Ivancevic_Applied-Diff-Geom

22 Applied Differential Geometry: A Modern Introduction1.1.5.3 Application: Lagrangian MechanicsRiemannian manifolds are natural stage for the Lagrangian mechanics,which is a re–formulation of classical mechanics introduced by Joseph LouisLagrange in 1788. In Lagrangian mechanics, the trajectory of an object isderived by finding the path which minimizes the action, a quantity whichis the integral of the Lagrangian over time. The Lagrangian for classicalmechanics L is taken to be the difference between the kinetic energy T andthe potential energy V , so L = T − V . This considerably simplifies manyphysical problems.For example, consider a bead on a hoop. If one were to calculate themotion of the bead using Newtonian mechanics, one would have a complicatedset of equations which would take into account the forces thatthe hoop exerts on the bead at each moment. The same problem usingLagrangian mechanics is much simpler. One looks at all the possible motionsthat the bead could take on the hoop and mathematically finds theone which minimizes the action. There are fewer equations since one isnot directly calculating the influence of the hoop on the bead at a givenmoment.Lagrange’s EquationsThe equations of motion in Lagrangian mechanics are Lagrange’s equations,also known as Euler–Lagrange equations. Below, we sketch out thederivation of Lagrange’s equation from Newton’s laws of motion (see nextchapter for details).Consider a single mechanical particle with mass m and position vector⃗r. The applied force, ⃗ F , can be expressed as the gradient (denoted ∇) of ascalar potential energy function V (⃗r, t):⃗F = −∇V.A contracted curvature tensor is called the Ricci tensor. It is a symmetric second–ordertensor given by:R ik = ∂Γl ik∂x l− ∂Γl il∂x k+ Γl ikΓ m lm − Γ m ilΓ l km.Its further contraction gives the Ricci scalar curvature, R = g ik R ik . The Einstein tensorG ik is defined in terms of the Ricci tensor R ik and the Ricci scalar R,G ik = R ik − 1 2 g ikR.