Question on general solution to harmonic EoM

In summary, the general solution to harmonic EoM is a mathematical expression that represents all possible solutions to a harmonic equation. It is derived by solving the differential equation that represents the harmonic motion and includes both a particular solution, which satisfies the specific initial conditions of the problem, and a complementary solution, which accounts for all other possible solutions. A general solution can be used to solve any problem involving harmonic motion and is useful in practical applications for predicting future motion, analyzing the effects of changing parameters, and designing systems with specific harmonic behavior.
  • #1
Vitani11
275
3

Homework Statement


An equation of motion for a pendulum:

(-g/L)sinΦ = Φ(double dot)

Homework Equations


L = length
g = gravity
ω = angular velocity
Φο = initial Φ

The Attempt at a Solution


The solution is Φ=Asinωt+Bcosωt

solving for A and B by setting Φ and Φ(dot) equal to zero respectively gives:

Φ= Φοcosωt

My question is how can you just go ahead and write down Φ=Asinωt+Bcosωt? My professor said it was because this was the only function that when differentiated twice give -sinωt but I can't just take that for what it is because I need to know how to arrive there.
 
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  • #2
Vitani11 said:

Homework Statement


An equation of motion for a pendulum:

(-g/L)sinΦ = Φ(double dot)

Homework Equations


L = length
g = gravity
ω = angular velocity
Φο = initial Φ

The Attempt at a Solution


The solution is Φ=Asinωt+Bcosωt

solving for A and B by setting Φ and Φ(dot) equal to zero respectively gives:

Φ= Φοcosωt

My question is how can you just go ahead and write down Φ=Asinωt+Bcosωt? My professor said it was because this was the only function that when differentiated twice give -sinωt but I can't just take that for what it is because I need to know how to arrive there.

Your solution is incorrect: the differential equation ##\ddot{\Phi} = -k \sin(\Phi)## involves Elliptical functions. It is NOT of the form ##\Phi = A \sin \omega t + B \cos \omega t##. See, eg.,
https://en.wikipedia.org/wiki/Pendulum_(mathematics) .

For very small angles ##|\Phi| \ll 1## we have ##\sin(\Phi) \approx \Phi##, so for small angles the solution is of the form you want.

As for why that form applies: you can take several points of view.
(1) Two hundred years ago somebody discovered that form of solution, and we have been taught it ever since.
(2) You can recognize that the equation ##\ddot{x} = k x## has solution ##x = e^{\sqrt{k} t}.## When ##k = -c^2## that gives ##e^{\pm i c t}## and Euler's equation gives that as ##\cos ct \pm i \sin ct.## Taking the real and imaginary parts gives us the ##\sin## and ##\cos## forms.
(3) You can try to solve ##\ddot{x} = -c^2 x## as a power series ##x = a_0 + a_1 t + a_2 t^2 + \cdots## and use standard DE methods to determine the coefficients ##a_0, a_1, a_2, \ldots##. You will find that you get a combination of the series expansions of ##\sin ct## and ##\cos ct.##
(4) You can use the method of Laplace transforms.

There are other ways as well, but realisically, (1) is the easiest explanation. Your professor's explanation is 100% correct (except that both ##\sin## and ##\cos## will work).
 
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  • #3
Good explanation, thank you.
 
  • #4
When you say solve x(double dot) = -c2x as a power series in (3) are you referring to perturbation theory? .
 
  • #5
Vitani11 said:
When you say solve x(double dot) = -c2x as a power series in (3) are you referring to perturbation theory? .

No, it has nothing to do with perturbation theory. Development of series solutions to DEs is one of the topics that gets covered in just about any DE course. Or, you can Google "series solution to differential equation".
 
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FAQ: Question on general solution to harmonic EoM

1. What is a general solution to harmonic EoM?

The general solution to harmonic EoM (equation of motion) is a mathematical expression that represents all possible solutions to a harmonic equation. It includes both the particular solution, which satisfies the specific initial conditions of the problem, and the complementary solution, which accounts for all other possible solutions.

2. How is a general solution to harmonic EoM derived?

A general solution to harmonic EoM is derived by solving the differential equation that represents the harmonic motion. This typically involves using mathematical techniques such as separation of variables or the method of undetermined coefficients.

3. Can a general solution to harmonic EoM be used to solve any harmonic motion problem?

Yes, a general solution to harmonic EoM can be used to solve any problem involving harmonic motion, as long as the initial conditions are known. It is a universal solution that can be applied to a wide range of problems, from simple harmonic oscillators to more complex systems.

4. What is the difference between a general solution and a particular solution to harmonic EoM?

A particular solution to harmonic EoM is a specific solution that satisfies the given initial conditions of a problem. It is unique to that particular problem. On the other hand, a general solution includes all possible solutions to the harmonic equation, including the particular solution and the complementary solution.

5. How is a general solution to harmonic EoM useful in practical applications?

A general solution to harmonic EoM is useful in practical applications because it provides a complete understanding of the behavior of a harmonic system. It can be used to predict future motion, analyze the effects of changing parameters, and design systems that exhibit specific harmonic behavior. It is also a fundamental concept in fields such as physics, engineering, and mathematics.

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