# Determining force of constraint

## Homework Statement

Consider a particle moving over the curve ##z=a-bx^2## under the force of gravity. If the particle starts from rest at point ##(0,0)## (I'm guessing it means point ##(0,a)##), tell if the particle ever separates from the curve; if yes, find the point at which it does.

## Homework Equations

$$\frak{L} = T-U$$

$$\frac{\partial\frak{L}}{\partial q_i}-\frac{d}{dt}\left(\frac{\partial\frak{L}}{\partial \dot q_i}\right)+\sum_{k=1}^n \lambda_k\frac{\partial f_k}{\partial q_i} = 0$$

## The Attempt at a Solution

Well, clearly it's a problem suited for Lagrangian mechanics. We have the coordinates to be ##x, z## and the Lagrangian to be

$$\frak{L} = \frac{m}{2}(\dot x^2+\dot z^2)-mgz$$

Then the Lagrange equations are:

$$m\ddot x+2b\lambda x=0$$

$$-mg-m\ddot z-\lambda = 0$$

Then applying the constraint we obtaint the equations:

$$m\ddot x+2b\lambda x=0$$

$$mg+2bm(x\ddot x+\dot x^2) = \lambda$$[/B]

How do I proceed to obtain ##\lambda##? I'm seriously stuck on here, so I'd appreciate if you could tell whether I am on the right track or not, and if yes, how to proceed.

Thanks!

Orodruin
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Homework Helper
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You also need to use the constraint equation, which is the EL equation for ##\lambda##.

Yea, I wasn’t very explicit on the original post. But the constraint equation is

$$f(x,z)=a-z-bx^2=0$$

Using that and plugging them on the EL equations (of the first kind) we arrive at the last two equations I typed

Orodruin
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You have two different differential equations, both involving ##\lambda##. You would typically eliminate ##\lambda## from those and solve the differential equations. Then you can find ##\lambda## by inserting the solution into one of the equations.

However, I suggest that you replace one of your EL equations by a suitable constant of motion.

You have two different differential equations, both involving ##\lambda##. You would typically eliminate ##\lambda## from those and solve the differential equations. Then you can find ##\lambda## by inserting the solution into one of the equations.

However, I suggest that you replace one of your EL equations by a suitable constant of motion.
However nothing is a constant of motion in this problem, right?

Orodruin
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However nothing is a constant of motion in this problem, right?
Why would you think so?

If I recall,##\dot q_i## is a constant of motion if ##\frac{\partial L}{\partial q_i}=0## and that is not the case here.

Orodruin
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If I recall,##\dot q_i## is a constant of motion if ##\frac{\partial L}{\partial q_i}=0## and that is not the case here.
This is only a sufficient condition for a constant of motion to exist, not a necessary one.

Hint: Does your Lagrangian depend explicitly on time?

This is only a sufficient condition for a constant of motion to exist, not a necessary one.

Hint: Does your Lagrangian depend explicitly on time?
It doesn't, but forgive me but I do not understand at all what does that mean for my problem?

Orodruin
Staff Emeritus