First integral of the motion

In summary, the conversation discusses finding a first integral for a given ODE, which involves finding an equation of the form F(\dot{r},r)=\mbox{const.} and its physical significance. The equation can be obtained by multiplying the given equation by \dot r and integrating. The physical problem involves a mass orbiting a hole in a table with a rope passing through it. However, the discussion also brings up the problem of finding a first integral for a simpler system of equations, which may not have a continuous first integral due to the growth of phase volumes over time.
  • #1
quasar987
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I have found the following ODE in the context of a mechanics problem and am now asked to find a first integral of this equation.

[tex](m_1+m_2)\ddot{r}-m_1Cr^{-3}+m_2g=0[/tex]

I know this means that I'm supposed to find an equation of the form [itex]F(\dot{r},r)=\mbox{const.}[/itex] but I don't know how to achieve that.

Am I expected to guess a coordinate transformation whose associated constant of the motion (in the sense of Noether's theorem) is of the form [itex]F(\dot{r},r)=\mbox{const.}[/itex]? Or is there a more direct approach? Certainly the equation cannot be integrated directly because what's [itex]\int r^{-3}dt[/itex]??

Thanks for the help!
 
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  • #2
Multiply the equation by [itex]\dot r[/itex], then integrate.

That "trick" works for almost all the 2nd order dynamics equations that turn up in textbook and exam questions.
 
  • #3
Thanks for the tip! :tongue2:
 
  • #4
I have found the first inetegral of the motion; it is

[tex]\frac{1}{2}(m_1+m_2)\dot{r}^2+\frac{m_1C}{2}r^{-2}+m_2gr=\mbox{const.}[/tex]

and I am now asked to give its physical significance.

My best shot was to remark that the equation above is an equation of energy conservation for a mass [itex]m_1+m_2[/itex] free to move in one dimension and subject to move in a ficticious potential [itex]'V'(r)= \frac{m_1C}{2}r^{-2}+m_2gr[/itex]. But I'm thinking, maybe this is too mathematical and not enough physical?

Otherwise, how can the equation be interpreted physically?
 
  • #5
Mmmh, this potential is dubious too because it won't let r go to zero*... but the actual problem is that of a table with a hole in it and two masses linked by a rope passing through the hole in the table, such that one mass slides w/o friction on the table and the other is hanging in the air below it.

This kinds of baffles physical intuition doesn't it? Instead of the mass spiraling towards the hole, the mass on the table will behave like a kind of planet orbiting the hole... This doesn't make sense; I can't imagine that the mass hanging in the air will ever lift up! :grumpy:*unless C is zero, which corresponds to a null initial angular velocity according to my previous calculations, so this makes sense
 
  • #6
From the description of the physical problem it should be clear what
[tex]\frac{1}{2}(m_1+m_2)\dot{r}^2[/tex]
and
[tex]m_2gr[/tex]
represent (they are both energies).

You didn't give the full question, but if it involves the mass on the table "orbiting" round the hole, shouldn't there be some theta terms in the equations of motion?

OTOH if C = 0, you don't need to explain what a zero term represents!
 
Last edited:
  • #7
Indeed there initially were two coupled equations of motion, but I was able to uncouple them by showing that the second one meant

[tex]\dot{\phi}^2=Cr^{-4}[/tex]

Then inject that in the first to get the equation of post #1.
 
  • #8
So the discussion in this thread is really advanced and here I come with a simple question:

say we have the system

x' = x
y' = y

How do we find a first integral for this?

I know that f(x) = e^x and f(y) = e^y are solutions for this system...

But I have no idea where to start with this..
 
  • #9
well, formally one can divide the two equations

dx/dy=x/y,

so dx/x=dy/y, yielding ln(abs(x/y))=const.

HOWEVER, a first integral should be a continuous function, and the above isn't.
Intuitively, your equations describe a source at x=y=0, and therefore phase volumes grow in time (i.e. there is no conserved quantity that is continuous and is not constant on an open subset of the x-y plane).

Mathador
 

What is a "First Integral of the Motion"?

A "First Integral of the Motion" is a mathematical concept used in classical mechanics to describe a quantity that remains constant as a system evolves over time. It is often referred to as the "conserved quantity" or "conserved quantity of motion".

What is the significance of the First Integral of the Motion?

The First Integral of the Motion is significant because it helps us understand the behavior of a system over time. It allows us to make predictions about the future state of the system based on its current state and the conserved quantity. Additionally, it provides a deeper understanding of the underlying laws and principles that govern the system.

How is the First Integral of the Motion calculated?

The First Integral of the Motion is calculated by finding a function of the system's position and velocity that remains unchanged as the system evolves over time. This function is often represented as a mathematical equation and is derived using the principles of classical mechanics.

What are some examples of First Integrals of the Motion?

Some common examples of First Integrals of the Motion include energy, momentum, and angular momentum. These quantities are conserved in various physical systems, including collisions, pendulums, and planetary motion.

What is the difference between a First Integral of the Motion and a constant of motion?

A First Integral of the Motion is a particular type of constant of motion. While all constants of motion remain unchanged as a system evolves, the First Integral of the Motion is specifically a function of the system's position and velocity. This means that it can be used to make predictions about the future state of the system, while other constants of motion may not have this predictive power.

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