- #1

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Thanks in advance for all the help (and please, if it's not too much to ask, bear in mind I'm not a physics student, so I'd appreciate simplicity).

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- #1

- 33

- 1

Thanks in advance for all the help (and please, if it's not too much to ask, bear in mind I'm not a physics student, so I'd appreciate simplicity).

- #2

Twigg

Gold Member

- 337

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Examples:

1. Find the trajectory of a ball rolling on a parabolic incline given it's initial height and the equation of the parabola.

It would be a pain to solve this with Newtonian mechanics because the normal force's line of action depends on the surface normal, which changes at every point on the incline (because it's a curved surface). But in Lagrangian/Hamiltonian mechanics, it's no big deal, because we only work in scalar quantities, and we can use a single parametric coordinate (say, x) instead of using both the x and y dimensions where they aren't necessary. It's just more efficient.

2. Find the orbital period of a charged particle moving in a circular path of given radius in a uniform magnetic field.

Langrangian mechanics is serious overkill here, as all you need is the centripetal acceleration and the magnetic of the magnetic force to figure it out. The force-based analysis is much faster.

In short, the reason we use three theories instead of one is for greater flexibility.

- #3

- 33

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Examples:

1. Find the trajectory of a ball rolling on a parabolic incline given it's initial height and the equation of the parabola.

It would be a pain to solve this with Newtonian mechanics because the normal force's line of action depends on the surface normal, which changes at every point on the incline (because it's a curved surface). But in Lagrangian/Hamiltonian mechanics, it's no big deal, because we only work in scalar quantities, and we can use a single parametric coordinate (say, x) instead of using both the x and y dimensions where they aren't necessary. It's just more efficient.

2. Find the orbital period of a charged particle moving in a circular path of given radius in a uniform magnetic field.

Langrangian mechanics is serious overkill here, as all you need is the centripetal acceleration and the magnetic of the magnetic force to figure it out. The force-based analysis is much faster.

In short, the reason we use three theories instead of one is for greater flexibility.

So they're only different in their practical use? There's no conceptual difference whatsoever between them?

I'm very ignorant of these issues, but I thought I heard Newton's framework was eventually proved to work only in very special circumstances. Or does that have to do with Einstein's relativity? Again, I apologize if I'm saying something ridiculously wrong. I seriously haven't been able to study this.

- #4

Twigg

Gold Member

- 337

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So they're only different in their practical use? There's no conceptual difference whatsoever between them?

I'm very ignorant of these issues, but I thought I heard Newton's framework was eventually proved to work only in very special circumstances. Or does that have to do with Einstein's relativity? Again, I apologize if I'm saying something ridiculously wrong. I seriously haven't been able to study this.

No need to apologize. As theories of

- #5

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Newtonian methods focus on forces.

Lagrangian and Hamiltonian methods are energy methods with a focus on (respectively) "position"-and-velocity and "position"-and-momentum.

Some systems are better suited to one method over another.

Symmetry and conserved quantities (Noether theorem) and constrained systems are aspects that Lagrangian and Hamiltonian methods are better at than Newtonian methods. Some numerical methods exploit symmetry to manage numerical errors.

Lagrangian and Hamiltonian methods also allow "generalized coordinates" that need not be rectangular or spherical polar, etc... which can make problems easier than what their Newtonian version would suggest.

In fact, Lagrangian and Hamiltonian methods are not restricted to mechanics....

One can exploit dynamical analogies.... and solve problems in optics, circuits, electromagnetism, and other classical field theories.

In many cases, Lagrangian and Hamiltonian methods generalize to quantum mechanics more easily than using Newtonian methods.

Of course, these energy methods are more abstract and require more mathematical ability than Newtonian methods.

- #6

wrobel

Science Advisor

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For beginning one should just understand that the Hamiltonian formalism does not work in general for nonholonomic systems and for the systems with dissipative forces like friction etc

- #7

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- #8

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They don't dispel the notion of force.

Lagrangian mechanics uses https://en.wikipedia.org/wiki/Generalized_forces .

As I said earlier, it gives different viewpoints on the dynamics.

Each method has its strengths and its weaknesses.

In simple cases, they may seem equivalent... but they're not in general.

So, you can't really give one up.

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