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 Nov 19 at 10:24
qcc101
456113
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 Nov 19 at 10:24
qcc101
456113
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
Nov 19 at 10:33
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
Nov 19 at 11:08
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
Nov 19 at 16:44
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 Nov 19 at 10:24
qcc101
456113
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 Nov 19 at 10:24
qcc101
456113
asked Nov 19 at 10:24
qcc101
456113
asked Nov 19 at 10:24
qcc101
456113
asked Nov 19 at 10:24
qcc101
456113
asked Nov 19 at 10:24
qcc101
456113
456113
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
Nov 19 at 10:33
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
Nov 19 at 11:08
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
Nov 19 at 16:44
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
Nov 19 at 10:33
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
Nov 19 at 11:08
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
Nov 19 at 16:44
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
Nov 19 at 10:33
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
Nov 19 at 10:33
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
Nov 19 at 11:08
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
Nov 19 at 11:08
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
Nov 19 at 16:44
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
Nov 19 at 16:44
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
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
Nov 19 at 10:33
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
Nov 19 at 11:08
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
Nov 19 at 16:44