# Difference between revisions of "CDS 140b Spring 2014 Homework 2"

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</li> | </li> | ||

<li>'''Perko, Section 3.6, problem 4.''' | <li>'''Perko, Section 3.6, problem 4.''' | ||

+ | </li> | ||

+ | <li> | ||

+ | Consider the stability of the Lagrange points (with some simplifying steps). | ||

+ | With the mass of the sun as $1-\mu$ and planet as $\mu$, then in the rotating coordinate system the Hamiltonian is given by | ||

+ | <center><amsmath> | ||

+ | H=\frac{(p_x+y)^2+(p_y-x)^2}{2}+\Omega(x,y) | ||

+ | </amsmath></center> | ||

+ | Show that the equilibrium points are given by the critical points of the (messy) function $\Omega$. | ||

+ | (This leads to 5 solutions, for L1 through L5, with L1, L2, and L3 collinear, i.e., y=0.) | ||

+ | To explore the linearized dynamics it is sufficient to retain only quadratic terms in $H$ (why?). For the collinear Lagrange points this leads to | ||

+ | <center><amsmath> | ||

+ | H=\frac{(p_x+y)^2+(p_y-x)^2}{2}-ax^2+by^2 | ||

+ | </amsmath></center> | ||

+ | for $a>0$ and $b>0$. E.g., for the L1 point in the Earth-Jupiter system then a=9.892, b=3.446. Describe the linearized dynamics about this Lagrange point; are periodic orbits stable? | ||

</li> | </li> | ||

</ol> | </ol> |

## Revision as of 00:06, 6 April 2014

R. Murray, D. MacMartin | Issued: 9 Feb 2014 (Wed) |

CDS 140b, Spring 2014 | Due: 16 Feb 2014 (Wed) |

**Note:** In the upper left hand corner of the *second* page of your homework set, please put the number of hours that you spent on
this homework set (including reading).

WARNING: UNDER CONSTRUCTION, DO NOT START |

- A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
**Perko, Section 3.3, problem 5.**Show that<amsmath>\aligned \dot x &=y+y(x^2+y^2)\\ \dot y &=x-x(x^2+y^2)

\endaligned</amsmath>is a Hamiltonian system with $4H(x,y)=(x^2+y^2)^2-2(x^2-y^2)$. Show that $dH/dt=0$ along solution curves of this system and therefore that solution curves of this system are given by

<amsmath> (x^2+y^2)^2-2(x^2-y^2)=C

</amsmath>Show that the origin is a saddle for this system and that $(\pm 1,0)$ are centers for this system. (Note the symmetry with respect to the $x$-axis.) Sketch the two homoclinic orbits corresponding to $C=0$ and sketch the phase portrait for this system, noting the occurrence of a compound separatrix cycle.

**Perko, Section 3.6, problem 4.**-
Consider the stability of the Lagrange points (with some simplifying steps).
With the mass of the sun as $1-\mu$ and planet as $\mu$, then in the rotating coordinate system the Hamiltonian is given by
<amsmath> H=\frac{(p_x+y)^2+(p_y-x)^2}{2}+\Omega(x,y)

</amsmath>Show that the equilibrium points are given by the critical points of the (messy) function $\Omega$. (This leads to 5 solutions, for L1 through L5, with L1, L2, and L3 collinear, i.e., y=0.) To explore the linearized dynamics it is sufficient to retain only quadratic terms in $H$ (why?). For the collinear Lagrange points this leads to

<amsmath> H=\frac{(p_x+y)^2+(p_y-x)^2}{2}-ax^2+by^2

</amsmath>for $a>0$ and $b>0$. E.g., for the L1 point in the Earth-Jupiter system then a=9.892, b=3.446. Describe the linearized dynamics about this Lagrange point; are periodic orbits stable?