CDS 240, Spring 2016: HW 3: Difference between revisions

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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__
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'''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).


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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.
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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.
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{{warning|The problems below are still under development}}
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<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><math>
\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\
\begin{align}
\dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2
  \dot x &= a_{11}x + a_{12} y + Ax^2 - 2 B x y + C y^2 \\
\endaligned</amsmath></center>
  \dot y &=a_{21}x-a_{11}y+Dx^2-2Axy+By^2
is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
\end{align}
<br>
</math></center>
(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 $x\in E$.
is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function <math>H(x,y)</math> for this system.
</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 <math>\reals^5</math> given by
</li>
<center><math>
<li> '''Perko, Section 2.14, problem 5 a,b,c.'''
\left[ \begin{matrix}  0&  1 & \rho \sin q_5 & \rho \cos q_3 & \cos q_5 \end{matrix} \right] \dot{q} = 0
<br>
</math></center>
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.
is nonholonomic.
<br>
(a) $H(x,y)=x^2+2y^2$
<br>
(b) $H(x,y)=x^2-y^2$
<br>
<!-- (c) $H(x,y)=y\sin{x}$ -->
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<!--
<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>

Latest revision as of 01:23, 8 May 2016

R. Murray, J. Doyle Issued: 2 May 2016 (Mon)
(updated 4 May)
CDS 240, Spring 2016 Due: 11 May 2016 (Thu)
  1. MLS, Problem 4.1
    Pendulum-wire.png

    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.

  2. MLS, Problem 6.1
    Spherical-pendulum.png

    Calculate the dynamics of a spherical pendulum using the Lagrange-d'Alambert equations. The system consists of a mass 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.

  3. (updated 4 May) Consider a constrained Lagrangian system with

    Show that substituting the constraints into the Lagrangian and then applying Lagrange's equations (without constraints) gives the incorrect equations of motion.

  4. A planar pendulum (in the x-z plane) of mass and length hangs from a support point that moves according to . Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
  1. Perko, Section 2.14, problem 1(a)
    Show that the system

    is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function for this system.

  2. MLS, Problem 7.2
    Show that the differential constraint in given by

    is nonholonomic.

  3. MLS, Problem 7.7
    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.
    Mls-firetruck.png

    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.

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