# Can Newton's Method Solve Freer Motion?

• A
• Juli
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.

#### Juli

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?

Delta2
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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vanhees71
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.

vanhees71