- #1
jleon008
- 1
- 1
- Homework Statement
- A general two-body problem (sun and planet) is given by the Hamiltonian
$$H(p,p_S,q, q_S)=\frac{1}{2M}p^T_S p_S +\frac{1}{2m}p^Tp-\frac{GmM}{\left\| q-q_S \right\| }$$
where ##q_S,q\in \mathbb{R}^3## are the positions of the sun (mass M) and the planet (mass m), ##p_S,p\in \mathbb{R}^3## are their momenta, and G is the gravitational constant.
Prove that in heliocentric coordinates ##Q:= q - q_s##, the equations of motion are
$$\ddot Q=-G(M+m)\frac{Q}{\left\| Q\right\|}$$.
- Relevant Equations
- For computing the motion of two bodies which attract each other, we choose one of the bodies as the centre of our coordinate system; the motion will stay in the plane and we can use two-dimensional coordinates ##q=(q_1,q_2)## for the position of the second body. Newton's Laws, with a suitable normalization, then yield the following differential equations
$$\ddot q_1=-\frac{q_1}{(q^2_1+q^2_2)^\frac{3}{2}}, \ddot q_2=-\frac{q_2}{(q^2_1+q^2_2)^\frac{3}{2}}$$.
This is equivalent to a Hamiltonian system with the Hamiltonian
$$H(p_1,p_2,q_1,q_2)=\frac{1}{2}(p^2_1+p^2_2)-\frac{1}{\sqrt{q^2_1+q^2_2}}, p_i=\dot q_i$$
It seems as though in general, the equations of motion are described with two equations which result from the definition of a Hamiltonian problem, where the problems are of the form:
$$\dot p=-H_q(p,q), \dot q=H_p(p,q)$$
It is a little confusing to me how the equations of motion go from two equations, to one equation involving the second derivative of ##Q##
My main plan of attack for this problem has been to try to transform the original Hamiltonian from ##H(p,p_S,q, q_S)## to ##H(P, Q)## where ##Q=q-q_S## and ##P=\dot Q = \dot q - \dot q_S = p - p_S##
I understand that from the Hamiltonian, I get
$$\dot q = \frac{\partial H}{\partial p} = p$$ and
$$\dot p = - \frac{\partial H}{\partial q} = -\frac{q}{\left\|q\right\|^3}$$
$$\dot p=-H_q(p,q), \dot q=H_p(p,q)$$
It is a little confusing to me how the equations of motion go from two equations, to one equation involving the second derivative of ##Q##
My main plan of attack for this problem has been to try to transform the original Hamiltonian from ##H(p,p_S,q, q_S)## to ##H(P, Q)## where ##Q=q-q_S## and ##P=\dot Q = \dot q - \dot q_S = p - p_S##
I understand that from the Hamiltonian, I get
$$\dot q = \frac{\partial H}{\partial p} = p$$ and
$$\dot p = - \frac{\partial H}{\partial q} = -\frac{q}{\left\|q\right\|^3}$$