)=\mathbf{v}\left(t_0\right)+\mathbf{g} \cdot\left(t-t_0\right) \ &\mathbf{r}(t)=\mathbf{r}\left(t_0\right)+\mathbf{v}\left(t_0\right)\left(t-t_0\right)+\frac{1}{2} \mathbf{g} \cdot\left(t-t_0\right)^2 . \end{aligned}

## 物理代写|理论力学代写Theoretical Mechanics代考|Linear Differential Equations

We refer to
$$x^{(n)}(t)=\frac{d^n}{d t^n} x(t)$$
as the $n$-th derivative of the function $x(t)$. A relation which contains one or more derivatives of a given function, where the $n$-th derivative appears as the highest,
$$f\left(x^{(n)}, x^{(n-1)}, \ldots, \dot{x}, x, t\right)=0,$$

is called a differential equation of $n$-th order. The goal is to derive the solution function $x(t)$ from such a relation. The basic dynamical equation of Classical Mechanics (2.43) written in Cartesian coordinates, e.g., has just this shape:
$$m \ddot{x}_i-F_i\left(\dot{x}_1, \dot{x}_2, \dot{x}_3, x_1, x_2, x_3, t\right)=0, \quad i=1,2,3 .$$
This is a coupled system of three differential equations of second order for the three functions $x_1(t), x_2(t), x_3(t)$.

Let us first focus, however, on a general relation of the type (2.95). The central statement is formulated in the following

Theorem 2.3.1 The general solution of a differential equation of $n$-th order (2.95) is an ensemble of solutions
$$x=x\left(t \mid \gamma_1, \gamma_2, \ldots, \gamma_n\right),$$
which depends on $n$ independent parameters $\gamma_1, \gamma_2, \ldots, \gamma_n$. Every set of $\gamma_i$ ‘s which are fixed in advance then leads to a special (particular) solution.

## 物理代写|理论力学代写Theoretical Mechanics代考|Motion in the Homogeneous Gravitational Field

$$\ddot{\mathbf{r}}=\mathbf{g} ; \quad \mathbf{g}=(0,0,-g) .$$

2.1.2. 我们可以直接取前一个结果 $(2.30)$ 和 $(2.31)$ :
$$\mathbf{v}(t)=\mathbf{v}\left(t_0\right)+\mathbf{g} \cdot\left(t-t_0\right) \quad \mathbf{r}(t)=\mathbf{r}\left(t_0\right)+\mathbf{v}\left(t_0\right)\left(t-t_0\right)+\frac{1}{2} \mathbf{g} \cdot\left(t-t_0\right)^2 .$$

## 物理代写|理论力学代写Theoretical Mechanics代考|Linear Differential Equations

$$x^{(n)}(t)=\frac{d^n}{d t^n} x(t)$$

$$f\left(x^{(n)}, x^{(n-1)}, \ldots, \dot{x}, x, t\right)=0,$$

$$m \ddot{x}_i-F_i\left(\dot{x}_1, \dot{x}_2, \dot{x}_3, x_1, x_2, x_3, t\right)=0, \quad i=1,2,3 .$$

$$x=x\left(t \mid \gamma_1, \gamma_2, \ldots, \gamma_n\right),$$

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