Derivation of Kepler's laws- differential equation question

[ddoctor]

Hi group,

Could someone 'remind me' why the equation u" + u = km/L^2 has the solution of the form u(theta) = km/L^2 + C cos(theta - theta(o)).
Any references would be appreciated.
Thanks

Dave

 

[shyboy]

a linear differential equation with non zero additional term has a soulution which is a sum of a general solution of this equation with zero additional term and a particular solution of this equation. To be more clear (I assume that theta is independent variable) the equation

$$u^{''}+u=0$$
has solution
$$u_0=C\times cos(\theta-\theta_0)$$

on the other hand, your equation has a particular solution

$$u_p=km/L^2=const$$
so the general solution is

$$u=u_0+u_p$$

 

[charng]

Hi Dave,

The equation u" + u = km/L^2 is known as the harmonic oscillator equation and it can be derived from Newton's second law for a body moving under the influence of a central force, such as the gravitational force. This equation is also known as the equation of motion for a simple harmonic oscillator.

To solve this differential equation, we can use the method of variation of parameters. This involves assuming a solution of the form u(theta) = A cos(theta) + B sin(theta), where A and B are constants to be determined. Substituting this into the equation, we get:

u" + u = -A cos(theta) - B sin(theta) + A cos(theta) + B sin(theta) = 0

This means that A and B must satisfy the equations A = km/L^2 and B = 0. Therefore, the solution to the differential equation is u(theta) = km/L^2 cos(theta) = km/L^2 + C cos(theta - theta(o)).

This solution is in the form of a superposition of a constant term and a cosine function, which represents the oscillatory behavior of the system. The constant term represents the equilibrium position of the system, while the cosine term represents the oscillations around this equilibrium position.

I hope this helps to clarify the solution to the differential equation. For further references, you can refer to any textbook on classical mechanics or celestial mechanics, which will provide a detailed derivation of Kepler's laws from the harmonic oscillator equation.