CDS 240, Spring 2016: HW 3 - Revision history

Murray at 01:23, 8 May 2016

2016-05-08T01:23:33Z

← Older revision		Revision as of 01:23, 8 May 2016
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	<br>		<br>
	Show that the system		Show that the system
	<center>		<center><math>
	{\|		\begin{align}
	\|-		\dot x &= a_{11}x + a_{12} y + Ax^2 - 2 B x y + C y^2 \\
	~~\| align=left \| <math>~~\dot x = a_{11}x + a_{12} y + Ax^2 - 2 B x y + C y^2~~</math>~~		\dot y &=a_{21}x-a_{11}y+Dx^2-2Axy+By^2
	\|-		\end{align}
	~~\| align=left \| <math>~~\dot y =a_{21}x-a_{11}y+Dx^2-2Axy+By^2 </math>		</math></center>
	\|}		is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function <math>H(x,y)</math> for this system.
	</center>
	is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
	</li>		</li>
	<li>'''MLS, Problem 7.2''' <br>		<li>'''MLS, Problem 7.2''' <br>

Murray at 22:25, 5 May 2016

2016-05-05T22:25:55Z

← Older revision		Revision as of 22:25, 5 May 2016
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	\| issued = 2 May 2016 (Mon) <br> (updated 4 May)		\| issued = 2 May 2016 (Mon) <br> (updated 4 May)
	\| due = 11 May 2016 (Thu)		\| due = 11 May 2016 (Thu)
	}} ~~__MATHJAX__~~		}}

	~~{{warning\|I'm still trying to resolve the mathematics formatting on this page. If you don't read latex, let me know and I'll e-mail you a PDF with the equations -richard}}~~

	~~'''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>		<ol>
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	<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 <math>m</math> 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>		<br clear=both>
	</li>		</li>
	<li> (updated 4 May) Consider a constrained Lagrangian system with		<li> (updated 4 May) Consider a constrained Lagrangian system with
	<center>		<center><math>
	L(q, \dot q) = \frac{1}{2} \dot q^T \dot q, \qquad \omega(q) \dot q = \dot q_3 - q_2 \dot q_1 = 0.		L(q, \dot q) = \frac{1}{2} \dot q^T \dot q, \qquad \omega(q) \dot q = \dot q_3 - q_2 \dot q_1 = 0.
	</center>		</math></center>
	Show that substituting the constraints into the Lagrangian and then applying Lagrange's equations (without constraints) gives the incorrect equations of motion.		Show that substituting the constraints into the Lagrangian and then applying Lagrange's equations (without constraints) gives the incorrect equations of motion.
	</li>		</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.		A planar pendulum (in the x-z plane) of mass <math>m</math> and length <math>\ell</math> hangs from a support point that moves according to <math>x=a\cos (\omega t)</math>. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
	</li>		</li>
	</ol>		</ol>
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	<br>		<br>
	Show that the system		Show that the system
	<center><~~amsmath~~>~~\aligned~~		<center>
	\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>		\| align=left \| <math>\dot x = a_{11}x + a_{12} y + Ax^2 - 2 B x y + C y^2</math>
			\|-
			\| align=left \| <math>\dot y =a_{21}x-a_{11}y+Dx^2-2Axy+By^2 </math>
			\|}
			</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>		</li>
	<li>'''MLS, Problem 7.2''' <br>		<li>'''MLS, Problem 7.2''' <br>
	Show that the differential constraint in $\reals^5$ given by		Show that the differential constraint in <math>\reals^5</math> given by
	<center>		<center><math>
	\begin{~~displaymath~~}		\left[ \begin{matrix} 0& 1 & \rho \sin q_5 & \rho \cos q_3 & \cos q_5 \end{matrix} \right] \dot{q} = 0
	~~\bmatrix~~ 0& 1 & \rho \sin q_5 & \rho \cos q_3 & \cos q_5 \~~endbmatrix~~ \dot{q} = 0		</math></center>
	~~\end{displaymath}~~
	</center>
	is nonholonomic.		is nonholonomic.
	</li>		</li>

