CDS 240, Spring 2016: HW 3

R. Murray, J. Doyle	Issued: 2 May 2016 (Mon) (updated 4 May)
CDS 240, Spring 2016	Due: 11 May 2016 (Thu)

__MATHJAX__

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1. MLS, Problem 4.1
   

   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
   

   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.
   

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

    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.
   

   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 $m$ 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.

WARNING: The problems below are still under development

5. Perko, Section 2.14, problem 1
   (a) 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.
   (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 $x\in E$.

6. Perko, Section 2.14, problem 4. Given the function $U(x)$ in the text, sketch the phase portrait for the Newtonian system with Hamiltonian $H(x,y)=y^2/2+U(x)$
7. 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) $H(x,y)=x^2-y^2$
   
8. 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