Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 65For example, in standard spherical coordinates x i = {ρ, θ, ϕ}, we havethe components of the acceleration vector given by (2.11), if we now reinterpretoverdot as the time derivative,a ρ = ¨ρ − ρ˙θ 2 − ρ cos 2 θ ˙ϕ 2 , a θ = ¨θ + 2 ρ ˙ρ ˙ϕ + sin θ cos θ ˙ϕ2 ,a ϕ = ¨ϕ + 2 ρ ˙ρ ˙ϕ − 2 tan θ ˙θ ˙ϕ.Now, using (2.12), the Newton’s fundamental equation of motion, thatis the basis of all science, F = m a, gets the following tensorial formF i = ma i = m ˙¯v i = m(v i ;kẋ k ) ≡ m( ˙v i + Γ i jkv j v k ) = m(ẍ i + Γ i jkẋ j ẋ k ),(2.13)which defines Newtonian force as a contravariant vector.However, modern Hamiltonian dynamics reminds us that: (i) Newton’sown force definition was not really F = m a, but rather F = ṗ, wherep is the system’s momentum, and (ii) the momentum p is not really avector, but rather a dual quantity, a differential one–form 3 . Consequently,the force, as its time derivative, is also a one–form (see Figure 2.1; also,compare with Figure Figure 5.2 above). This new force definition includesthe precise definition of the mass distribution within the system, by meansof its Riemannian metric tensor g ij . Thus, (2.13) has to be modified asF i = mg ij a j ≡ mg ij ( ˙v j + Γ j ik vi v k ) = mg ij (ẍ j + Γ j ikẋi ẋ k ), (2.14)where the quantity mg ij is called the material metric tensor, or inertiamatrix. Equation (2.14) generalizes the notion of the Newtonian force F,from Euclidean space R n to the Riemannian manifold M.2.1.4 Application: Covariant MechanicsRecall that a material system is regarded from the dynamical standpointas a collection of particles which are subject to interconnections and constraintsof various kinds (e.g., a rigid body is regarded as a number ofparticles rigidly connected together so as to remain at invariable distancesfrom each other). The number of independent coordinates which determinethe configuration of a dynamical system completely is called the number ofdegrees of freedom (DOF) of the system. In other words, this number, n,3 For example, in Dirac’s < bra|ket > formalism, kets are vectors, while bras areone–forms; in matrix notation, columns are vectors, while rows are one–forms.