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How can I prove that a state of equilibrium is unstable?
The Next CEO of Stack OverflowDoes the induced charge on a conductor stay at the surface?Given a Magnetic Field vector, what is the Electric Field, charge density, & current density?Checking for equilibrium in a square configuration of chargesWhy can't charge be in a stable equilibrium in electrostatic field?Charge distribution: electrostatic equilibrium in conducting sphereSituation of Stable, Neutral and Unstable EquilibriumWhat is the formal accepted definition of gravitational potential energy (or potential energy in general)?Explanation of the negative integralMagnitude of an electric field on a long rodUnstable equilibrium due to an arbitrary electrostatic configuration
$begingroup$
In the particular problem I encountered, an electric field was zero at the origin and we were meant to prove that a particle at the origin was in an unstable state of equilibrim.
Is it enough to state that for any none-nule coordinates, the electric field isn't zero, ergo the equilibrium is unstable? Or is there a more elegant way of proving it?
electric-fields soft-question oscillators equilibrium
$endgroup$
add a comment |
$begingroup$
In the particular problem I encountered, an electric field was zero at the origin and we were meant to prove that a particle at the origin was in an unstable state of equilibrim.
Is it enough to state that for any none-nule coordinates, the electric field isn't zero, ergo the equilibrium is unstable? Or is there a more elegant way of proving it?
electric-fields soft-question oscillators equilibrium
$endgroup$
add a comment |
$begingroup$
In the particular problem I encountered, an electric field was zero at the origin and we were meant to prove that a particle at the origin was in an unstable state of equilibrim.
Is it enough to state that for any none-nule coordinates, the electric field isn't zero, ergo the equilibrium is unstable? Or is there a more elegant way of proving it?
electric-fields soft-question oscillators equilibrium
$endgroup$
In the particular problem I encountered, an electric field was zero at the origin and we were meant to prove that a particle at the origin was in an unstable state of equilibrim.
Is it enough to state that for any none-nule coordinates, the electric field isn't zero, ergo the equilibrium is unstable? Or is there a more elegant way of proving it?
electric-fields soft-question oscillators equilibrium
electric-fields soft-question oscillators equilibrium
asked 6 hours ago
RyeRye
287
287
add a comment |
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
In the centre of a bowl there is equilibrium.
Put a ping pong ball in it. If this ball is ever shaken slightly away from equilibrium, it will immediately roll back. A "shake-proof" equilibrium is called stable.
Now, turn the bowl around and put the ball on the top. This is an equilibrium. But at the slightest shake, the ball rolls down. This "non-shake-proof" equilibrium is called unstable.
It is about the potential energy. Because, systems always tend towards lowest potential energy. The bottom of the bowl is of lowest potential energy, so the ball wants to move back when it is slightly displaced. The top of the flipped bowl is of highest potential energy, and any neighbour point is of lower energy. So the ball has no tendency to roll back up.
Mathematically, it is thus all about figuring out if the equilibrium is a minimum or a maximum. Only a minimum is stable.
You might for many practical/physical purposes be able to determine this by simply looking at the graph of the potential energy.
But mathematically, this can be solved directly from the potential energy expression $U$. Just look at the sign of the double derivative (derived to position).
- If it is positive, $U''_xx>0$, then the value at the equilibrium is about to increase - so it is a minimum.
- If it is negative, $U''_xx<0$, then the value at the equilibrium is about to decrease - so it is a maximum.
If you have a 2D function, then you have more than one double derivative, $U''_xx$, $U''_xy$, $U''_yx$ and $U''_yy$. In this case, you must collect them into a so-called Hessian matrix and look at the eigenvalues of that matrix. If both positive, then the point is a minimum; if both negative, then the point is a maximum. (And if a mix, then the point is neither a minimum nor a maximum, but a saddle point).
This may be a bit more than you expected - but it is the rather elegant, mathematical method.
$endgroup$
1
$begingroup$
@AaronStevens Then it is a saddle point :) So, neither max nor min.
$endgroup$
– Steeven
5 hours ago
1
$begingroup$
Yep, exactly :) Also, I thought you wanted negative eigenvalues for minima
$endgroup$
– Aaron Stevens
5 hours ago
1
$begingroup$
@AaronStevens If you are evaluating energy you want the function concave up (so positive 2nd derivative). If you are evaluating feedback you want it to be negative.
