I Number of integration constants

AI Thread Summary
The discussion centers on the number of integration constants in a system of ordinary differential equations related to mechanics. It highlights the challenge of having two initial conditions while deriving a solution that typically requires three constants. The participants debate whether to set one constant to zero to reconcile the discrepancy, emphasizing that initial conditions for velocity and acceleration are crucial for determining constants. They also explore the implications of integrating the equations and how initial conditions affect the constants in the solutions. Ultimately, the conversation underscores the complexity of deriving a general solution from differential equations with limited initial conditions.
LagrangeEuler
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If we have system of 3 ordinary differential equation in mechanics and we have two initial condition ##\vec{r}(t=0)=0## and ##\vec{v}(t=0)=\vec{v}_0 \vec{i}##. If we somehow get
\frac{d^2v_x}{dt^2}=-\omega^2v_x
then v_x(t)=A\sin(\omega t)+B\cos(\omega t)
Two integration constants and one initial condition for velocity. What to do? Should we put that one constant is equal to zero? So ##A=0##, ##B=v_0##?
 
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From your solution of velocity,
cooridinate:
x=A\frac{1-\cos \omega t}{\omega}+B \frac{\sin \omega t}{\omega}
accerelation:
a=A\omega \cos \omega t - B\omega \sin \omega t
So
A= \frac{a_0}{\omega}
B= v_0
 
Yes, I agree. But suppose that I have only two initial conditions in the beginning. I do not know acceleration. When I integrate ##v_x(t)## I will get
x(t)=C\cos(\omega t)+D\sin (\omega t)+E
so 3 integration constants. And because ##x(t)## satisfy some differential equation of second order, general solution should have only 2 integration constants. And I can put ##D=0##. Is it good reasoning?
 
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My formula of x in #2 which includes initial coordinate condition is different from yours. Please criticize it.
 
I made a typo. From ##v_x(t)=A\sin \omega t+B \cos \omega t## you can write
##\frac{dx}{dt}=A\sin \omega t+B \cos \omega t##
by integrating
##x(t)=-\frac{A}{\omega}\cos \omega t+\frac{B}{\omega}\sin \omega t+E##.
Of course I can say that ##\frac{A}{\omega}=C## and ##\frac{B}{\omega}=D##
Initial condition is ##x(0)=0##. From that
##0=-\frac{A}{\omega}+E##.
I will get the same result as you.
 
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LagrangeEuler said:
anuttarasammyak said:
From your solution of velocity,
cooridinate:
x=A\frac{1-\cos \omega t}{\omega}+B \frac{\sin \omega t}{\omega}
accerelation:
a=A\omega \cos \omega t - B\omega \sin \omega t
So
A= \frac{a_0}{\omega}
B= v_0
I do not think also that this is a general solution. Because by putting ##t=0## first term is always zero, regardless of value of ##A##.
 
LagrangeEuler said:
by integrating
sin and cos are to be exchanged.

I agree with your observation that initial coordinate condition does not decide A nor B. As said in #2 initial velocity and initial acceleration decide them.

Popular oscillation equation for x has two constnts to decide the motion, initial cooridinate and initial velocity. Your oscillation equation for v, which has three times derivative of coordinates thus is not Newton's equation of motion, has two constants to decide the motion, initial velocity and initial acceleration. All one rank up in time derivative.
 
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