observer1 said:
Why should I learn calculus on manifolds (ConM)?
What impact do they have in Lagrange's equation for a dynamical system?
What can ConM do that my “normal” calculus cannot?
I've been studying differential geometry for 3-4 years now, and I can count on one hand the number of times it's been essential in solving a practical problem in physics or engineering. The examples that come to mind are the Foucault pendulum (a basic grasp of Lie theory allows you to transform to the rotating frame at the initial point), calculating the precession of mercury (the metric tensor showed up in the first few lines, that's it), and making sense of time evolution operators and operator algebra in quantum mechanics (another application of basic Lie theory). There are far more problems that fully-general differential geometry
can be applied to than it
must be applied to. My advice for problem solving is to avoid the general mathematical theories of differential geometry like the plague wherever possible.
The tell-tale sign of a problem that necessitates the use of methods beyond the scope of calculus on Euclidean space is the absence of any global coordinate chart. These problems are truly rare and extremely messy. In physics, I've only encountered them in problems involving the use of non-inertial or gravitational frames or an interaction picture. In applied mathematical problems, I've also run into them in the analysis of symmetries of differential and algebraic equations and variational/optimization problems when you look at solution in a neighborhood of a point without any prior knowledge of the shape of the hypersurface that solves your problem. For example, suppose I wanted to know the radiation field produced by a thin rotating dipole antenna. I can use a Lie group created by the rotating frame of the antenna to connect a surface element of a wavefront to an inertial frame, in which I know exactly how the radiation field will evolve using Green's functions.
Noether's theorem in Lagrangian mechanics is based on Lie theory. In theory, all of variational calculus is applied Lie theory, but that doesn't make it any more powerful than it was when the Bernoullis, Euler, Fermat, Lagrange, Hamilton, and others developed the subject well before Lie's publications. I single out Noether's theorem in particular because it involves the direct use of symmetry generators to demonstrate.
The Bill said:
The phase space for the Hamiltonian formulation of a system in classical mechanics with a finite number of degrees of freedom can be viewed as a symplectic manifold. The techniques of differential geometry, including what is classed as calculus on manifolds, can then be used to examine the invariants and phase portraits of the system.
I would point out that the whole point of Hamiltonian mechanics is to provide you with a Poisson bracket so you don't have to deal with the full differential geometric analysis of invariants over jet bundles. For instance, if you want to find invariants in a system in Lagrangian mechanics, you have to take a first-order partial differential operator on the tangent bundle of position space (not a vector field in the tangent bundle, a vector field ON the whole tangent bundle). This requires some algebraic heavy lifting, as their are intricate expressions constraining differential operators on the tangent bundle obtained by prolonging transformations on position space. The Poisson bracket is a convenient short-cut to the analysis of invariants that really depends on nothing but the chain rule on phase space and Hamilton's equations. The development of the fully general Poisson bracket is demonstrated without reference to differential geometry (except for a passing comment in retrospect about local contact transformations at the end) in the second section of Dirac's Lectures on Quantum Mechanics.
observer1 said:
Why do forms matter?
How do form really help do math on the tangent space?
As stated above, the general theory of differential forms isn't necessary until you start talking about a hypersurface that doesn't have any global coordinate charts, which forces you to state your problem locally (i.e., your problem becomes a family of subproblems each restricted to the domain of a certain local coordinate chart). In the problem I stated above about the thin rotating dipole antenna, the "subproblems" are the radiation field patterns around each surface element of the wavefront in question. You know the shape of the wavefronts of the radiation pattern in the inertial frame from integrating the Green's functions over the length of the antenna. Surface elements of the wavefront (which themselves are very simplistic differential 2-forms) can be "pulled back" to more complicated locally defined differential forms in the rotating frame via the inverse of the rotation transformation that took us to the inertial frame in the first place. That is a case where the full theory of differential forms would mean something more complicated than you'd see in ordinary calculus on Euclidean space, because you're working in more than three dimensions on a geometric shape you pretend to know nothing about except that it's locally related to the inertial frame by a transformation that varies from point to point.