$endgroup$
– dmckee♦
3 hours ago
1
$begingroup$
Also a saddle point is generally classified as "unstable" unless you are asked explicitly about the response in a particular direction.
$endgroup$
– dmckee♦
3 hours ago
$begingroup$
@dmckee Ah yes. I was getting mixed up with the Jacobian in linear stability analysis
$endgroup$
– Aaron Stevens
3 hours ago
add a comment |
$begingroup$
Is it enough to state that for any none-null coordinates,
the electric field isn't zero, ergo the equilibrium is unstable?
No, that is not enough.
You are right with: At the point of equilibrium the electric force needs to be null.
But furthermore: The direction of the electric force in the surroundings of the equilibrium position is important.
- If the electric force points towards the equilibrium position,
then the equilibrium is stable. - If the electric force points away from the equilibrium position,
then the equilibrium is instable.
Or is there a more elegant way of proving it?
It is usually easier to analyze equilibrium with potential energy,
instead of with forces.
- If the potential energy is a minimum,
then the equilibrium is stable. - If the potential energy is a maximum,
then the equilibrium is instable.
$endgroup$
1
$begingroup$
+1, but I believe your first bullet list is too general. What if there is a mix? Or in your second list you can have unstable equilibria when the potential is not a local maximum.
$endgroup$
– Aaron Stevens
5 hours ago
add a comment |
$begingroup$
While checking for any point that its for unstable or stable equilibrium, graphical method can be used. If slope at that point is negative i.e. with increase in one coordinate other decreases and graph goes back to same point again,then its stable and for unstable its vice-versa.
New contributor
$endgroup$
add a comment |
$begingroup$
If there is no charge at the origin (that is producing the electric field), then by Gauss's law (in derivative form), the divergence of the electric field there is 0: $fracpartial ^2 Vpartial x^2 + fracpartial ^2 Vpartial y^2 + fracpartial ^2 Vpartial z^2= 0$. Unless all three of these quantities are zero, then one of them must be negative, which means that in that direction, the equilibrium is unstable.
$endgroup$
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
In the centre of a bowl there is equilibrium.
Put a ping pong ball in it. If this ball is ever shaken slightly away from equilibrium, it will immediately roll back. A "shake-proof" equilibrium is called stable.
Now, turn the bowl around and put the ball on the top. This is an equilibrium. But at the slightest shake, the ball rolls down. This "non-shake-proof" equilibrium is called unstable.
It is about the potential energy. Because, systems always tend towards lowest potential energy. The bottom of the bowl is of lowest potential energy, so the ball wants to move back when it is slightly displaced. The top of the flipped bowl is of highest potential energy, and any neighbour point is of lower energy. So the ball has no tendency to roll back up.
Mathematically, it is thus all about figuring out if the equilibrium is a minimum or a maximum. Only a minimum is stable.
You might for many practical/physical purposes be able to determine this by simply looking at the graph of the potential energy.
But mathematically, this can be solved directly from the potential energy expression $U$. Just look at the sign of the double derivative (derived to position).
- If it is positive, $U''_xx>0$, then the value at the equilibrium is about to increase - so it is a minimum.
- If it is negative, $U''_xx<0$, then the value at the equilibrium is about to decrease - so it is a maximum.
If you have a 2D function, then you have more than one double derivative, $U''_xx$, $U''_xy$, $U''_yx$ and $U''_yy$. In this case, you must collect them into a so-called Hessian matrix and look at the eigenvalues of that matrix. If both positive, then the point is a minimum; if both negative, then the point is a maximum. (And if a mix, then the point is neither a minimum nor a maximum, but a saddle point).
This may be a bit more than you expected - but it is the rather elegant, mathematical method.
$endgroup$
1
$begingroup$
@AaronStevens Then it is a saddle point :) So, neither max nor min.
$endgroup$
– Steeven
5 hours ago
1
$begingroup$
Yep, exactly :) Also, I thought you wanted negative eigenvalues for minima
$endgroup$
– Aaron Stevens
5 hours ago
1
$begingroup$
@AaronStevens If you are evaluating energy you want the function concave up (so positive 2nd derivative). If you are evaluating feedback you want it to be negative.
