# Difference between revisions of "CDS 140a Winter 2013 Homework 6"

(Created page with "{{warning|This homework set is still being written. Do not start working on these problems until this banner is removed.}}") |
|||

Line 1: | Line 1: | ||

{{warning|This homework set is still being written. Do not start working on these problems until this banner is removed.}} | {{warning|This homework set is still being written. Do not start working on these problems until this banner is removed.}} | ||

{{CDS homework | |||

| instructor = R. Murray, D. MacMartin | |||

| course = ACM 101/AM 125b/CDS 140a | |||

| semester = Winter 2013 | |||

| title = Problem Set #6 | |||

| issued = 12 Feb 2013 (Tue) | |||

| due = 19 Feb 2013 (Tue) | |||

}} __MATHJAX__ | |||

'''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). | |||

<ol> | |||

<li>'''Perko, Section 2.14, problem 1''' | |||

<ol> | |||

<li> Show that the system | |||

<center><amsmath>\aligned | |||

\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\ | |||

\dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2 | |||

\endaligned</amsmath></center> | |||

is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system. | |||

</li> | |||

<li> Given $f\in C^2(E)$, where $E$ is an open, simply connected subset of $\mathbb R^2$, show that the system $\dot{x}=f(x)$ is a Hamiltonian system on $E$ iff $\nabla\cdot f(x)=0$ for all $x\in E$. | |||

</li> | |||

</ol> | |||

</li> | |||

<li>'''Perko, Section 2.14, problem 12''' | |||

Show that the flow defined by a Hamiltonian system with one degree of freedom is area preserving. '''Hint:''' Cf. Problem 6 in Section 2.3 | |||

</li> | |||

<li>'''Perko, Section 3.3, problem 5''' | |||

Show that | |||

<center><amsmath>\aligned | |||

\dot x &=y+y(x^2+y^2)\\ | |||

\dot y &=x-x(x^2+y^2) | |||

\endaligned</amsmath></center> | |||

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 | |||

<center><amsmath> | |||

(x^2+y^2)^2-2(x^2-y^2)=C | |||

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

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. (You need not comment on the compound separatrix cycle.) | |||

</li> | |||

<li> 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. | |||

</li> | |||

</ol> |

## Revision as of 04:39, 11 February 2013

WARNING: This homework set is still being written. Do not start working on these problems until this banner is removed. |

R. Murray, D. MacMartin | Issued: 12 Feb 2013 (Tue) |

ACM 101/AM 125b/CDS 140a, Winter 2013 | Due: 19 Feb 2013 (Tue) |

__MATHJAX__

**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).

**Perko, Section 2.14, problem 1**- Show that the system
<amsmath>\aligned \dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\ \dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2

\endaligned</amsmath>is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.

- Given $f\in C^2(E)$, where $E$ is an open, simply connected subset of $\mathbb R^2$, show that the system $\dot{x}=f(x)$ is a Hamiltonian system on $E$ iff $\nabla\cdot f(x)=0$ for all $x\in E$.

- Show that the system
**Perko, Section 2.14, problem 12**Show that the flow defined by a Hamiltonian system with one degree of freedom is area preserving.**Hint:**Cf. Problem 6 in Section 2.3**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. (You need not comment on the compound separatrix cycle.)

- 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.