Throw whatever you have against these 2 equations

  • Thread starter andrewm
  • Start date
In summary, the conversation discusses the problem of solving two equations with four unknowns in the context of physics research. The individual is seeking a way to solve the equations numerically or by hand and is considering different approaches and constraints. The conversation ultimately concludes with the suggestion to pick a value for one of the unknowns and use the equations to solve for the remaining unknowns.
  • #1
andrewm
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This isn't my homework: I'm doing some physics research and I'm stuck at a simple 2 equations. I want to solve these equations

[tex] A \cos(\gamma) \sinh(\theta) = \lambda - B \cosh(\theta) [/tex]
[tex] A \cos(\gamma) \cosh(\theta) = A \sin(\gamma) - B \sinh(\theta) [/tex]

I'd like to know if there's any way I can find [tex]\lambda[/tex] if I start with A and B known. I'd be happy to do this numerically, but I can't see how I would. I've tried monkeying with the algebra for a while.

All that my undergrad math tells me is that there should be a solution since there are 2 equations, 2 unknowns.

Is there any way I can solve this numerically, or approximate the solution by hand, or even show that there is a solution?

I'm stumped!
 
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  • #2
There are 4 unknowns in this. A, B, Gamma, Theta.

There may be a way of solving this but not with any form of simulataneous equations of matrices i wouldn't think. Don't hold me to that though.
 
  • #3
A and B are known parameters. Indeed, I have already tabulated them numerically.
 
  • #4
Sorry that was me misreading the post. Another problem is still there. Even though we have two equations we have three variables. (I missed out lambda last time). You would need 3 equations for this to work.
 
  • #5
Of course, thanks. I'll try to find a new constraint.
 
  • #6
andrewm said:
This isn't my homework: I'm doing some physics research and I'm stuck at a simple 2 equations. I want to solve these equations

[tex] A \cos(\gamma) \sinh(\theta) = \lambda - B \cosh(\theta) [/tex]
[tex] A \cos(\gamma) \cosh(\theta) = A \sin(\gamma) - B \sinh(\theta) [/tex]

I'd like to know if there's any way I can find [tex]\lambda[/tex] if I start with A and B known. I'd be happy to do this numerically, but I can't see how I would. I've tried monkeying with the algebra for a while.

All that my undergrad math tells me is that there should be a solution since there are 2 equations, 2 unknowns.

Is there any way I can solve this numerically, or approximate the solution by hand, or even show that there is a solution?

I'm stumped!
At the very least, you may square&subtract:
[tex]A^{2}\cos^{2}(\gamma) \cosh^{2}(\theta)-A^{2} \cos^{2}(\gamma) \sinh^{2}(\theta)= (A \sin(\gamma) - B \sinh(\theta))^{2}-(\lambda - B \cosh(\theta) )^{2}[/tex]
Which can be simplified to:
[tex]A^{2}\cos(2\gamma)=2B\lambda\cosh(\theta)-2AB\sin(\gamma)\sinh(\theta)-(B^{2}+\lambda^{2})[/tex]
Maybe this equation can be used for something, or maybe not.
 
  • #7
You still have three unknowns, [itex]\lambda[/itex], [itex]\theta[/itex], and [itex]\gamma[/itex] with only two equations.
 
  • #8
If you're just looking for any solution...

To start, pick gamma=pi/2. Then your equations reduce to

[tex]\lambda -Bcosh( \theta ) = 0[/tex]
[tex]A - Bsinh (\theta ) = 0[/tex]

Based on the second equation, you find theta, then use the first equation to find lambda.
 
  • #9
I'd bring all thetas on one side and then square and subtract. The result is
[tex]
\cos 2\gamma=\frac{B^2-\lambda^2}{A^2}
[/tex]
but please check it for yourself.
 

1. What are the two equations that I should throw everything against?

The two equations are commonly known as the equation of motion and the energy conservation equation. The equation of motion is used to calculate the position, velocity, and acceleration of an object in motion. The energy conservation equation is used to determine the total amount of energy in a system and how it is conserved.

2. Why is it important to throw everything against these equations?

These equations are fundamental in understanding the behavior of objects in motion and the conservation of energy in various systems. By throwing everything against these equations, scientists can accurately predict and analyze the motion and energy of objects in a variety of scenarios.

3. Can these equations be applied to real-life situations?

Yes, these equations have been extensively tested and proven to accurately describe the motion and energy of objects in real-life situations. They are widely used in fields such as physics, engineering, and astronomy to solve complex problems and make predictions.

4. Are there any limitations to these equations?

While these equations are extremely useful, they do have limitations. They are based on certain assumptions and may not accurately describe the behavior of objects in extreme conditions, such as at the speed of light or in very small scales. Additionally, the equations may not apply to systems with complex interactions or non-conservative forces.

5. Can these equations be modified or combined with other equations?

Yes, scientists often modify and combine these equations with other equations to better describe and analyze specific systems or scenarios. For example, in the field of relativity, the equations of motion and energy conservation are modified to account for the effects of gravity and high speeds. In this way, these equations serve as a foundation for further scientific discoveries and advancements.

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