Can Newton's Method Solve Freer Motion?

In summary, the conversation discusses whether there are cases where Newton's method is the only viable option for solving equations of motion, particularly in situations with no restrictions or constraints. The conversation also touches on the difference between Lagrange 1 and 2 formulations, and the applicability of Lagrange mechanics to systems without constraints. It is noted that Lagrange mechanics can be more efficient in deriving constants of motion, but may face challenges with dissipative forces.
  • #1
Juli
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Hello everyone,

my question is, if there is a case, where you can't you Langrange (1 or 2) but only Newton to solve the equation of motion?
My guess is, that it might be, when we have no restrictions at all, so a totally free motion.
Does anybody know?
 
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  • #2
What do you mean by Lagrange 1 and 2? That does not seem like standard nomenclature to me. Please be specific.

Generally, the equations of motion are differential equations and whether they can be solved or not does not depend on the theory you used to derive them. Where you could fail is in arriving at a set of equations of motion.
 
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  • #3
Usually "Lagrange 1" is the formulation with the (holonomic) constraints treated with Lagrange multipliers, while "Lagrange 2" is the formulation in terms of an appropriate set of "generalized coordinates".
 
  • #4
Regardless, it should probably be pointed out that Lagrange mechanics is perfectly applicable to systems without constraints. It could even be argued it does better in ease of deriving constants of motion etc. Where you can run into issues is when there are dissipative forces (eg, friction) acting on the system.
 
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FAQ: Can Newton's Method Solve Freer Motion?

Can Newton's Method solve for all types of free motion?

No, Newton's Method can only be applied to systems with a known, deterministic force function. It cannot be used for systems with random or chaotic motion.

How accurate is Newton's Method in solving free motion problems?

The accuracy of Newton's Method depends on the initial guess and the number of iterations performed. In general, it is a highly accurate method for solving free motion problems.

Can Newton's Method handle complex systems with multiple particles?

Yes, Newton's Method can be applied to complex systems with multiple particles. However, it becomes more computationally intensive and may require more iterations to reach a solution.

Is Newton's Method the only method for solving free motion problems?

No, there are other methods such as Euler's Method and Runge-Kutta methods that can also be used to solve free motion problems. Each method has its own advantages and limitations.

Can Newton's Method be used to predict the future motion of a system?

Yes, Newton's Method can be used to predict the future motion of a system by iterating the process and calculating the position and velocity at each time step. However, the accuracy of the prediction depends on the accuracy of the initial conditions and the force function used.

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