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Physics+ Lagrangian, Noether and Momentum: UNIZOR.COM - Physics+ 4 All - Lagrangian

UNIZOR.COM - Creative Mind through Art of Mathematics Read full text of notes for this lecture on UNIZOR.COM - Physics+ 4 All - Lagrangian - Noether Theorem and Momentum Conservation Notes to a video lecture on UNIZOR.COM Noether's Theorem Symmetry and Momentum Conservation Let's introduce a concept of momentum in generalized coordinates. We are familiar with a vector of momentum in Euclidean three-dimensional space with Cartesian coordinates (x,y,z) for a point-mass m, as a vector with three components px(t) = m·x'(t) py(t) = m·y'(t) pz(t) = m·z'(t) Another approach, that uses the kinetic energy of this object T=½m·(x'²+y'²+z'²) would be to define px = ∂T/∂x' py = ∂T/∂y' pz = ∂T/∂z' Both definitions are equivalent, but the latter leads us to the third definition using the Lagrangian L=T−U instead of just kinetic energy T, because potential energy U does not depend on velocity: px = ∂L/∂x' py = ∂L/∂y' pz = ∂L/∂z' Since the Lagrangian of a mechanical system is usable in both Cartesian and non-Cartesian (generalized) coordinates, we can define a generalized momentum (p1,...,pn) as a set of partial derivatives of the Lagrangian L by corresponding component of generalized velocity (∂L(...)/q1',...,∂L(...)/qn'). The time-dependent function pk(t) = ∂L(...)/∂qk' is called the kth component of the generalized momentum. Consider a mechanical system with n degrees of freedom and its trajectory in generalized coordinates q(t) = (q1(t),...,qn(t)). Theorem If the Lagrangian of this system L(q1,...,qn,q1',...,qn',t) is invariant under translation of qk by infinitesimal value ε qk → qk + ε then the kth coordinate of the generalized momentum pk = ∂L/∂qk' is conserved.