$endgroup$
– dmckee♦
3 hours ago
1
$begingroup$
Also a saddle point is generally classified as "unstable" unless you are asked explicitly about the response in a particular direction.
$endgroup$
– dmckee♦
3 hours ago
$begingroup$
@dmckee Ah yes. I was getting mixed up with the Jacobian in linear stability analysis
$endgroup$
– Aaron Stevens
3 hours ago
add a comment |
$begingroup$
In the centre of a bowl there is equilibrium.
Put a ping pong ball in it. If this ball is ever shaken slightly away from equilibrium, it will immediately roll back. A "shake-proof" equilibrium is called stable.
Now, turn the bowl around and put the ball on the top. This is an equilibrium. But at the slightest shake, the ball rolls down. This "non-shake-proof" equilibrium is called unstable.
It is about the potential energy. Because, systems always tend towards lowest potential energy. The bottom of the bowl is of lowest potential energy, so the ball wants to move back when it is slightly displaced. The top of the flipped bowl is of highest potential energy, and any neighbour point is of lower energy. So the ball has no tendency to roll back up.
Mathematically, it is thus all about figuring out if the equilibrium is a minimum or a maximum. Only a minimum is stable.
You might for many practical/physical purposes be able to determine this by simply looking at the graph of the potential energy.
But mathematically, this can be solved directly from the potential energy expression $U$. Just look at the sign of the double derivative (derived to position).
- If it is positive, $U''_xx>0$, then the value at the equilibrium is about to increase - so it is a minimum.
- If it is negative, $U''_xx<0$, then the value at the equilibrium is about to decrease - so it is a maximum.
If you have a 2D function, then you have more than one double derivative, $U''_xx$, $U''_xy$, $U''_yx$ and $U''_yy$. In this case, you must collect them into a so-called Hessian matrix and look at the eigenvalues of that matrix. If both positive, then the point is a minimum; if both negative, then the point is a maximum. (And if a mix, then the point is neither a minimum nor a maximum, but a saddle point).
This may be a bit more than you expected - but it is the rather elegant, mathematical method.
$endgroup$
1
$begingroup$
@AaronStevens Then it is a saddle point :) So, neither max nor min.
$endgroup$
– Steeven
5 hours ago
1
$begingroup$
Yep, exactly :) Also, I thought you wanted negative eigenvalues for minima
$endgroup$
– Aaron Stevens
5 hours ago
1
$begingroup$
@AaronStevens If you are evaluating energy you want the function concave up (so positive 2nd derivative). If you are evaluating feedback you want it to be negative.
$endgroup$
– dmckee♦
3 hours ago
1
$begingroup$
Also a saddle point is generally classified as "unstable" unless you are asked explicitly about the response in a particular direction.
$endgroup$
– dmckee♦
3 hours ago
$begingroup$
@dmckee Ah yes. I was getting mixed up with the Jacobian in linear stability analysis
$endgroup$
– Aaron Stevens
3 hours ago
add a comment |
$begingroup$
In the centre of a bowl there is equilibrium.
Put a ping pong ball in it. If this ball is ever shaken slightly away from equilibrium, it will immediately roll back. A "shake-proof" equilibrium is called stable.
Now, turn the bowl around and put the ball on the top. This is an equilibrium. But at the slightest shake, the ball rolls down. This "non-shake-proof" equilibrium is called unstable.
It is about the potential energy. Because, systems always tend towards lowest potential energy. The bottom of the bowl is of lowest potential energy, so the ball wants to move back when it is slightly displaced. The top of the flipped bowl is of highest potential energy, and any neighbour point is of lower energy. So the ball has no tendency to roll back up.
Mathematically, it is thus all about figuring out if the equilibrium is a minimum or a maximum. Only a minimum is stable.
You might for many practical/physical purposes be able to determine this by simply looking at the graph of the potential energy.
But mathematically, this can be solved directly from the potential energy expression $U$. Just look at the sign of the double derivative (derived to position).
- If it is positive, $U''_xx>0$, then the value at the equilibrium is about to increase - so it is a minimum.
- If it is negative, $U''_xx<0$, then the value at the equilibrium is about to decrease - so it is a maximum.
