A Can Newton's Method Solve Freer Motion?

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Newton's method can be used to solve equations of motion even in cases of free motion, contrary to the assumption that Lagrangian mechanics is necessary. Lagrange formulations, whether with constraints or generalized coordinates, are applicable to unconstrained systems and may simplify the derivation of constants of motion. The main challenge arises when dissipative forces, such as friction, are present, which complicates the equations of motion. The discussion clarifies that the ability to solve these equations is not inherently tied to the choice of theoretical framework. Ultimately, both Newton's and Lagrangian methods have their merits depending on the specific conditions of the motion being analyzed.
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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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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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".
 
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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