Understanding Lagrangian Mechanics: Equations of Motion and Applications

[Fascheue]

I’m a bit confused about what exactly lagranigian mechanics is.

I know that L = Ke - Pe

I also know the equation d/dt(∂L/∂x’) - ∂L/∂x = 0

1.) Apparentaly solving this equation gives the equations of motion. What exactly does that mean though? I solved a very simple problem and got the acceleration of the system.

2.) How exactly is this a reformulation of classical mechanics? It doesn’t seem like much more than just an equation.

3.) In which circumstances is this useful. The simple problem that I solved using lagrangian mechanics seemed like it could have been solved much faster with F = ma.

 

[hilbert2]

The usual Newtonian mechanics is based on forces, while the Lagrangian approach starts with the concept of energy, and derives the trajectories of point masses or other objects from the principle of least (or more generally, extremal) action. Action is defined as the time integral of Lagrangian over some interval: $S = \int_{t_1}^{t_2}Ldt$.

The usefulness of the Lagrangian method is that it's easy to apply it in non-cartesian coordinate systems (spherical, cylindrical, etc.), and constraint forces (such as what keeps a block sliding on an inclined plane from "sinking" in the plane) can be handled by simply defining coordinates in such a way that "forbidden" trajectories are not even mathematically possible.

EDIT: Here's an example problem that's best handled with Lagrangian mechanics:

http://electron6.phys.utk.edu/PhysicsProblems/Mechanics/5-Lagrangian/inclined planes.html

There's an object sliding on a wedge that acts as an inclined plane, and the wedge itself is on a frictionless surface and starts sliding too (because of momentum conservation in the horizontal direction).

 

[pixel]

To answer your questions:
1) If you know the acceleration, then you can integrate and find the position as a function of time. That is the equation of motion, for instance, the orbit of a planet around the sun.
2) Isn't classical mechanics just an equation, F=ma? The real value is in your question #3.
3) Read up on "generalized coordinates." For example, with a simple pendulum, instead of having equations for the horizontal and vertical coordinates, you just have to deal with one equation involving the angle with respect to the vertical. Also, it's usually easier to write down the scalars T and V as opposed to dealing with vector forces.

 

[Fascheue]

To answer your questions:
1) If you know the acceleration, then you can integrate and find the position as a function of time. That is the equation of motion, for instance, the orbit of a planet around the sun.
2) Isn't classical mechanics just an equation, F=ma? The real value is in your question #3.
3) Read up on "generalized coordinates." For example, with a simple pendulum, instead of having equations for the horizontal and vertical coordinates, you just have to deal with one equation involving the angle with respect to the vertical. Also, it's usually easier to write down the scalars T and V as opposed to dealing with vector forces.

So the Lagrange equations always gives the acceleration function of the system, and then the velocity and position functions can be found by integrating?

I’ve been thinking of classical mechanics as all of physics excluding quantum mechanics, and there’s a lot more to that than just F = ma. What if you were finding the momentum of a system given the mass and velocity? This seems like a classical mechanics question, but it doesn’t seem like the Newtonian and lagrangian “formulations of classical mechanics” are different. You just use the equation p = mv. The only thing that seems different is one substituted equation.

 

[Dr.D]

Newton and Lagrange are alternate ways to obtain the equation of motion. Neither of them provides the solution to the equation of motion, but both will enable you to evaluate the accelerations which is the starting point for integrating to obtain the solution.

Energy methods, such as Lagrange, can be much, much easier to apply for complex systems with many moving parts. Being much simpler, Lagrange is less prone to errors, and this is important. Time spent later solving the wrong equation of motion is pure waste!

 

[PeroK]

2.) How exactly is this a reformulation of classical mechanics? It doesn’t seem like much more than just an equation.

It's much more than an equation. It's an entirely different method of solving physical problems without the concept of a force.

The importance of Lagrange is that in both Quantum Mechanics and General Relativity there is no concept of a force. A generalisation of Newton's laws, therefore, is not directly possible. But, the Lagrangian Method can be generalised, in the sense that nature tends to behave such as to maximise or minimise certain key quantities.

In General Relativity, for example, the replacement for Newton's second law is that a particle moves so as to the optimise the proper time it experiences. And, the motion of a particle in curved spacetime is then calculated using the Lagrangian method of optimising the integral that represents the proper time of the particle.