If you have a 2D function, then you have more than one double derivative, $U''_xx$, $U''_xy$, $U''_yx$ and $U''_yy$. In this case, you must collect them into a so-called Hessian matrix and look at the eigenvalues of that matrix. If both positive, then the point is a minimum; if both negative, then the point is a maximum. (And if a mix, then the point is neither a minimum nor a maximum, but a saddle point).
This may be a bit more than you expected - but it is the rather elegant, mathematical method.
$endgroup$
In the centre of a bowl there is equilibrium.
Put a ping pong ball in it. If this ball is ever shaken slightly away from equilibrium, it will immediately roll back. A "shake-proof" equilibrium is called stable.
Now, turn the bowl around and put the ball on the top. This is an equilibrium. But at the slightest shake, the ball rolls down. This "non-shake-proof" equilibrium is called unstable.
It is about the potential energy. Because, systems always tend towards lowest potential energy. The bottom of the bowl is of lowest potential energy, so the ball wants to move back when it is slightly displaced. The top of the flipped bowl is of highest potential energy, and any neighbour point is of lower energy. So the ball has no tendency to roll back up.
Mathematically, it is thus all about figuring out if the equilibrium is a minimum or a maximum. Only a minimum is stable.
You might for many practical/physical purposes be able to determine this by simply looking at the graph of the potential energy.
But mathematically, this can be solved directly from the potential energy expression $U$. Just look at the sign of the double derivative (derived to position).
- If it is positive, $U''_xx>0$, then the value at the equilibrium is about to increase - so it is a minimum.
- If it is negative, $U''_xx<0$, then the value at the equilibrium is about to decrease - so it is a maximum.
If you have a 2D function, then you have more than one double derivative, $U''_xx$, $U''_xy$, $U''_yx$ and $U''_yy$. In this case, you must collect them into a so-called Hessian matrix and look at the eigenvalues of that matrix. If both positive, then the point is a minimum; if both negative, then the point is a maximum. (And if a mix, then the point is neither a minimum nor a maximum, but a saddle point).
This may be a bit more than you expected - but it is the rather elegant, mathematical method.
edited 5 hours ago
answered 5 hours ago
SteevenSteeven
27.4k765112
27.4k765112
1
$begingroup$
@AaronStevens Then it is a saddle point :) So, neither max nor min.
$endgroup$
– Steeven
5 hours ago
1
$begingroup$
Yep, exactly :) Also, I thought you wanted negative eigenvalues for minima
$endgroup$
– Aaron Stevens
5 hours ago
1
$begingroup$
@AaronStevens If you are evaluating energy you want the function concave up (so positive 2nd derivative). If you are evaluating feedback you want it to be negative.
$endgroup$
– dmckee♦
3 hours ago
1
$begingroup$
Also a saddle point is generally classified as "unstable" unless you are asked explicitly about the response in a particular direction.
$endgroup$
– dmckee♦
3 hours ago
$begingroup$
@dmckee Ah yes. I was getting mixed up with the Jacobian in linear stability analysis
$endgroup$
– Aaron Stevens
3 hours ago
add a comment |
1
$begingroup$
@AaronStevens Then it is a saddle point :) So, neither max nor min.
$endgroup$
– Steeven
5 hours ago
1
$begingroup$
Yep, exactly :) Also, I thought you wanted negative eigenvalues for minima
$endgroup$
– Aaron Stevens
5 hours ago
1
$begingroup$
@AaronStevens If you are evaluating energy you want the function concave up (so positive 2nd derivative). If you are evaluating feedback you want it to be negative.
$endgroup$
– dmckee♦
3 hours ago
1
$begingroup$
Also a saddle point is generally classified as "unstable" unless you are asked explicitly about the response in a particular direction.
$endgroup$
– dmckee♦
3 hours ago
$begingroup$
@dmckee Ah yes. I was getting mixed up with the Jacobian in linear stability analysis
$endgroup$
– Aaron Stevens
3 hours ago
1
1
$begingroup$
@AaronStevens Then it is a saddle point :) So, neither max nor min.
$endgroup$
– Steeven
5 hours ago
$begingroup$
@AaronStevens Then it is a saddle point :) So, neither max nor min.