Murray at 06:17, 5 May 2016

2016-05-05T06:17:57Z

← Older revision		Revision as of 06:17, 5 May 2016
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	\| due = 11 May 2016 (Thu)		\| due = 11 May 2016 (Thu)
	}} __MATHJAX__		}} __MATHJAX__

			{{warning\|I'm still trying to resolve the mathematics formatting on this page. If you don't read latex, let me know and I'll e-mail you a PDF with the equations -richard}}

	'''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		'''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
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	</li>		</li>
	</ol>		</ol>
	~~{{warning\|The problems below are still under development}}~~
	<ol start=5>		<ol start=5>
	<li>'''Perko, Section 2.14, problem 1'''		<li>'''Perko, Section 2.14, problem 1(a)'''
	<br>		<br>
	~~(a)~~ Show that the system		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.
	~~<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>
	<li> '''~~Perko~~, ~~Section 2.14, problem 4~~.~~'''~~		<li>'''MLS, Problem 7.2''' <br>
	~~Given the function $U(x)$ in the text, sketch the phase portrait for the Newtonian system with Hamiltonian $H(x,y)=y^2/2+U(x)$~~		Show that the differential constraint in $\reals^5$ given by
	~~</li>~~		<center>
	~~<li> '''Perko, Section~~ 2~~.14, problem 5 a,b,c.~~'''		\begin{displaymath}
	<br>		\bmatrix 0& 1 & \rho \sin q_5 & \rho \cos q_3 & \cos q_5 \endbmatrix \dot{q} = 0
	~~For each of~~ the ~~following Hamiltonian functions, sketch the phase portraits for the Hamiltonian system and the gradient system orthogonal to it. Draw both phase portraits on the same phase plane.~~		\end{displaymath}
	~~<br>~~		</center>
	~~(a)~~ $~~H(x,y)=x~~^~~2+2y^2~~$		is nonholonomic.
	<br>
	~~(b) $H(x,y)~~=~~x^2-y^2$~~
	~~<br>~~
	~~<!-- (c) $H(x,y)=y~~\~~sin~~{x}~~$ --~~>
	</li>		</li>
	~~<!--~~		<li>'''MLS, Problem 7.7''' <br>
	<li> '''~~Perko~~, ~~Section 2~~.~~14, problem~~ 7.'''		A firetruck can be modeled as a car with one trailer, with the difference that the trailer is steerable, as shown in the figure below.
	~~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 $\dot x=-\mathrm{grad}V(x)$~~.		<center>[[Image:mls-firetruck.png]]</center>
	-->		The constraints on the system are similar to that of the car in Section 7.3 of MLS, with the difference that back wheels are steerable. Derive the nonlinear control system for a firetruck corresponding to the control inputs for driving the cab and steering both the cab and the trailer, and show that it represents a controllable system.
	~~<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>
	</ol>		</ol>

Murray at 06:02, 5 May 2016

2016-05-05T06:02:25Z

← Older revision		Revision as of 06:02, 5 May 2016
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	</li>		</li>
	<li> (updated 4 May) Consider a constrained Lagrangian system with		<li> (updated 4 May) Consider a constrained Lagrangian system with
	<center~~><math~~>		<center>
	L(q, \dot q) = \frac{1}{2} \dot q^T \dot q, \qquad \omega(q) \dot q = \dot q_3 - q_2 \dot q_1 = 0.		L(q, \dot q) = \frac{1}{2} \dot q^T \dot q, \qquad \omega(q) \dot q = \dot q_3 - q_2 \dot q_1 = 0.
	~~</math>~~</center>		</center>
	Show that substituting the constraints into the Lagrangian and then applying Lagrange's equations (without constraints) gives the incorrect equations of motion.		Show that substituting the constraints into the Lagrangian and then applying Lagrange's equations (without constraints) gives the incorrect equations of motion.
	</li>		</li>

