Mixing
Draw phase diagram of $x'' + V(x) = 0$ conservative system, with $V(x) = omega^2 cos{x} - alpha x$
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 18 hours ago
qcc101
445111
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 18 hours ago
qcc101
445111
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
18 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
18 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
12 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 18 hours ago
qcc101
445111
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 18 hours ago
qcc101
445111
asked 18 hours ago
qcc101
445111
asked 18 hours ago
qcc101
445111
asked 18 hours ago
qcc101
445111
asked 18 hours ago
qcc101
445111
445111
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
18 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
18 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
12 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
18 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
18 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
12 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
18 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
18 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
18 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
18 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
12 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
12 hours ago
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004756%2fdraw-phase-diagram-of-x-vx-0-conservative-system-with-vx-omega%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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
18 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
18 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
12 hours ago