Forms do math on the tangent space in the same way the a surface element acts on an electric field in Gauss's law. They enable you to evaluate fluxes through hypersurfaces by integrating vector fields and higher dimensional "geometric flux densities." In fancy lingo, these flux densities are contravariant tensor densities, such as the electromagnetic stress-energy tensor in relativistic electrodynamics (##T^{\mu\nu}##), which is like a 4-dimensional generalization of the Maxwell stress tensor in 3-dimensional electrodynamics, or the Faraday electromagnetic field tensor (##F^{\mu\nu}##), which is like the 4-dimensional analogue of the electric flux density (##\vec{D}##) and magnetic flux density (##\vec{B}##) combined. If the basis differential forms are n-1 dimensional surface elements and parallelotopes formed with parallel families of surface elements for edges (##dx \wedge dy## is like taking the flux of a field through level sets of x followed by taking the flux through level sets of y on the boundary of an infinitesimal rectangle with edge of length |dx| and |dy|), then the action of these "generalized oriented differential parallelotopes" on a same-dimensional flux density is the flux of said field around the parallelotope.
observer1 said:
What is the relationship between forms and vectors?
If vector fields represent slope fields on your space, then 1-forms represent a field of infinitesimal flat hyperplanes that appear locally parallel to each other and everywhere normal to the vector field to which they are dual. If you happen to have a metric tensor (which is guaranteed on smooth manifolds of finite dimension), then duality happens to coincide with index raising/lowering operations because of the wonders of doing linear algebra in an inner product space. Tensorial acrobatics aside, the relationship between forms and vectors is that of orthogonal complement (which as linear algebra operation in flat space would be expressed as the transpose). This is why surface integrals in 3D Euclidean space can be represented as a magnitude times a normal vector.
observer1 said:
Why is it that all treatments on this also treat Lie Algebra and Groups?
Lie groups are just smooth groups, so where is the manifold.?
Lie algebras and Lie groups enjoy a prominent role in differential geometric theory because on any space, curved or flat, (of finite dimension, as far as I know), every vector field is an infinitesimal transformation. More formally, the set of all vector fields in the tangent bundle of any smooth manifold (of finite dimension) belongs to an infinite dimensional Lie algebra. The corresponding one-parameter Lie group associated with a given vector field is the group of translations along integral curves of that field. It turns out you can't do or prove much of anything without invoking this particular Lie algebra. The basic reason for this is that this Lie group, of translations along curves, is what connects geometric points to their neighboring points without leaving the manifold; it represents motion restricted to the manifold, at least locally. How would you even begin to talk about tangent spaces, let alone metrics, if you couldn't connect points? (If you have a local transformation group connecting points in a neighborhood, then the differential, or "push-forward" of that transformation connects tangent planes of these points).
Smoothness doesn't mean anything if it isn't on a manifold. Globally-defined Lie groups (as opposed to local Lie groups, which are much less common in modern treatments but what you're most likely to come across in a problem that needs Lie theory to solve) are usually written in terms of parameters. For example, SO(2) is written in terms of the parameter ##\theta## which identifies each element of SO(2) with an element of the unit circle and gives SO(2) the same smooth structure as the unit circle. SO(3) has the same smooth structure as the real projective space ##\mathbb{RP}^{3}##. SU(2) has the same structure as the 3-sphere. In other words, if I pick a point on the 3-sphere (any point), I also pick a transformation in SU(2). Any calculus-ish statement about one can be taken as an identical statement about the other. The operation of group multiplication, the differentials of multiplication by elements, the Lie algebra, and anything else about the structure of SU(2) can be taken to mean something geometric on the 3-sphere.
I apologize for the lateness of the post. I put this up anyways because I felt this could be helpful to the OP and many others studying the field.
