- #1
NeutronStar
- 419
- 1
How would I go about finding a solution to this differential equation?
[tex]r\ddot\theta-g\sin\theta=0[/tex]
Where r and g are constants.
[tex]r\ddot\theta-g\sin\theta=0[/tex]
Where r and g are constants.
Solving 2nd Order ODE: r\ddot\theta-g\sin\theta=0
- Thread starter NeutronStar
- Start date
In summary, you can approximate small angles using sin and cos, and you can use the Jacobi elliptic function to approximate large angles.
- #2
- 2,606
- 800
Numerically, use Runge-Kutta.
- #3
NeutronStar
- 419
- 1
Oh pooh,...
That's what I didn't want to hear!
That's what I didn't want to hear!
- #4
Tide
Science Advisor
Homework Helper
- 3,087
- 0
Did you intend for the second term to have a plus sign?
IIRC, you should be able to reduce the solution to quadratures expressed in terms of elliptic integrals.
IIRC, you should be able to reduce the solution to quadratures expressed in terms of elliptic integrals.
- #5
NeutronStar
- 419
- 1
Yes, it should be plus. Sorry about that.Tide said:
Hey, thanks for the tip about the elliptic integrals! That may be just what I'm looking for!
- #6
Clausius2
Science Advisor
Gold Member
- 1,440
- 7
NeutronStar said:
If you want to solve the equations of your Lagrange Dynamics problems, you could also post it ¡n that thread you wrote. I didn't mention it to you, but the next step after writing the equations is solving them analytically. The usual assumptions made here by phsicists and engineers are to consider small displacements (i.e [tex] \theta\rightarrow 0[/tex]). Then you could remove [tex] sen\theta[/tex] and [tex]cos\theta[/tex] of your equations and made it quasi-linears. Try to go about that, because it is the usual estrategy in Lagrange Dynamic courses.
- #7
Galileo
Science Advisor
Homework Helper
- 1,995
- 7
Such an equation usually appears for oscillating motions.
You can get a reasonably good approximation for small angles, where:
[tex]\sin \theta \approx \theta[/tex]
and
[tex]\cos \theta \approx 1-\frac{1}{2}\theta^2[/tex]
You can get a reasonably good approximation for small angles, where:
[tex]\sin \theta \approx \theta[/tex]
and
[tex]\cos \theta \approx 1-\frac{1}{2}\theta^2[/tex]
- #8
- 13,353
- 3,168
NeutronStar said:
The solution should be sinus amplitudinis,the Jacobi elliptic function.I'm sure of it.
- #9
- 13,353
- 3,168
I was curious as well to learn the solution of the simple pendulum ODE.The best approach i came across is the one in
Derek F.Lawden:"Elliptic Functions and Applications",Springer Verlag,1989,p.114 pp.117.
But the chapters 1 pp.3 (p.1 pp.94) are essential to understanding properly what he's doing when speaking of the simple pendulum.
Derek F.Lawden:"Elliptic Functions and Applications",Springer Verlag,1989,p.114 pp.117.
But the chapters 1 pp.3 (p.1 pp.94) are essential to understanding properly what he's doing when speaking of the simple pendulum.
Related to Solving 2nd Order ODE: r\ddot\theta-g\sin\theta=0
1. What is a second order ODE?
A second order ODE (ordinary differential equation) is a mathematical equation that contains a second derivative of an unknown function. It is commonly used to model physical phenomena and can be solved to find the behavior of the unknown function.
2. What is the meaning of r, θ, and g in the equation?
In this equation, r represents the displacement of a pendulum, θ represents the angle of the pendulum from its resting position, and g represents the acceleration due to gravity.
3. How do I solve a second order ODE?
To solve a second order ODE, you need to use mathematical techniques such as separation of variables, substitution, or undetermined coefficients. Additionally, you may need to apply initial conditions or boundary conditions to get a specific solution.
4. What is the significance of solving this specific second order ODE?
This particular second order ODE is known as the pendulum equation, which is commonly used to model the motion of a pendulum. By solving it, you can determine the behavior of a pendulum and understand factors such as its period, amplitude, and stability.
5. Can this equation be solved analytically or numerically?
Yes, this equation can be solved both analytically and numerically. Analytical solutions involve finding a closed-form expression for the unknown function, while numerical solutions use computational methods to approximate the solution. The method used will depend on the complexity of the equation and the desired level of accuracy.
Similar threads
-
Differential Equations
-
Differential Equations
-
Differential Equations
-
Introductory Physics Homework Help
-
Differential Equations
-
Differential Equations
-
Differential Equations
-
Differential Equations
-
Differential Equations