CDS 140b Spring 2014 Homework 1 - Revision history

Murray: Reverted edits by Murray (talk) to last revision by Macmardg

2016-05-03T01:26:29Z

Reverted edits by Murray (talk) to last revision by Macmardg

@@ Line 13: / Line 13: @@
 <ol>
 <li>'''Perko, Section 2.14, problem 1'''
 <br>
@@ Line 62: / Line 43: @@
 <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>
 </ol>

Murray at 00:12, 3 May 2016

2016-05-03T00:12:53Z

← Older revision		Revision as of 00:12, 3 May 2016
Line 16:		Line 16:
	[[Image:pendulum-wire.png\|right]]		[[Image:pendulum-wire.png\|right]]
	Derive the equations of motion for a pendulum on a wire: an idealized planar pendulum whose pivot is free to slide along a horizontal wire. Assume that the top of the pendulum can move freely on the wire.		Derive the equations of motion for a pendulum on a wire: an idealized planar pendulum whose pivot is free to slide along a horizontal wire. Assume that the top of the pendulum can move freely on the wire.
			<br clear=both>
	</li>		</li>
	<li>'''MLS, Problem 6.1'''		<li>'''MLS, Problem 6.1'''
	[[Image:spherical-pendulum.png\|right]]		[[Image:spherical-pendulum.png\|right]]
	Calculate the dynamics of a spherical pendulum using the Lagrange-d'Alambert equations. The system consists of a mass $m$ suspended from a rigid wire that is free to pivot in any direction at its point of attachment (a spherical joint). Choose as your primary coordinates the $xy$ position of the bottom of the pendulum.		Calculate the dynamics of a spherical pendulum using the Lagrange-d'Alambert equations. The system consists of a mass $m$ suspended from a rigid wire that is free to pivot in any direction at its point of attachment (a spherical joint). Choose as your primary coordinates the $xy$ position of the bottom of the pendulum.
			<br clear=both>
	</li>		</li>
	<li>'''MLS, Problem 6.2'''		<li>'''MLS, Problem 6.2'''

Murray at 00:01, 3 May 2016

2016-05-03T00:01:40Z

← Older revision		Revision as of 00:01, 3 May 2016
Line 27:		Line 27:
	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.		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>		</li>
			</ol>
			{{warning\|The problems below are still under development}}
			<ol start=5>
	<li>'''Perko, Section 2.14, problem 1'''		<li>'''Perko, Section 2.14, problem 1'''
	<br>		<br>

Murray at 00:00, 3 May 2016

2016-05-03T00:00:26Z

@@ Line 13: / Line 13: @@
 <ol>
 <li>'''Perko, Section 2.14, problem 1'''
 <br>
@@ Line 43: / Line 57: @@
 <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>
 </ol>

Macmardg at 03:29, 2 April 2014

2014-04-02T03:29:59Z

← Older revision		Revision as of 03:29, 2 April 2014
Line 35:		Line 35:
	(b) $H(x,y)=x^2-y^2$		(b) $H(x,y)=x^2-y^2$
	<br>		<br>
	(c) $H(x,y)=y\sin{x}$		<!-- (c) $H(x,y)=y\sin{x}$ -->
	</li>		</li>
			<!--
	<li> '''Perko, Section 2.14, problem 7.'''		<li> '''Perko, Section 2.14, problem 7.'''
	Show that if $x_0$ is a strict local minimum of $V(x)$ then the function $V(x)-V(x_0)$ is a strict Lyapunov function (i.e., $\dot{V}<0$ for $x\neq 0$) for the gradient system $\dot x=-\mathrm{grad}V(x)$.		Show that if $x_0$ is a strict local minimum of $V(x)$ then the function $V(x)-V(x_0)$ is a strict Lyapunov function (i.e., $\dot{V}<0$ for $x\neq 0$) for the gradient system $\dot x=-\mathrm{grad}V(x)$.
			-->
	<li>'''Perko, Section 2.14, problem 12.'''		<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		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

Murray at 18:56, 31 March 2014

2014-03-31T18:56:55Z

← Older revision		Revision as of 18:56, 31 March 2014
Line 6:		Line 6:
	\| title = Problem Set #1		\| title = Problem Set #1
	\| issued = 2 Apr 2014 (Wed)		\| issued = 2 Apr 2014 (Wed)
	\| due = 9 Apr 2014 (~~Wed~~)		\| due = 10 Apr 2014 (Thu)
	}} __MATHJAX__		}} __MATHJAX__

Macmardg at 22:38, 29 March 2014

2014-03-29T22:38:15Z

← Older revision		Revision as of 22:38, 29 March 2014
Line 42:		Line 42:
	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		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>
	<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> ''Preview for homework 2:''
			<br>
			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>		</li>
	</ol>		</ol>

Macmardg at 22:37, 29 March 2014

2014-03-29T22:37:08Z

@@ Line 23: / Line 23: @@
 <br>
 (b) 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>
 <li> '''Perko, Section 2.14, problem 7.'''

Macmardg: Created page with " {{CDS homework | instructor = R. Murray, D. MacMartin | course = CDS 140b | semester = Spring 2014 | title = Problem Set #1 | issued = 2 Apr 2014 (Wed) | due = 9 Apr 20..."

2014-03-25T22:25:11Z

Created page with " {{CDS homework | instructor = R. Murray, D. MacMartin | course = CDS 140b | semester = Spring 2014 | title = Problem Set #1 | issued = 2 Apr 2014 (Wed) | due = 9 Apr 20..."

New page

{{CDS homework
| instructor = R. Murray, D. MacMartin
| course = CDS 140b
| semester = Spring 2014
| title = Problem Set #1
| issued = 2 Apr 2014 (Wed)
| due = 9 Apr 2014 (Wed)
}} __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'''
<br>
(a) 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.
<br>
(b) 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>
<li> '''Perko, Section 2.14, problem 7.'''
Show that if $x_0$ is a strict local minimum of $V(x)$ then the function $V(x)-V(x_0)$ is a strict Lyapunov function (i.e., $\dot{V}<0$ for $x\neq 0$) for the gradient system $\dot x=-\mathrm{grad}V(x)$.
<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> 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>