How Is the Constraint Function in Equation 727 Used in Lagrangian Mechanics?

[cmmcnamara]

So in my internet readings on Lagrangian mechanics I started researching applications with non-potential and/or non-conservative forces and came across this page:

http://farside.ph.utexas.edu/teaching/336k/Newton/node90.html

This page is fascinating but I'm having a bit of difficulty understanding a piece of the first example. Can some one explain to me the constraint function they came up with? Its labeled as equation 727. I feel like I'm missing something obvious but I just can't figure it out. Thank you!

 

[Studiot]

Isn't this the no slip condition?

That is in words

The distance (χ) down the plane = the angle turned through by the cylinder times the radius

 

[vanhees71]

Precisely: If the cylinder is rolling without slipping the distance, traveled of a fixed point from the initial time, where we set $\phi=0$ is given by the circumference along the cylinder boundary, i.e., it's $a \phi$, where $a$ is the cylinder's radius. The same distance the cylinder's center axis has travelled, i.e., we must have the constraint
$$x=a \phi,$$
where we have chosen $x=0$ as an initial condition. The constraint function is then given (up to a sign and an overall multiplicative constant, which both are irrelevant for the solution of the problem) thus reads in this case as given by Eq. (727).

All the scripts of Fitzpatrick's are just mavelous by the way!

 

[cmmcnamara]

Sigh...I knew it was something obvious, thanks a lot guys!

 

[PJC01]

I would be happy to provide some clarification on the constraint function in equation 727. In Lagrangian mechanics, a constraint function is used to represent any restrictions on the motion of a system. In the example on the website you provided, the system consists of two masses connected by a rigid rod, and the constraint function represents the length of the rod.

Equation 727 is derived from the constraint that the length of the rod must remain constant. This is represented by the equation:

L = sqrt((x2 - x1)^2 + (y2 - y1)^2)

where L is the length of the rod, x1 and y1 are the coordinates of the first mass, and x2 and y2 are the coordinates of the second mass.

To incorporate this constraint into the Lagrangian, we use the method of Lagrange multipliers. This involves introducing a new variable, λ (lambda), and adding it to the Lagrangian as a multiplier. The Lagrangian then becomes:

L' = T - V + λ(L - L0)

where T is the kinetic energy, V is the potential energy, and L0 is the desired length of the rod. This equation is then used to derive the equations of motion for the system.

In summary, the constraint function in equation 727 represents the length of the rod in the system and is used to incorporate the constraint into the Lagrangian. I hope this helps clarify the concept. If you have any further questions, please don't hesitate to ask.