Exercise 1 Choose a set of generalized coordinates that totally specify the mechanical state of the following systems:

Exercise 2 Derive the transformation equations for the double pendulum.
It is , , and 
Exercise 3 Show that .

Exercise 4 Consider a system of particles that experiences an increment on its generalized coordinates. Derive the following expression for the total work done by the force and indicate the physical meaning of .
First note that For it is
with being the generalized force. 
Exercise 5 Show that .
We have and . Hence . Since are linearly independent it is . 
Exercise 6 Derive the lagrangian for a simple pendulum and obtain an equation to describe its motion.
The generalized coordinate for the simple pendulum is and the transformation equations are and . For the kinetic energy it is . for the potential . Hence the Lagrangian is . and . Hence the Euler Lagrange equation is

Exercise 7 Two particles of mass are connected with each other and to two points and by springs with constant factor . The particles are free to slide along the direction of and . Use the EulerLagrange equations to derive the equations of motion of the particles.
The kinetic energy is . The potential energy is . Hence the Lagrangian is . The partial derivatives of the Lagrangian are: And the EulerLagrange equations are: 
Exercise 8 A particle of mass moves subject to a conservative force field. Use cylindrical coordinates to derive:

Exercise 9 A double pendulum oscillates on a vertical plane.
Calculate:

Exercise 10 A particle moves along the plane subject to a central force that is a function of the distance between the particle and the origin.

Exercise 11 A particle describes a one dimensional motion subject to a force
where and are positive constants. Find the lagrangian and the hamiltonian. Compare the hamiltonian with the total energy and discuss energy conservation for this system. Since it follows . For the kinetic energy it is . Hence the lagrangian is . Now . For the Hamiltonian it is . Since the system isn’t conservative. Since it is . 
Exercise 12 Consider two functions of the generalized coordinates and the generalized momenta, and . The Poisson brackets are defined as:
Show the following properties of the Poisson brackets:
