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Energy and Momentum LL2 Section 9 From mechanics, the momentum of a particle is given by v2 = va va For v << c, or c , p mv. If the force on the particle is perpendicular to the velocity, then the velocity changes only in direction, and If the force on the particle is in the same direction as the velocity, the velocity changes only in magnitude, and HW. (Note typos in new printing of book.) The ratio of force to acceleration, dp/dt : dv/dt, is different in the two cases. In classical mechanics, F/a = m. Energy Energy of a free particle does not go to zero as v 0. For v/c << 1 Classical expression Relativistic e is completely definite and positive. Classical e is defined only to within and arbitrary additive constant. It may be positive, zero, or negative. An atom. Composite body at rest: Rest energies of constituents + their kinetic energy + their interaction energies = M c2 Total energy of composite particle at rest, including internal kinetic energy and interactions. Sum of rest energies of parts, which does not include kinetic energy or interactions No conservation of mass in relativistic mechanics. Only conservation of energy (including rest energies) is valid. The relation between momentum p and energy E: = e2/c2 Hamiltonian For Usual classical expression Relation between This will be used a lot If Thus, if the particle cannot move at speed c. Particles with m = 0 can have velocity c. Then p = e/c. For ultra relativistic particles with mass HW The four dimensional form of the equation of motion The action for a free particle Principle of least action: Integrate by parts Variation is of trajectories for the same end points A free particle has constant 4-velocity Equation of motion for a free particle To find the momentum, we need S as a function of xi. We need the variation of S with respect to a change in the coordinates. We had Now consider the actual trajectory, so Take the initial point to be fixed Allow the final point to vary Momentum 4-vector From Mechanics 4-gradient of S is = (e/c, -p) Momentum & energy are components of a single 4-vector. Transformation formulas follow immediately (6.1) Square of 4-momentum (7.3) Force 4-vector For free particle, force = 0 We already had 4-acceleration Relativistic Hamilton-Jacobi Equation of a free particle From mechanics Transition to classical mechanics