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
21
5
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?
 
  • Wow
Likes Delta2
Physics news on Phys.org
  • #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.
 
Last edited:
  • Like
Likes vanhees71
  • #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.
 
  • Like
Likes vanhees71

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.

Similar threads

I Using reference frames for derivation of Lagrangian
Replies
25
Views
2K
A Equations of Motion for Torsional Inverted Pendulum on a moving cart
Replies
2
Views
910
B What is the Absolute Value of the Normal force on a block on this wedge?
Replies
3
Views
491
I Why is it Impossible to Solve the Three Body Problem Analytically?
Replies
6
Views
3K
Learning DEs: Solving 2nd Order Differential Equations
Replies
1
Views
779
Why were Newton's laws of motion discovered so late?
2
Replies
66
Views
8K
Equations of motion of damped oscillations due to kinetic friction
Replies
6
Views
1K
Can analytical mechanics be deduced from Newtonian mechanics?
Replies
22
Views
2K
I Acceleration in Newton's second law
Replies
5
Views
1K
Is Mach's Principle truly a paradox in rotational frames?
Replies
2
Views
936

Hot Threads

Recent Insights

Back
Top