$endgroup$
– Steeven
5 hours ago
1
1
$begingroup$
Yep, exactly :) Also, I thought you wanted negative eigenvalues for minima
$endgroup$
– Aaron Stevens
5 hours ago
$begingroup$
Yep, exactly :) Also, I thought you wanted negative eigenvalues for minima
$endgroup$
– Aaron Stevens
5 hours ago
1
1
$begingroup$
@AaronStevens If you are evaluating energy you want the function concave up (so positive 2nd derivative). If you are evaluating feedback you want it to be negative.
$endgroup$
– dmckee♦
3 hours ago
$begingroup$
@AaronStevens If you are evaluating energy you want the function concave up (so positive 2nd derivative). If you are evaluating feedback you want it to be negative.
$endgroup$
– dmckee♦
3 hours ago
1
1
$begingroup$
Also a saddle point is generally classified as "unstable" unless you are asked explicitly about the response in a particular direction.
$endgroup$
– dmckee♦
3 hours ago
$begingroup$
Also a saddle point is generally classified as "unstable" unless you are asked explicitly about the response in a particular direction.
$endgroup$
– dmckee♦
3 hours ago
$begingroup$
@dmckee Ah yes. I was getting mixed up with the Jacobian in linear stability analysis
$endgroup$
– Aaron Stevens
3 hours ago
$begingroup$
@dmckee Ah yes. I was getting mixed up with the Jacobian in linear stability analysis
$endgroup$
– Aaron Stevens
3 hours ago
add a comment |
$begingroup$
Is it enough to state that for any none-null coordinates,
the electric field isn't zero, ergo the equilibrium is unstable?
No, that is not enough.
You are right with: At the point of equilibrium the electric force needs to be null.
But furthermore: The direction of the electric force in the surroundings of the equilibrium position is important.
- If the electric force points towards the equilibrium position,
then the equilibrium is stable. - If the electric force points away from the equilibrium position,
then the equilibrium is instable.
Or is there a more elegant way of proving it?
It is usually easier to analyze equilibrium with potential energy,
instead of with forces.
- If the potential energy is a minimum,
then the equilibrium is stable. - If the potential energy is a maximum,
then the equilibrium is instable.
$endgroup$
1
$begingroup$
+1, but I believe your first bullet list is too general. What if there is a mix? Or in your second list you can have unstable equilibria when the potential is not a local maximum.
$endgroup$
– Aaron Stevens
5 hours ago
add a comment |
$begingroup$
Is it enough to state that for any none-null coordinates,
the electric field isn't zero, ergo the equilibrium is unstable?
No, that is not enough.
You are right with: At the point of equilibrium the electric force needs to be null.
But furthermore: The direction of the electric force in the surroundings of the equilibrium position is important.
- If the electric force points towards the equilibrium position,
then the equilibrium is stable. - If the electric force points away from the equilibrium position,
then the equilibrium is instable.
Or is there a more elegant way of proving it?
It is usually easier to analyze equilibrium with potential energy,
instead of with forces.
- If the potential energy is a minimum,
then the equilibrium is stable. - If the potential energy is a maximum,
then the equilibrium is instable.
$endgroup$
1
$begingroup$
+1, but I believe your first bullet list is too general. What if there is a mix? Or in your second list you can have unstable equilibria when the potential is not a local maximum.
$endgroup$
– Aaron Stevens
5 hours ago
add a comment |
$begingroup$
Is it enough to state that for any none-null coordinates,
the electric field isn't zero, ergo the equilibrium is unstable?
No, that is not enough.
You are right with: At the point of equilibrium the electric force needs to be null.
But furthermore: The direction of the electric force in the surroundings of the equilibrium position is important.
- If the electric force points towards the equilibrium position,
then the equilibrium is stable. - If the electric force points away from the equilibrium position,
then the equilibrium is instable.
Or is there a more elegant way of proving it?
It is usually easier to analyze equilibrium with potential energy,
instead of with forces.
- If the potential energy is a minimum,
then the equilibrium is stable. - If the potential energy is a maximum,
then the equilibrium is instable.
$endgroup$
Is it enough to state that for any none-null coordinates,
the electric field isn't zero, ergo the equilibrium is unstable?
No, that is not enough.
You are right with: At the point of equilibrium the electric force needs to be null.
But furthermore: The direction of the electric force in the surroundings of the equilibrium position is important.
