- #1
spacetimedude
- 88
- 1
Homework Statement
A point like particle of mass m moves under gravity along a cycloid given in parametric form by
$$x=R(\phi+\sin\phi),$$
$$y=R(1-cos\phi),$$
where R is the radius of the circle generating the cycloid and ##\phi## is the parameter (angle). The particle is released at the point (##x=\pi R, y=2R##) from rest.
1) What types of constraint apply to this system and how many degrees of freedom are needed to describe the motion?
2) Show that the lagrangian of the system is
$$L=2mR^2\dot{\phi}\cos^2\frac{\phi}{2}-2mgR\sin^2\frac{\phi}{2}$$
3) Introduce a new generalised coordinate ##s=4Rsin\frac{\phi}{2}## and express the Lagrangian in terms of s.
Homework Equations
The Attempt at a Solution
1) I am not sure what the constraints are. It's just that the particle is forced to move in a cycloid. Could someone help me here?
The degrees of freedom is 2 since x and y are dependent on R and ##\phi##. So R and ##\phi## are the variables required.
2) The lagrangian is L=T-V.
Am I correct in thinking ##T=\frac{1}{2}m(\dot x^2+\dot y^2)##? I have tried taking the time derivative of the right hand side of the given equations of x and y. I got $$\dot x=R(\dot \phi+\dot \phi \cos\phi),$$
$$\dot y=R(\dot \phi \sin\phi).$$
Substituting into T, $$T=R^2m\dot \phi ^2 (1+cos\phi).$$
Similarly, $$V=mgy=mgR(1-\cos\phi)$$
Then $$L=R^2m\dot \phi ^2 (1+cos\phi)-mgR(1-\cos\phi)$$
I'm looking through the double angle formulas and can't really determine how to get to the given lagrangian.
3) What to do?
Thanks in advance