Lagrange & Hamiltonian mech => Newtonia mech.

[MathematicalPhysicist]

My question is simple is every classical mechanics problem which is solvable by Lagrangian & Hamiltonian methods also solvable by Newtonian methods of forces and torques?

And why does it seem that LH make solutions to be a lot more easier than Newtonian methods, and is it always this way?

 

[vanhees71]

The principle of least action is the common description of all physical theories at a fundamental level. All fundamental theories are most simply described as such a variational principle (standard model of elementary particle physics, classical field theory (e.g., classical electromagnetics, general relativity), and point-particle mechanics).

The reason for this is that it most easily allows to study symmetries of the fundamental laws of nature, such as space-time symmetries (Galileo, Poincare, general covariance) and external symmetries leading to conservation laws for engery, momentum, angular momentum, center-of-mass velocity, and various charge-like quantities (electrical charge, baryon number, lepton number, etc.), respectively.

Of course, on the classical (i.e., non-quantum) level of point-particle mechanics, the action principle is equivalent to the Newton (or relativistic if necessary) equations of motion for the particle, and you can of course write down this equations from the very beginning.

What makes the action principle more convenient from a practical point of view is that it is way easier to express the action (or the Lagrangian/Hamiltonian) of the system in general coordinates, adapted to the problem at hand, than using directly the equations of motion and performing the transformation from Cartesian to generalized coordinates.

 

[juanrga]

My question is simple is every classical mechanics problem which is solvable by Lagrangian & Hamiltonian methods also solvable by Newtonian methods of forces and torques?

And why does it seem that LH make solutions to be a lot more easier than Newtonian methods, and is it always this way?

If by "classical" you mean non-relativistic classical, then the three formulations are formally equivalent. You can, however, find computational or analytical difficulties. That is why one chooses the formulation more adequate to the problem.

 

[dextercioby]

It only <seems> that way, well said. Indeed, the lagrangian methods seem simpler, but in some cases, they are really not (here I mean dissipative systems, where the task to find a lagrangian is not simple). It normally all boils down to solving ODE's. To get to them can be simpler using one method or another, but it's not a general rule.

 

[PJC01]

I would like to clarify that both Lagrangian and Hamiltonian mechanics are alternative mathematical formulations of classical mechanics, which is the study of motion and forces in a physical system. These methods are based on different mathematical principles and can provide different insights into a problem.

To answer your question, yes, every classical mechanics problem that can be solved using Lagrangian and Hamiltonian methods can also be solved using Newtonian methods. This is because all three methods are based on the same fundamental principles of classical mechanics and can be used to describe the same physical systems. However, the approach and mathematical tools used in each method may differ, and some problems may be more easily solved using one method over the other.

It may seem that Lagrangian and Hamiltonian methods make solutions easier because they often involve fewer equations and are more elegant mathematically. These methods also allow for the use of generalized coordinates, which can simplify the representation of complex systems. However, it is not always the case that these methods are easier, as some problems may require more advanced mathematical techniques or may be more easily solved using Newtonian methods. It ultimately depends on the specific problem at hand and the skills and familiarity of the scientist with each method.

In conclusion, Lagrangian and Hamiltonian mechanics are powerful tools for solving classical mechanics problems, but they are not the only methods available. Both methods have their advantages and limitations, and it is important for a scientist to have a good understanding of all methods in order to choose the most appropriate one for a given problem.