 

[Dr.D]

there’s a lot more to that than just F = ma.

This is certainly true. If you go back to Hamilton's Principle which underlies the Lagrange equation, you have a very broad, unifying principle that connects solid mechanics, fluids, electricity, magnetism, acoustics, etc. It is extremely powerful for hybrid systems.

 

[pixel]

[QUOTE="Fascheue, post: 5879733, member: 632695"I’ve been thinking of classical mechanics as all of physics excluding quantum mechanics, and there’s a lot more to that than just F = ma. What if you were finding the momentum of a system given the mass and velocity? This seems like a classical mechanics question, but it doesn’t seem like the Newtonian and lagrangian “formulations of classical mechanics” are different. You just use the equation p = mv. The only thing that seems different is one substituted equation.[/QUOTE]

The conservation of momentum comes out of F=ma, which can be rewritten as F = dp/dt .

Also, as a matter of terminology, the Lagrange formulation is considered part of classical mechanics.

 

[hilbert2]

Something that wasn't mentioned yet is that the Lagrangian and Hamiltonian methods are straightforward to extent to the case of field systems, where you define a Lagrangian density $\mathcal{L}$ which gives the actual Lagrangian function when integrated over all space (and enforces the field equation to be local with no instant faster-than-light interactions between points). Symmetries and conservation laws can also be easily found from the functional form of the Lagrange density.

 

[PeroK]

This seems like a classical mechanics question, but it doesn’t seem like the Newtonian and lagrangian “formulations of classical mechanics” are different. You just use the equation p = mv. The only thing that seems different is one substituted equation.

They are fundamentally different:

Newton:

Write down all the forces on a particle, then solve the resulting second-order differential equation. One of the problems is that you don't always know things like restraining forces: e.g. if you have a ball sliding on a frictionless surface.

Lagrange:

Express the Lagrangian (KE - PE) as a function of position and velocity and minimise this. In this method, you do not directly consider the forces on a particle.

Also, theoretically, the concept that force is not necessarily the fundamental driver for motion, but instead the interplay of Potential and Kinetic Energies, is a huge departure from pure Newtonian mechanics.

 

[Dr.D]

Express the Lagrangian (KE - PE) as a function of position and velocity and minimise this. In this method, you do not directly consider the forces on a particle.

Also, theoretically, the concept that force is not necessarily the fundamental driver for motion, but instead the interplay of Potential and Kinetic Energies, is a huge departure from pure Newtonian mechanics.

This is not entirely true. Nonconservative forces must still be included via a generalized force term on the right side of the equation.

 

[PeroK]

This is not entirely true. Nonconservative forces must still be included via a generalized force term on the right side of the equation.

Since you are being pedantic, the phrase "not necessarily" in my post means you missed the point. To say that forces are "not necessarily the fundamental driver" doesn't say that "forces never have to be considered".

I never said you could do everything in Classical Mechanics without forces. But, many problems you can do without forces, which is pretty fundamental.

 

[Dr.D]

Since you are being pedantic, the phrase "not necessarily" in my post means you missed the point. To say that forces are "not necessarily the fundamental driver" doesn't say that "forces never have to be considered".

I never said you could do everything in Classical Mechanics without forces. But, many problems you can do without forces, which is pretty fundamental.

Wow! "Pedantic" no less! What would you expect from a pedagogue? We try to get things correct; that is a big part of teaching.

I really don't see anything profoundly fundamental about the fact that some problems require the inclusion of nonconservative force terms while others do not. That's just the way the world is; some systems are conservative and others are not. If you wish to see that as "pretty fundamental," then who am I to object?

 

[PeroK]

Wow! "Pedantic" no less! What would you expect from a pedagogue? We try to get things correct; that is a big part of teaching.

I really don't see anything profoundly fundamental about the fact that some problems require the inclusion of nonconservative force terms while others do not. That's just the way the world is; some systems are conservative and others are not. If you wish to see that as "pretty fundamental," then who am I to object?

You've missed the point (again!). It's the Lagrangian formulation that is (pretty) fundamentally different from Newton.

I, for one, was certainly struck by the Lagrangian method. The OP, on the other hand, claims he can hadly see a difference. Whereas, I thought it was quite profound to solve a mechanics problem using the Lagrange approach.