Draw phase diagram of $x'' + V(x) = 0$ conservative system, with $V(x) = omega^2 cos{x} - alpha x$
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I want to draw the phase diagram of $x'' + V(x) = 0$, with $V(x) = omega^2 cos{x} - alpha x$,
with all constants greater than $0$.
I write this second order differential equation as a system in this way:
$$ x' = y $$
$$ y' = omega^2 cos{x} - alpha x $$
Then I linearize the second equation finding:
$$ y = omega^2 x + alpha $$
The equilibrium point is $(-frac{alpha}{omega^2},0)$
I then find the matrix A:begin{bmatrix}
0 & 1 \
omega^2 & 0
end{bmatrix}
The eigenvalues are w and -w, so I find that the equilibrium point is unstable because I have an Eigen value with real part positive.
Is this procedure right?
From now on I am stuck, because I do not know how to draw the phase diagram.
differential-equations
asked 21 hours ago
qcc101
447111
add a comment |
up vote
0
down vote
favorite
I want to draw the phase diagram of $x'' + V(x) = 0$, with $V(x) = omega^2 cos{x} - alpha x$,
with all constants greater than $0$.
I write this second order differential equation as a system in this way:
$$ x' = y $$
$$ y' = omega^2 cos{x} - alpha x $$
Then I linearize the second equation finding:
$$ y = omega^2 x + alpha $$
The equilibrium point is $(-frac{alpha}{omega^2},0)$
I then find the matrix A:begin{bmatrix}
0 & 1 \
omega^2 & 0
end{bmatrix}
The eigenvalues are w and -w, so I find that the equilibrium point is unstable because I have an Eigen value with real part positive.
Is this procedure right?
From now on I am stuck, because I do not know how to draw the phase diagram.
differential-equations
asked 21 hours ago
qcc101
447111
2
Hint Plot level sets of $frac{dot{x}^2}{2} + int_{0}^{x} V(z), dz$. All trajectories lie on level sets of this first integral.
– Evgeny
21 hours ago
From what I understand that is the energy which is constant for a conservative system. However I am not sure how follow up on that.
– qcc101
20 hours ago
How to follow up on what exactly? You can just differentiate this function by $t$ and see for yourself that derivative vanishes, hence this system is conservative. Did you try to plot level sets of the first integral? Try ones that go through equilibria first.
– Evgeny
15 hours ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I want to draw the phase diagram of $x'' + V(x) = 0$, with $V(x) = omega^2 cos{x} - alpha x$,
with all constants greater than $0$.
I write this second order differential equation as a system in this way:
$$ x' = y $$
$$ y' = omega^2 cos{x} - alpha x $$
Then I linearize the second equation finding:
$$ y = omega^2 x + alpha $$
The equilibrium point is $(-frac{alpha}{omega^2},0)$
I then find the matrix A:begin{bmatrix}
0 & 1 \
omega^2 & 0
end{bmatrix}
The eigenvalues are w and -w, so I find that the equilibrium point is unstable because I have an Eigen value with real part positive.
Is this procedure right?
From now on I am stuck, because I do not know how to draw the phase diagram.
differential-equations
asked 21 hours ago
qcc101
447111
I want to draw the phase diagram of $x'' + V(x) = 0$, with $V(x) = omega^2 cos{x} - alpha x$,
with all constants greater than $0$.
I write this second order differential equation as a system in this way:
$$ x' = y $$
$$ y' = omega^2 cos{x} - alpha x $$
Then I linearize the second equation finding:
$$ y = omega^2 x + alpha $$
The equilibrium point is $(-frac{alpha}{omega^2},0)$
I then find the matrix A:begin{bmatrix}
0 & 1 \
omega^2 & 0
end{bmatrix}
The eigenvalues are w and -w, so I find that the equilibrium point is unstable because I have an Eigen value with real part positive.
Is this procedure right?
From now on I am stuck, because I do not know how to draw the phase diagram.
differential-equations
differential-equations
asked 21 hours ago
qcc101
447111
asked 21 hours ago
qcc101
447111
asked 21 hours ago
qcc101
447111
asked 21 hours ago
qcc101
447111
asked 21 hours ago
qcc101
447111
447111
2
Hint Plot level sets of $frac{dot{x}^2}{2} + int_{0}^{x} V(z), dz$. All trajectories lie on level sets of this first integral.
– Evgeny
21 hours ago
From what I understand that is the energy which is constant for a conservative system. However I am not sure how follow up on that.
– qcc101
20 hours ago
How to follow up on what exactly? You can just differentiate this function by $t$ and see for yourself that derivative vanishes, hence this system is conservative. Did you try to plot level sets of the first integral? Try ones that go through equilibria first.
– Evgeny
15 hours ago
add a comment |
2
Hint Plot level sets of $frac{dot{x}^2}{2} + int_{0}^{x} V(z), dz$. All trajectories lie on level sets of this first integral.
– Evgeny
21 hours ago
From what I understand that is the energy which is constant for a conservative system. However I am not sure how follow up on that.
– qcc101
20 hours ago
How to follow up on what exactly? You can just differentiate this function by $t$ and see for yourself that derivative vanishes, hence this system is conservative. Did you try to plot level sets of the first integral? Try ones that go through equilibria first.
– Evgeny
15 hours ago
2
2
Hint Plot level sets of $frac{dot{x}^2}{2} + int_{0}^{x} V(z), dz$. All trajectories lie on level sets of this first integral.
– Evgeny
21 hours ago
Hint Plot level sets of $frac{dot{x}^2}{2} + int_{0}^{x} V(z), dz$. All trajectories lie on level sets of this first integral.
– Evgeny
21 hours ago
From what I understand that is the energy which is constant for a conservative system. However I am not sure how follow up on that.
– qcc101
20 hours ago
From what I understand that is the energy which is constant for a conservative system. However I am not sure how follow up on that.
– qcc101
20 hours ago
How to follow up on what exactly? You can just differentiate this function by $t$ and see for yourself that derivative vanishes, hence this system is conservative. Did you try to plot level sets of the first integral? Try ones that go through equilibria first.
– Evgeny
15 hours ago
How to follow up on what exactly? You can just differentiate this function by $t$ and see for yourself that derivative vanishes, hence this system is conservative. Did you try to plot level sets of the first integral? Try ones that go through equilibria first.
– Evgeny
15 hours ago
add a comment |
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Hint Plot level sets of $frac{dot{x}^2}{2} + int_{0}^{x} V(z), dz$. All trajectories lie on level sets of this first integral.
– Evgeny
21 hours ago
From what I understand that is the energy which is constant for a conservative system. However I am not sure how follow up on that.
– qcc101
20 hours ago
How to follow up on what exactly? You can just differentiate this function by $t$ and see for yourself that derivative vanishes, hence this system is conservative. Did you try to plot level sets of the first integral? Try ones that go through equilibria first.
– Evgeny
15 hours ago