数学代写|常微分方程代写ordinary differential equation代考|Some Examples of Nonlinear Systems

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数学代写|常微分方程代写ordinary differential equation代考|Some Examples of Nonlinear Systems

We now give several examples of systems which are described by some rather well-known differential equations which are not necessarily linear equations, as were the preceding cases. To simplify our discussion and to limit it to a manageable scope, we concentrate on second order differential equations of the form
$$\frac{d x^2}{d t^2}+p\left(t, x, x^{\prime}\right)=q(t), \quad t \geq 0,$$
where $x(0)$ and $x^{\prime}(0)$ are specified, and where the functions $p(\cdot)$ and $q(\cdot)$ are specified. If we let $x_1=x$ and $x_1^{\prime}=x_2$, then Eq. (2.15) can equivalently be represented by
$$\left[\begin{array}{l} x_1^{\prime} \ x_2^{\prime} \end{array}\right]=\left[\begin{array}{c} x_2 \ -p\left(t, x_1, x_2\right) \end{array}\right]+\left[\begin{array}{c} 0 \ q(t) \end{array}\right]$$
with $\left[x_1(0) x_2(0)\right]^{\top}=\left[x(0) x^{\prime}(0)\right]^{\top}$.
Example 2.5. An important special case of (2.15) is the Lienard equation given by
$$\frac{d^2 x}{d t^2}+f(x) \frac{d x}{d t}+g(x)=0,$$

where $f: R \rightarrow R$ and $g: R \rightarrow R$ are continuously differentiable functions with $f(x) \geq 0$ for all $x \in R$ and with $x g(x)>0$ for all $x \neq 0$. This equation can be used to represent, for example, $R L C$ circuits with nonlinear circuit elements $(R, L, C)$

数学代写|常微分方程代写ordinary differential equation代考|Hamiltonian Systems

Conservative dynamical systems are those systems which contain no energy dissipating elements. Such systems, with $n$ degrees of freedom, can be characterized by means of a Hamiltonian function $H(p, q)$, where $q^{\mathrm{T}}=$ $\left(q_1, \ldots, q_n\right)$ denotes $n$ generalized position coordinates and $p^{\mathrm{T}}=\left(p_1, \ldots, p_n\right)$ denotes $n$ generalized momentum coordinates. We assume $H(p, q)$ is of the form
$$H(p, q)=T\left(q, q^{\prime}\right)+W(q),$$
where $T$ denotes the kinetic energy and $W$ denotes the potential energy of the system. These energy terms are obtained from the path independent line integrals
\begin{aligned} T\left(q, q^{\prime}\right) & =\int_0^{q^{\top}} p\left(q, \xi^{\top}\right)^{\top} d \xi=\int_0^{q^{\prime}} \sum_{i=1}^n p_i\left(q, \xi^{\xi}\right) d \xi_i, \ W(q) & =\int_0^q f(\eta)^{\top} d \eta=\int_0^4 \sum_{i=1}^n f_i(\eta) d \eta_i, \end{aligned}
where $f_i, i=1, \ldots, n$, denote generalized potential forces.

In order that the integral in (2.34) be path independent, it is necessary and sufficient that
$$\frac{\partial p_i\left(q, q^{\prime}\right)}{\partial q_j^{\prime}}=\frac{\partial p_j\left(q, q^{\prime}\right)}{\partial q_i^{\prime}}, \quad i, j=1, \ldots, n .$$
Conservative dynamical systems are described by the system of $2 n$ ordinary differential equations
$$\begin{array}{ll} q_i^{\prime}=\frac{\partial H}{\partial p_i}(p, q), & i=1, \ldots, n \ p_i^{\prime}=-\frac{\partial H}{\partial q_i}(p, q), & i=1, \ldots, n . \end{array}$$

常微分方程代写

数学代写|常微分方程代写ordinary differential equation代考|Some Examples of Nonlinear Systems

$$\frac{d x^2}{d t^2}+p\left(t, x, x^{\prime}\right)=q(t), \quad t \geq 0,$$

$$\left[\begin{array}{l} x_1^{\prime} \ x_2^{\prime} \end{array}\right]=\left[\begin{array}{c} x_2 \ -p\left(t, x_1, x_2\right) \end{array}\right]+\left[\begin{array}{c} 0 \ q(t) \end{array}\right]$$

$$\frac{d^2 x}{d t^2}+f(x) \frac{d x}{d t}+g(x)=0,$$

有限元方法代写

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