What Parameters Are Needed for a System to Reach a Saddle Point?

[enot]

Hello, this is my first post here, so if I do problems, please correct me and do not be upset = )
I have one small theoretical and one greater question.
small one first:

Homework Statement

I have a potential energy :
$$W(L)= -\frac{1}{4}k_4(L_x^4+L_y^4+L_z^4)$$
How can describe my potential, replacing L_z with L_x,L_y.
My question is: what the situation should be, that I can replace L_z with L_x, L_y

Homework Equations

$$W(L)= -\frac{1}{4}k_4(L_x^4+L_y^4+L_z^4)$$,
k_4 - constant

The Attempt at a Solution

I thought to write something like :
L_x_tilda^2 + L_y_tilda^2 + L_z_tilda^2 = constant(scalar)
L_x_tilda = (L_x + L_x(0))
L_y_tilda = (L_y + L_y(0))
L_z_tilda = (L_z + L_z(0))
if my special point is L(0) is (0,0,1) then:
L_x^2 + L_y^2 + L_z^2 + 2L_z L_z(0) = 1
1 + 2L_z L_z(0) = 1;
but what does it mean. I do not know. I stuck on this...big one

Homework Statement

I have a potential energy again :
$$W(L)= -\frac{1}{4}k_4(L_x^4+L_y^4+L_z^4)$$
I have a system of diff. eq. that describe movement of a part(now without a z dimension)
$$ \ddot{L_x} = - \omega_{af}^2 L_x - 2 \gamma_{af} \dot{L_x} - j L_y $$.

$$ \ddot{L_y} = - \omega_{af}^2 L_y - 2 \gamma_{af} \dot{L_y} + j L_x $$.
where j - is the current ,
My question is: how can I find the value of a current, and other parameters like \omega_{af} , \gamma_{af} so that my system will go to the saddle point.

Homework Equations

?

The Attempt at a Solution

the idea is here the following I solve the system by naming
L_x = A exp(iwt)
L_y = Bexp(iwt)
then I gather
$$ - A w^2 e^{iwt}= - \omega_{af}^2 A e^{iwt} - 2 \gamma_{af} A iw e^{iwt} - j B e^{iwt} $$

$$ - B w^2 e^{iwt}= - \omega_{af}^2 B e^{iwt} - 2 \gamma_{af} B iw e^{iwt} + j A e^{iwt} $$

then I get the following :
$$ (( w^2 - \omega_{af}^2 - 2 \gamma_{af} iw )^2 + ( j p_z )^2 )B= 0 $$
and I solve such equations:
$$w^2 - \omega_{af}^2 - 2 \gamma_{af} iw = - i j $$
$$w^2 - \omega_{af}^2 - 2 \gamma_{af} iw = i j $$
So for the first eq. I get
$$w = \gamma_{af} i \pm \sqrt{ (- \gamma_{af}^2 + \omega_{af}^2 - i j)} $$
Then I use De Moivre's formula
$$ z^n = |z| ^n \exp(i \phi n)$$
$$\sqrt { |z| } = \sqrt[4] { (- \gamma_{af}^2 + \omega_{af}^2)^2 + ( - j )^2 }$$
if current is small
$$\sqrt { |z| } = \sqrt {- \gamma_{af}^2 + \omega_{af}^2}$$
and
$$\phi = \arctan \frac {b}{a} = \arctan ( \frac { - j p_z} {- \gamma_{af}^2 + \omega_{af}^2} ) $$.
and as $$\omega_{af}, \gamma_{af} > 0$$, $$\gamma_{af}^2 \ll \omega_{af}^2$$ we get:
$$\phi = \arctan ( -\frac { j } { \omega_{af}^2 } ) $$
$$j \ll \omega_{af}^2$$ and $$\tan (\phi) = \frac {-j} {\omega_{af}^2} \approx \phi $$
$$ \exp(\frac {i \phi} {2} ) \approx 1 + \left( \frac {i \phi} {2} \right) + O\left( \left( \frac {i \phi} {2} \right)^2\right) $$
$$ \exp \left( - \frac {i j} {2\omega_{af}^2} \right ) \approx 1 + \left( - \frac {i j} {2\omega_{af}^2} \right) + O\left( \left( - \frac {i j} {2\omega_{af}^2} \right)^2 \right) $$

$$ \sqrt z = \sqrt{ (- \gamma_{af}^2 + \omega_{af}^2 - i j p_z)} = \omega_{af} \left( 1 - \frac {i j} {2\omega_{af}^2} \right)$$

$$w = \pm \omega_{af} + i \left( \gamma_{af} \mp \frac { j } {2\omega_{af}} \right)$$
$$ L_x = A \exp( iwt ) = A \exp \left ( \left ( -\gamma_{af} \pm \frac { j } {2\omega_{af}} \right) t \right) \times \exp \left ( \pm i \omega_{af} t \right ) $$
FINALY, when
$$ -\gamma_{af} \pm \frac { j } {2\omega_{af}} < 0$$.
we should get the stable situation.
But how to gather from here parameters I need to have a saddle point. People, help me please.

 

[mark726]

Hello there! Welcome to the forum and thanks for your post. I would be happy to help you with your questions.

For the first question, it seems like you are trying to find a relation between Lz and Lx, Ly. In this case, you can use the Pythagorean theorem to find the magnitude of Lz in terms of Lx and Ly. So, Lz = sqrt(Lx^2 + Ly^2). This way, you can replace Lz with Lx and Ly in your potential energy equation.

For the second question, it seems like you are trying to find the values of current and other parameters for your system to reach a saddle point. This is a bit more complicated and would require more information about your system and its equations of motion. It would be helpful if you could provide more context and information about your system.