- If the electric force points towards the equilibrium position,
then the equilibrium is stable. - If the electric force points away from the equilibrium position,
then the equilibrium is instable.
Or is there a more elegant way of proving it?
It is usually easier to analyze equilibrium with potential energy,
instead of with forces.
- If the potential energy is a minimum,
then the equilibrium is stable. - If the potential energy is a maximum,
then the equilibrium is instable.
answered 5 hours ago
Thomas FritschThomas Fritsch
1,445415
1,445415
1
$begingroup$
+1, but I believe your first bullet list is too general. What if there is a mix? Or in your second list you can have unstable equilibria when the potential is not a local maximum.
$endgroup$
– Aaron Stevens
5 hours ago
add a comment |
1
$begingroup$
+1, but I believe your first bullet list is too general. What if there is a mix? Or in your second list you can have unstable equilibria when the potential is not a local maximum.
$endgroup$
– Aaron Stevens
5 hours ago
1
1
$begingroup$
+1, but I believe your first bullet list is too general. What if there is a mix? Or in your second list you can have unstable equilibria when the potential is not a local maximum.
$endgroup$
– Aaron Stevens
5 hours ago
$begingroup$
+1, but I believe your first bullet list is too general. What if there is a mix? Or in your second list you can have unstable equilibria when the potential is not a local maximum.
$endgroup$
– Aaron Stevens
5 hours ago
add a comment |
$begingroup$
While checking for any point that its for unstable or stable equilibrium, graphical method can be used. If slope at that point is negative i.e. with increase in one coordinate other decreases and graph goes back to same point again,then its stable and for unstable its vice-versa.
New contributor
$endgroup$
add a comment |
$begingroup$
While checking for any point that its for unstable or stable equilibrium, graphical method can be used. If slope at that point is negative i.e. with increase in one coordinate other decreases and graph goes back to same point again,then its stable and for unstable its vice-versa.
New contributor
$endgroup$
add a comment |
$begingroup$
While checking for any point that its for unstable or stable equilibrium, graphical method can be used. If slope at that point is negative i.e. with increase in one coordinate other decreases and graph goes back to same point again,then its stable and for unstable its vice-versa.
New contributor
$endgroup$
While checking for any point that its for unstable or stable equilibrium, graphical method can be used. If slope at that point is negative i.e. with increase in one coordinate other decreases and graph goes back to same point again,then its stable and for unstable its vice-versa.
New contributor
New contributor
answered 5 hours ago
sk9298sk9298
163
163
New contributor
New contributor
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If there is no charge at the origin (that is producing the electric field), then by Gauss's law (in derivative form), the divergence of the electric field there is 0: $fracpartial ^2 Vpartial x^2 + fracpartial ^2 Vpartial y^2 + fracpartial ^2 Vpartial z^2= 0$. Unless all three of these quantities are zero, then one of them must be negative, which means that in that direction, the equilibrium is unstable.
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$begingroup$
If there is no charge at the origin (that is producing the electric field), then by Gauss's law (in derivative form), the divergence of the electric field there is 0: $fracpartial ^2 Vpartial x^2 + fracpartial ^2 Vpartial y^2 + fracpartial ^2 Vpartial z^2= 0$. Unless all three of these quantities are zero, then one of them must be negative, which means that in that direction, the equilibrium is unstable.
$endgroup$
add a comment |
$begingroup$
If there is no charge at the origin (that is producing the electric field), then by Gauss's law (in derivative form), the divergence of the electric field there is 0: $fracpartial ^2 Vpartial x^2 + fracpartial ^2 Vpartial y^2 + fracpartial ^2 Vpartial z^2= 0$. Unless all three of these quantities are zero, then one of them must be negative, which means that in that direction, the equilibrium is unstable.
$endgroup$
If there is no charge at the origin (that is producing the electric field), then by Gauss's law (in derivative form), the divergence of the electric field there is 0: $fracpartial ^2 Vpartial x^2 + fracpartial ^2 Vpartial y^2 + fracpartial ^2 Vpartial z^2= 0$. Unless all three of these quantities are zero, then one of them must be negative, which means that in that direction, the equilibrium is unstable.
answered 2 hours ago
Faraz MasroorFaraz Masroor
298115
298115
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