Constant solutions to the Lagrange equations of motion

In summary, the conversation discusses the equations of motion for a particle moving on the surface of an inverted cone, with a given Lagrangian. The question asks for a solution where the radial coordinate and its derivative take on constant values. This leads to the concept of uniform circular motion and the need to verify it using the equations of motion. The differentiation of a constant is also mentioned in relation to this problem.
  • #1
bobred
173
0

Homework Statement


A particle moves on the surface of an inverted cone. The Lagrangian is given by

t%29%5Cdot%7Br%7D%5E%7B2%7D+r%5E%7B2%7D%5Cdot%7B%5Cphi%7D%5E%7B2%7D%20%5Cright%29-mg%5Calpha%20r.png


Show that there is a solution of the equations of motion where
png.latex?r.png
and
png.latex?%5Cdot%7B%5Cphi%7D.png
take constant values
png.latex?r_0.png
and
png.latex?%5COmega.png
respectively

Homework Equations


The equations of motion
png.latex?r.png
and
png.latex?%5Cphi.png
are

tex?%5Cddot%7Br%7D+%5Calpha%5E%7B2%7D%5Cddot%7Br%7D-r%5Cdot%7B%5Cphi%7D%5E%7B2%7D+g%5Calpha%20=0.png
(1)

png.latex?r%5E2%5Cdot%7B%5Cphi%7D=K.png
(2)

So
png.latex?%5Cdot%7B%5Cphi%7D.png
can be eliminated from equation (1)

The Attempt at a Solution


Would I go about this by setting
png.latex?%5Cddot%7Br%7D.png
equal to a constant,
png.latex?r_0.png
, and similarly for
png.latex?%5Cdot%7B%5Cphi%7D.png
?
 
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  • #2
bobred said:
Would I go about this by setting
png.png
equal to a constant,
png.png
, and similarly for
png.png
?

It is not [itex] \ddot{r}, \dot{\phi}[/itex] that you were asked to show provide a solution, but [itex] {r}, \dot{\phi}[/itex]. These requirements define uniform circular motion. The question is thus asking you to use the equations to check that uniform circular motion is a solution.
 
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  • #3
MarcusAgrippa said:
It is not [itex] \ddot{r}, \dot{\phi}[/itex] that you were asked to show provide a solution, but [itex] {r}, \dot{\phi}[/itex]. These requirements define uniform circular motion. The question is thus asking you to use the equations to check that uniform circular motion is a solution.

If ##r(t) = r_0## (a constant), what is ##\ddot{r}(t)##?
 
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  • #4
What do you get when you differentiate a constant?
 
  • #5
Differentiating a constant is zero.
I was thinking about the equations in isolation to each other and not as part of a system. So

[tex]v=r \dot{\phi}=r_0 \Omega[/tex]

I just need to show that the equations of motion lead to this.
 
Last edited:
  • #6
bobred said:
Differentiating a constant is zero.
I was thinking about the equations in isolation to each other and not as part of a system. So

[tex]v=r \dot{\phi}=r_0 \Omega[/tex]

I just need to show that the equations of motion lead to this.

No: they do not "lead to" this; they merely allow this.

There is a big difference, since (due to the lack of friction) a solution with non-constant ##r## and ##\dot{\phi}## never settles down to a limiting orbit of constant ##r## and ##\dot{\phi}##. However, if you start off with the correct initial conditions, your orbit maintains constant ##r## and ##\dot{\phi}## forever. Essentially, that is what you are being asked to verify.
 
  • #7
So setting the acceleration [itex]\ddot{r}=0[/itex] leaves a constant [itex]r[/itex] and [itex]\dot{\phi}[/itex] which allows the uniform motion.
 

1. What are constant solutions to the Lagrange equations of motion?

Constant solutions to the Lagrange equations of motion refer to the solutions where the position and velocity of a system do not change over time. This means that the system is in a state of equilibrium and there is no net force acting on it.

2. How do constant solutions to the Lagrange equations of motion relate to Newton's first law of motion?

Constant solutions to the Lagrange equations of motion are in line with Newton's first law of motion, which states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. In other words, constant solutions imply that there is no external force acting on the system.

3. What is the significance of finding constant solutions to the Lagrange equations of motion?

The presence of constant solutions to the Lagrange equations of motion indicates that the system is in a state of stability and will not change over time. This can be useful in predicting the behavior of a system and understanding the forces acting on it.

4. Can a system have multiple constant solutions to the Lagrange equations of motion?

Yes, a system can have multiple constant solutions to the Lagrange equations of motion. This means that there are multiple states of equilibrium for the system, where the position and velocity remain constant.

5. How are constant solutions to the Lagrange equations of motion different from periodic solutions?

Constant solutions are characterized by a lack of change in the system's position and velocity, while periodic solutions involve repetitive motion of the system over time. In other words, constant solutions represent a static state, while periodic solutions represent a dynamic state.

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