How Do You Solve the 3D Non-Isotropic Oscillator Numerically?

[quantumdude]

OK, no problem.

Once again, I haven't read the thread, so I don't know what you did.

Given:
V(x,y,z)=\frac{1}{2}k\left(x^2+4y^2+9z^2\right)
\vec{r}(0)=0\hat{i}+0\hat{j}+0\hat{k}
\vec{v}(0)=\frac{v_0}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})

Try to follow these steps.

1.) Calculate the force \vec{F} from the potential V(x,y,z).
2.) Set \vec{F} equal to m\vec{\ddot{x}} (mass times acceleration).
3.) Separate the vector equation into 3 scalar equations.
4.) Solve the scalar equations.

You should have 3 second order differential equations, which will require 6 pieces of initial data to solve completely. Fortunately, you were given that. Also fortunately, all 3 equations are basically the same.

Try to follow those steps. It shouldn't be too difficult. If you get stuck, post what you've done and where you got stuck.

 

[emptymaximum]

F = -kx\hat{x} -4ky \hat{y} -9kz \hat{z}
recall k = ╥²m, k/m = w² = ╥²
then the accelerations are:
\ddot{x}= -╥²Asin(╥t)
\ddot{y}=-4╥²Bsin(2╥t)
\ddot{z}=-9╥²Csin(3╥t)
my problem is finding the amplitudes of the motion. I know that A = \dot{x_i} / ╥
the problem is is I'm supposed to find x(t),y(t), and z(t) NUMERICALLY.
i'm mostly confused what this solve numerically means.
this is not a computer problem. no mathematica.
how do i find the amplitudes numerically?

 

[quantumdude]

F = -kx\hat{x} -4ky \hat{y} -9kz \hat{z}
recall k = ╥²m, k/m = w² = ╥²
then the accelerations are:
\ddot{x}= -╥²Asin(╥t)
\ddot{y}=-4╥²Bsin(2╥t)
\ddot{z}=-9╥²Csin(3╥t)

How did you get these expressions for the acceleration? They certainly don't come out of the differential equations you were supposed to have set up.

my problem is finding the amplitudes of the motion. I know that A = \dot{x_i} / ╥

Finding the amplitudes is trivial, once you have the general solution. All you have to do is apply the initial values that were given.

the problem is is I'm supposed to find x(t),y(t), and z(t) NUMERICALLY.
i'm mostly confused what this solve numerically means.
this is not a computer problem. no mathematica.
how do i find the amplitudes numerically?

You'll have to ask your teacher what is meant by that. This problem is easily solved by hand. No numerical analysis is required.

 

[emptymaximum]

question directly out of the textbook:
A small lead ball of mass m is suspended by six light springs. The stiffness constants are in the ratio 1:4:9, so that the potential energy function can be expressed as
V(x,y,z)=\frac{1}{2}k\left(x^2+4y^2+9z^2\right)
At time t = 0 the ball receives a push in the (1,1,1) direction that imparts to it a speed v_0 at the origin. If k = ╥²m, numerically find x, y, and z as functions of time t.

The expressions for acceleration come from the general solution and the IC. The general solution reads:
x = Asin(╥t+φ) + Dcos(╥t+φ)

From the IC at t=0 x=0. For the first term on the RHS to vanish φ = 0.
For the second term on the RHS to vanish D = 0. This leaves:
x = Asin(wt) so
\dot{x} = wAcos(wt) and
\ddot{x} = -w²sin(t)
where w = ╥

for y and z, w = 2╥ and 3╥ respectively.

is this wrong?

from what the question says does that mean numerical analysis?
or since w = ╥ i can get numbers for the amplitudes/initial speeds?

 

[quantumdude]

is this wrong?

Ach, no it's not wrong. You just worked it out a few steps more than I had expected you to.

from what the question says does that mean numerical analysis?
or since w = ╥ i can get numbers for the amplitudes/initial speeds?

You certainly can't get a number for the amplitudes, because you were given the initial velocity in terms of v_0. That implies that the symbol v_0 must appear in the amplitudes.

Again, I really have no idea of what is meant by a numerical solution in this case. You'll have to ask your teacher about that. You can however find the amplitudes easily by using the initial conditions. It looks like you've already successfully applied the condition \vec{r}(0)=0\hat{i}+0\hat{j}+0\hat{k} to eliminate the coefficients of the cosines. Now use the condition \vec{v}(0)=v_0(\hat{i}+\hat{j}+\hat{k}) to find the amplitudes of the sines, and you're done.

 

[emptymaximum]

the amplitudes are gotten by maximizing \dot{x}.
this is of course when the cosine term = 1, and this happens at t=0.
at t =0 \dot{x} = \dot{x_0} ; then:

\dot{x_0} = ╥A or A = \dot{x_0}/╥

right?

 

[quantumdude]

Right, but you should use the notation that was given.

Actually, there was a typo in my last post so I'll correct it now.

The initial velocity was given as:

\vec{v}(0)=\dot{\vec{x}}(0)=\frac{v_0}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})

I left out the \sqrt{3} last time. Anyway, the components of that vector are equal to \dot{x}(0), \dot{y}(0), and \dot{z}(0), respectively. Use the given symbol (v_0), and you will have it.

 

[emptymaximum]

thanks for the help