这是一份leeds利兹大学MATH335501作业代写的成功案例
Suppose that the position of a mechanical system with $d$ degrees of freedom is described by $q=\left(q_{1}, \ldots, q_{d}\right)^{T}$ as generalized coordinates (this can be for example Cartesian coordinates, angles, arc lengths along a curve, etc.). Consider the Lagrangian
$$
L=T-U,
$$
where $T=T(q, \dot{q})$ denotes the kinetic energy and $U=U(q)$ the potential energy. The motion of the system is described by Lagrange’s equation ${ }^{2}$
$$
\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}}\right)=\frac{\partial L}{\partial q},
$$
which are just the Euler-Lagrange equations of the variational problem $S(q)=$ $\int_{a}^{b} L(q(t), \dot{q}(t)) \mathrm{d} t \rightarrow \min .$
Hamilton ${ }^{3}$ simplified the structure of Lagrange’s equations and turned them into a form that has remarkable symmetry, by
- introducing Poisson’s variables, the conjugate momenta
$$
p_{k}=\frac{\partial L}{\partial \dot{q}_{k}}(q, \dot{q}) \quad \text { for } k=1, \ldots, d,
$$ - considering the Hamiltonian
$$
H:=p^{T} \dot{q}-L(q, \dot{q})
$$
MATH335501 COURSE NOTES ：
$$
\ddot{q}{1}=-\frac{q{1}}{\left(q_{1}^{2}+q_{2}^{2}\right)^{3 / 2}}, \quad \ddot{q}{2}=-\frac{q{2}}{\left(q_{1}^{2}+q_{2}^{2}\right)^{3 / 2}}
$$
This is equivalent to a Hamiltonian system with the Hamiltonian
$$
H\left(p_{1}, p_{2}, q_{1}, q_{2}\right)=\frac{1}{2}\left(p_{1}^{2}+p_{2}^{2}\right)-\frac{1}{\sqrt{q_{1}^{2}+q_{2}^{2}}}, \quad p_{i}=\dot{q}_{i} .
$$