Murray at 05:58, 5 May 2016

2016-05-05T05:58:27Z

← Older revision		Revision as of 05:58, 5 May 2016
Line 25:		Line 25:
	<li> (updated 4 May) Consider a constrained Lagrangian system with		<li> (updated 4 May) Consider a constrained Lagrangian system with
	<center><math>		<center><math>
	L(q, \dot q) = \frac{1}{2} \dot q^T \~~dotq~~, \qquad \omega(q) \dot q = \dot q_3 - q_2 \dot q_1 = 0.		L(q, \dot q) = \frac{1}{2} \dot q^T \dot q, \qquad \omega(q) \dot q = \dot q_3 - q_2 \dot q_1 = 0.
	</math></center>		</math></center>
	Show that substituting the constraints into the Lagrangian and then applying Lagrange's equations (without constraints) gives the incorrect equations of motion.		Show that substituting the constraints into the Lagrangian and then applying Lagrange's equations (without constraints) gives the incorrect equations of motion.

Murray at 05:58, 5 May 2016

2016-05-05T05:58:05Z

← Older revision		Revision as of 05:58, 5 May 2016
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	\| semester = Spring 2016		\| semester = Spring 2016
	\| title = Problem Set #3		\| title = Problem Set #3
	\| issued = 2 May 2016 (Mon)		\| issued = 2 May 2016 (Mon) <br> (updated 4 May)
	\| due = 11 May 2016 (Thu)		\| due = 11 May 2016 (Thu)
	}} __MATHJAX__		}} __MATHJAX__
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	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>		<br clear=both>
	</li>		</li>
	<li>~~'''MLS~~, ~~Problem 6.~~2~~'''~~		<li> (updated 4 May) Consider a constrained Lagrangian system with
	~~For the rolling disk example that we considered in class~~ (~~also example 6~~.~~2 in MLS), show~~ that substituting the constraints ~~on $\dot x$ and $\dot y$~~ into the Lagrangian and then applying Lagrange's equations (without constraints) gives the ~~wrong~~ equations of motion.		<center><math>
			L(q, \dot q) = \frac{1}{2} \dot q^T \dotq, \qquad \omega(q) \dot q = \dot q_3 - q_2 \dot q_1 = 0.
			</math></center>
			Show that substituting the constraints into the Lagrangian and then applying Lagrange's equations (without constraints) gives the incorrect equations of motion.
	</li>		</li>
	<li>		<li>

Murray: Created page with " {{CDS homework | instructor = R. Murray, J. Doyle | course = CDS 240 | semester = Spring 2016 | title = Problem Set #3 | issued = 2 May 2016 (Mon) | due = 11 May 2016 (..."

2016-05-03T01:27:36Z

Created page with " {{CDS homework | instructor = R. Murray, J. Doyle | course = CDS 240 | semester = Spring 2016 | title = Problem Set #3 | issued = 2 May 2016 (Mon) | due = 11 May 2016 (..."

New page

{{CDS homework
| instructor = R. Murray, J. Doyle
| course = CDS 240
| semester = Spring 2016
| title = Problem Set #3
| issued = 2 May 2016 (Mon)
| due = 11 May 2016 (Thu)
}} __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>'''MLS, Problem 4.1'''
[[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.
 
</li>
<li>'''MLS, Problem 6.1'''
[[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.
 
</li>
<li>'''MLS, Problem 6.2'''
For the rolling disk example that we considered in class (also example 6.2 in MLS), show that substituting the constraints on $\dot x$ and $\dot y$ into the Lagrangian and then applying Lagrange's equations (without constraints) gives the wrong equations of motion.
</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>
{{warning|The problems below are still under development}}
<ol start=5>
<li>'''Perko, Section 2.14, problem 1'''
 
(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.
 
(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 4.'''
Given the function $U(x)$ in the text, sketch the phase portrait for the Newtonian system with Hamiltonian $H(x,y)=y^2/2+U(x)$
</li>
<li> '''Perko, Section 2.14, problem 5 a,b,c.'''
 
For each of the following Hamiltonian functions, sketch the phase portraits for the Hamiltonian system and the gradient system orthogonal to it. Draw both phase portraits on the same phase plane.
 
(a) $H(x,y)=x^2+2y^2$
 
(b) $H(x,y)=x^2-y^2$
 

</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>
</ol>