Thanks for your insight. I may have misled you into thinking I needed to the differential equations because I asked for the Lagrangian. I'm trying to get the normal mode frequencies by solving the eigenvalue problem.
I was thinking your trick might help still but it seems \theta drops out of...
Homework Statement
One mass m constrained to the x-axis, another mass m constrained to the y-axis. Each mass has a spring connecting it to the origin with elastic constant k and they are connected together by elastic constant c. I.e. we have a right-angle triangle made from the springs with...
T = \frac{1}{2}(M+m)\dot{x}^2 + \frac{1}{2}m(l^2 \dot{\theta}^2 + 2 \dot{x} l \dot{\theta} \cos \theta)
V = \frac{1}{2} k x^2 - m g l \cos \theta
I want to find the normal mode frequencies. How do I handle the \cos \theta factor in the kinetic energy matrix when constructing the eigenvalue...
Homework Statement
A particle of mass m moves in a force field whose potential in spherical coordinates is,
U = \frac{-K \cos \theta}{r^3}
where K is constant.
Identify the two constants of motion of the system.
The Attempt at a Solution
L = T - V = \frac{1}{2} m (\dot{r}^2 + r^2...
Homework Statement
Spring-mass system on a frictionless surface. A pendulum hangs from the mass of the spring-mass system. Write the Lagrangian.
The Attempt at a Solution
Take x as the stretch from equilibrium of the spring and k its elastic constant. M is the mass on the spring.
Take...
Oops I misread the conversion for mdyne. The unit of time is,
>> sqrt(1.66e-27*1e-10/1e-8)
ans =
4.07430975749267e-015
and the resulting wavenumbers,
2788.1297896707 0 0
0 2709.85261638262...
For the unit of time calculation I get
>> (1.66e-27*1e-10/1e-11)^(1/2)
ans =
1.28840987267251e-013
Now, my intention was to convert the \omega values from the eigenvalue problem from [rad/whatever] to [rad/s]. Then I was using the speed of light in [cm/s] to convert these angular...
Here is how I'm setting up my eigenvalue problem:
I know how to construct the G matrix elements.
Here's how I construct the F matrix elements:
V(Q_1, Q_2, Q_3) = \sum_{i,k,j} K_{i,k,j} (Q_1)^i (Q_2)^j (Q_2)^k
K_{i,k,j} can be obtained from this table:
i j k Kijk Units
2 0 0 4.227 mdyne °A−1...
Homework Statement
I'm trying to do some numerical stuff with vibrations of H20 and I'm working in mdyne, angstroms, atomic mass units, and angles are given in radians. What would the corresponding unit of time be when I calculate my normal mode frequencies? femtosecond, 10e-15?
No I mean I get \omega^2_1 = 0,~\omega^2_2 = 3/m l^2, ~\omega^2_3 = 1/ m l^2
From
-.375 m l^2 \omega+.500 m^2 l^4 \omega^2-.125 m^3 l^6 \omega^3=0
I don't really get the idea of the natural units. Is it just convenience, i.e. solve the eigenvalue problem and then convert back?
Homework Statement
Three pendulums hand side-by-side and have there masses connected horizontally via springs. All lengths and masses are equal. Find the Lagrangian and put it in terms of "natural units".
The Attempt at a Solution
T = 1/2 m l^2 (\dot{\theta_1}^2 + \dot{\theta_2}^2 +...