# Going downhill - what's the constraint force? [closed]

We are going downhill on a path $$y=a(1-x^2),$$ and I need to calculate the constraint force-position function. What I've done is this:

The lagrangian of the system is $$L=\frac{1}{2}(\dot{x}^2+\dot{y}^2)-mgy+\lambda(t)(y-a(1-x^2)),$$ so the 2 equation are $$m\ddot{x}=2a\lambda x$$ and $$m\ddot{y}=-mg+\lambda.$$ Based on the equations, the constraint force is $$F_y=\lambda$$ and $$F_x=2a\lambda x$$, but I can't calculate the value of $$\lambda.$$ How should I do it?

## closed as off-topic by sammy gerbil, Jon Custer, John Rennie, Kyle Kanos, Aaron StevensNov 16 '18 at 13:35

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – sammy gerbil, Jon Custer, John Rennie, Kyle Kanos, Aaron Stevens
If this question can be reworded to fit the rules in the help center, please edit the question.

• Why not just find the normal force? – PiKindOfGuy Nov 7 '18 at 21:00
• Isn't the constraint force just the normal force? – PiKindOfGuy Nov 7 '18 at 21:02
• @PiKindOfGuy I think it is. But how would you calculate it? – J. Doe Nov 7 '18 at 21:03
• You have 2 equations for 3 unknowns. If you take the second time derivative of the constraint equation , you get one more equation to solve your problem – Eli Nov 7 '18 at 23:26
• Hints: 1. First solve for $x$: Energy conservation yields $E/m = (1+ (2ax)^2)\dot{x}^2/2+ga(1-x^2).$ What are the initial conditions? 2. Next solve for $y$ and $\lambda$. – Qmechanic Nov 8 '18 at 8:44