Overview of our progress so far
We have seen how to describe various mechanical systems in terms of Lagrangians. It is straightforward to find the Lagrangian for any system consisting of point masses, rigid sticks, ropes, rolling wheels, etc. Given a Lagrangian, it is very easy to derive the equations of motion (the Euler-Lagrange equations. Solving these equations is a technical task that may be accomplished using computers. In principle, the theoretical description of mechanical systems is now complete.
There is one caveat: Mechanical systems involving the force of friction generally cannot be easily described by the Lagrangian formalism! The Lagrangian formalism includes only conservative forces, i.e. forces which are gradients of a potential. (A constraint can be seen as an idealization of a very steep potential that effectively prohibits motion in some directions but allows motion in other directions.) However, friction is not a conservative force because it usually depends on velocity and position in a nontrivial way. In physics, the force of friction is not considered a fundamental force, but rather a force arising out of interactions with a large number of particles in the environment. Thus, effects of friction can be derived, in principle, from a more fundamental picture that involves only conservative forces. Of course, in practice it is much more convenient to introduce the force of friction phenomenologically, i.e. by guessing or experimentally measuring the formula for the friction. One well-known formula is , where is the normal force and is the friction coefficient; this formula approximately describes dynamic friction on rough surfaces. Another known formula is , where is the velocity of a body moving through a medium and is the coefficient that usually depends on the velocity and on the shape of the body in some complicated way. This formula can be used for bodies moving through air or water, although one needs to measure the function in each case.
What remains: applications
You still need to learn some practical applications of this mathematical theory to various important cases. In each case, one can apply the general theory, use suitable mathematical techniques, and extract important physical consequences. Students of theoretical mechanics need to learn these mathematical techniques, as well as the accompanying notions and their physical interpretations. Here are the major areas of interest:
- Describing the motion of a point mass in a central field of force. The most important example is the Kepler problem, which corresponds to a force that decays as with distance. This setup describes, for instance, the motion of planets and comets around the sun. In this case, one can solve the equations of motion analytically and derive important properties of the motion in a central field, such as periodicity, orbit parameters, escape velocity, Kepler laws, perihelion precession, etc. The theory of the Kepler problem is the foundation for celestial mechanics.
- Describing the motion of a rigid body under external forces. A rigid body is a collection of a very large number of point masses, which are spread in space and constrained to remain at fixed distances from each other. So, a rigid body may move as a whole or rotate as a whole, but it cannot be squeezed or deformed in any way. The concept of a "rigid body" is very important because it is an idealization of the behavior of rigid things that we use in real life. The motion of a rigid body is, of course, much simpler than the set of all possible motions of its constituent particles. So it is very useful to develop a special formalism describing the possible motions of a rigid body. This formalism involves such concepts as the tensor of inertia, torque, angular momentum, and rolling without sliding. I would like to stress that these concepts are not new fundamental axioms of mechanics; these concepts can be derived from the standard Lagrangian for the system consisting of many massive particles that are spread in space and constrained to remain at fixed distances from each other.
- Describing small oscillations around a static configuration. For a mechanical system that has oscillating degrees of freedom around a static equilibrium position (these systems range from molecules to bridges), we make an approximation that the system has only very small deviations from that position. In the limit when these deviations are very small, it is usually possible to derive a linear equation of motion for these deviations. These equations of motion describe oscillations and can be analyzed to verify that the position of the system is stable against small deviations. (The analysis proceeds most conveniently in the Lagrangian formalism.) The key concepts in this area are normal modes, normal frequences, and stability.
- Describing elastic scattering of point masses. More generally, one considers a point mass moving in a potential, such that the force is appreciably nonzero only in a small portion of space. A typical problem is to describe the motion of a point mass that has a given velocity far away from the interaction area. The particle flies in, is deflected (scattered) by the potential, and flies away with a the same speed but in a slightly different direction. A typical experimental situation is when one has an initial beam of particles with slightly different positions. In that case, the interesting question is not to describe the precise trajectory of each particle, but to predict how many outgoing particles will fly in a particular direction. In other words, one wants to characterize the final (outgoing) states at infinity in terms of the initial (ingoing) states at infinity. The key notions in this area are the differential and the total scattering cross-section.
What remains: theoretical developments
Besides these applications, there are certain theoretical developments that enrich the Lagrangian formalism and provide essential foundations for other areas of theoretical physics. At least some of these theoretical developments are usually included in courses of theoretical mechanics, even though some of them do not have a direct application in the field of mechanics proper. We shall only study the most important of these developments:
- General properties of Lagrangian formalism: invariance under coordinate changes, equivalence of systems with different Lagrangians, motivations for using the action principle, equivalence of Lagrangians with higher-order derivatives and first-order derivatives, etc. These are more or less formal developments that deepen our understanding and clarify the structure of the theory but only rarely help solve particular problems. For instance, the invariance under coordinate changes is an important conceptual fact which justifies why we can choose arbitrary generalized coordinates.
- Hamiltonian formalism. This is a very important mathematical development of the Lagrangian formalism, where a mathematical trick is used to include velocities as independent variables into the Lagrangian, thus making the equations of motion first-order in time derivatives. The Hamiltonian formalism involves such notions as Legendre transformation, Poisson brackets, and symplectic geometry. The most important application of the Hamiltonian formalism is in quantum mechanics; within classical mechanics, it largely remains a mathematical game which only occasionally has an advantage over the Lagrangian formalism in solving mechanical problems, although it has far-reaching consequences for the formal development of physics. For instance, such features as the presence of constraints, integrability, and a transition to chaos is most naturally expressed using the Hamiltonian formalism. In the minimal standard course of mechanics, only some basic ideas of the Hamiltonian formalism are covered: the Poisson brackets, canonical transformations, and the Hamiltoin-Jacobi equation.
- Perturbation theory. This is a method to describe a system approximately, if that system is only a small deviation from another system which we can solve exactly. Several methods of perturbation theory are known; only the most basic ones (anharmonic oscillations) are studied in the minimal standard course of mechanics.
- Symmetries and conservation laws. The discovery that the existence of a group of symmetry transformations is equivalent to the existence of a conservation law has become a very important foundation of particle physics and field theory. Keywords include: symmetry groups, infinitesimal transformations (also called "generators"), the Noether theorems, Galilei and Lorentz invariance. Only a qualitative understanding is intended at this point.