Question

This is for mechanics II.
We're working with a particle of mass \(m\) in one-dimensional motion.
The potential energy is given by
\(V(x)=-\frac12kx^2\)
and the force is anti-restoring at
\(F(x)=kx\)
It is also given that we see unstable equilibrium at \(x=0\).
We want to consider the initial conditions
\(\quad\)\(t=0\\x=x_0\\\dot x=0\)

And we want to show that the motion is an "exponential 'runaway'"
\(x(t)=\frac12x_0(e^{\alpha t}+e^{-\alpha t})\)
where \(\alpha=\sqrt{\frac km~~}\)

Answer 1

I know I should use \(F(x)=m\ddot x\) somwhere...

Answer 2

somewhere*

Answer 3

The \(e^{stuff}\) will probably come from integration with \(x\)'s in the denominator.

Answer 4

I looked at integrating \(F(x)\) from \(x_0\) to \(x\) so I could equate it to \(-V(x)\) to see if that would help. But I found that \(x_0=0\), I think because the interval of integration was meaningless.

Answer 5

I think simply just setting the two equal and solving the differential equation should do it. \[m x'' = k x\] I haven't solved it yet.

Answer 6

That might do it, thanks!

Answer 7

I'll look into it..

Answer 8

I'll start by integrating with respect to \(t\) :)

Answer 9

Or not.. I'll see...

Answer 10

So here's what I did to get you started: \[\frac{d^2x}{dt^2}=\frac{k}{m} x\]\[\frac{dv}{dt}=\frac{k}{m} x\] Chain rule here\[\frac{dv}{dx}\frac{dx}{dt}=\frac{k}{m} x\]but wait, dx/dt is v \[\frac{dv}{dx}v=\frac{k}{m} x\] Now you can integrate with respect to x and v separately and solve for your constant of integration by applying your initial conditions. Then continue onwards. You will get a trig substitution and I can help you get through that too and it works out pretty nicely. =)

Answer 11

The answer I found has the line
\(F(x)=-\dfrac{dV(x)}{dx}=kx=m\ddot x=m\dot x\dfrac{d\dot x}{dx}\)
Right now I can't see how the last two expressions are equivalent.

I'm looking at your post now!

Answer 12

Thanks! :)

Answer 13

When I said trig substitution I mean hyperbolic trig function btw haha.

Answer 14

\[\frac{d^2x}{dt^2}=\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}=\frac{dv}{dx}v\] Where along this are you stuck, or are you already past this and it makes sense now?

Answer 15

\(\ddot x-\alpha ^2 x=0\)
simply leads to solution :

\(x(t)=A\;e^{-\alpha x}+B\;e^{+\alpha x}\)

where A and B are determined by initial conditions.

Answer 16

Well...
From
\(\frac{dv}{dx}v=\frac{k}{m} x\)

I put in my variables to make it:
\(\dfrac{d\dot x}{dx}\dot x=\alpha^2x\)

Integrating with respect to \(x\),
\({\large\int_0^\dot x}\dfrac{d\dot x}{dx}\dot x\ dx={\large\int_{x_0}^x}\alpha^2x\ dx\)
\({\large\int_0^\dot x}\dot x\ d\dot x=\alpha^2{\large\int_{x_0}^x}x\ dx\)
\(\frac12\dot x^2=\alpha^2\left(\frac12x^2-\frac12x_0^2\right)\)
\(\frac12\dot x^2=\frac12\alpha^2\left(x^2-x_0^2\right)\)
\(\dot x^2=\alpha^2\left(x^2-x_0^2\right)\)
\(\dfrac{\dot x^2}{\alpha^2}=x^2-x_0^2\)
\(\dfrac{\dot x^2}{\alpha^2}+x_0^2=x^2\)
\(x(t)=\pm\sqrt{\dfrac{\dot x^2}{\alpha^2}+x_0^2~~}\)

Answer 17

@Vincent-Lyon.Fr Using my initial conditions, that should give me the answer. Thanks! I'll see what I can do!

It's the same equation that @Kainui , to which I found a different result than wolframalpha.com...

Answer 18

They should be the same, I think.

Answer 19

I suppose I'm really taking the long way, sorry! Haha. Yes, they're the same though.

Answer 20

You can alternatively use
\(x(t)=C\cosh (\alpha x) + D \sinh (\alpha x)\)

Answer 21

I'd really rather not use the hyperbolic trigonometric functions, haha!!

Answer 22

What you are trying to so is actually a maths problem, proving the nature of the solution of the differential equation.

Usually, the mathematicians input a solution that fits, then simply prove there can be no other one.

Answer 23

If initial velocity is zero, then the hyperbolic functions are simpler.

Answer 24

It's sort of obvious when you look at it actually haha. Since the hyperbolic sine and cosine functions are their own second derivatives, subtracting them from themselves will give you 0.

Here you really should use the hyperbolic trig functions. For instance: \[x(t)=x_0 \frac{e^{\alpha t}+e^{- \alpha t}}{2}=x_0 \cosh( \alpha t)\]

Answer 25

Sorry, I meant :

\(x(t)=C\cosh (\alpha t) + D \sinh (\alpha t)\)

Answer 26

So
\(\ddot x(t)=x(t)=C\cosh (\alpha x) + D \sinh (\alpha x)\)
?

Thanks! I'll look into that later! I mean, it's cool, but I have other problems to do and was never introduced to the hyperbolic trigonometric functions.

I think I'm getting somewhere since I got that general solution
\(x(t)=c_1e^{\alpha t}+c_2e^{-\alpha t}\)
and am using the initial conditions :)

Answer 27

Not exactly:

\(x(t)=C\cosh (\alpha t) + D \sinh (\alpha t)\)
and
\(\ddot x(t)=\alpha ^2 C\cosh (\alpha t) + \alpha ^2 D \sinh (\alpha t)\)

Hence \(\ddot x -\alpha ^2\,x=0\)

Answer 28

Ohhh, okay! Thanks! :)

Answer 29

And I got it with the initial conditions :) Thank you both! :)