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Dec23-11, 12:10 AM
P: 192
Quote Quote by D H View Post
In theory, yes. In practice, no. There are many applications in engineering, meteorology, and other areas of applied classical physics where one has no choice but to use an F=ma type formulation. How are you going to express in Lagrangian/Hamiltonian form the forces and torques exerted by a rocket that fires intermittently based on the whims of a control system, the atmospheric drag on a complex-shaped vehicle as it moves through an atmosphere with seemingly random winds, the behavior of a hurricane as it moves over a mountainous Caribbean island?
Well, just like the choice of coordinate system (rectangular or polar? fixed or rotating?), there are situations where a Newtonian analysis yields the equations of motion more easily, and situations where a Lagrangian or Hamiltonian analysis works better. In practice a Newtonian analysis is limited by the analyst's ability to guess the forces acting on the system elements, whereas the Lagrangianist (is that a word?) "merely" has to enumerate the energies.

Just as an example, consider a cylindrical tank of diameter D with rigid walls and a uniform flexible bottom, filled with water to a height H. Suppose the elastic parameters of the bottom are known, i.e. thickness, Young's modulus, Poisson ratio, boundary conditions, etc. Assuming the amplitude is small and therefore the resulting flow is a potential flow, what is the natural frequency of oscillation under gravity? In my view it is straightforward to write down the Lagrangian and to calculate the mass and spring terms (the coefficients of [itex]\frac{1}{2}\dot{x}^2[/itex] and [itex]\frac{1}{2}x^2[/itex] where x is the tank bottom coordinate). And though the problem can be solved in the Newtonian framework, one has to be fairly painstaking in computing and integrating the pressure distribution on the tank bottom, which varies with time and radial location. And since the water does not move at a uniform velocity, one has to be careful to compute the center-of-mass motion correctly. And in the Newtonian method one has to account for the distribution of inertia and elastic forces in the tank bottom, as well, while in the Lagrangian method one need merely know the elastic and kinetic energies of the bottom as a whole.

On the other hand, I've had to do force accounting for aeropropulsive interactions on supersonic aircraft. It's a bear to do correctly using Newtonian mechanics -- but I can't conceive of how one would do it using Lagrangian methods. Only by estimating the force first and then using the method of virtual work can this be hammered into the Lagrangian method.