CDS 140a Winter 2013 Homework 6 - Revision history

Macmardg at 22:53, 12 February 2013

2013-02-12T22:53:54Z

← Older revision		Revision as of 22:53, 12 February 2013
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	<ol>		<ol>
	<li>'''Perko, Section 2.14, problem 1'''		<li>'''Perko, Section 2.14, problem 1'''
	<~~ol type="a"~~>		<br>
	~~<li>~~ Show that the system		(a) Show that the system
	<center><amsmath>\aligned		<center><amsmath>\aligned
	\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\		\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\
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	\endaligned</amsmath></center>		\endaligned</amsmath></center>
	is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.		is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
	<~~/li~~>		<br>
	~~<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 $x\in E$.		(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>~~
	~~</ol>~~
	</li>		</li>
	<li> '''Perko, Section 2.14, problem 7.'''		<li> '''Perko, Section 2.14, problem 7.'''

Macmardg at 22:51, 12 February 2013

2013-02-12T22:51:41Z

← Older revision		Revision as of 22:51, 12 February 2013
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	<ol>		<ol>
	<li>'''Perko, Section 2.14, problem 1'''		<li>'''Perko, Section 2.14, problem 1'''
	<ol>		<ol type="a">
	<li> Show that the system		<li> Show that the system
	<center><amsmath>\aligned		<center><amsmath>\aligned

Macmardg at 22:42, 12 February 2013

2013-02-12T22:42:16Z

← Older revision		Revision as of 22:42, 12 February 2013
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	~~{{warning\|This homework set is still being written. Do not start working on these problems until this banner is removed.}}~~

	{{CDS homework		{{CDS homework
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	</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 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

Macmardg at 04:45, 11 February 2013

2013-02-11T04:45:15Z

← Older revision		Revision as of 04:45, 11 February 2013
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	</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 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 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

Macmardg at 04:44, 11 February 2013

2013-02-11T04:44:17Z

← Older revision		Revision as of 04:44, 11 February 2013
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	</ol>		</ol>
	</li>		</li>
	<li>'''Perko, Section 2.14, problem 12'''		<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 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		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>'''Perko, Section 3.3, problem 5'''		<li>'''Perko, Section 3.3, problem 5.'''
	Show that		Show that
	<center><amsmath>\aligned		<center><amsmath>\aligned

Macmardg at 04:39, 11 February 2013

2013-02-11T04:39:40Z

← Older revision		Revision as of 04:39, 11 February 2013
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	{{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>

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