Conservation Theorems: Sect. 2.5

2. Conservation Theorems:Sect. 2.5 Single Particle Only! • Discussion of conservation of • Linear Momentum • Angular Momentum • Total (Mechanical) Energy • Not new Laws! Direct consequences of Newton’s Laws! “Conserved”  “A constant”

3. Linear Momentum • The total linear momentum p = mv of a particle is conserved when the total force acting on it is zero. • Proof: Start with Newton’s 2nd Law: F = (dp/dt) (1) If F = 0 (1)  (dp/dt) = 0  p = constant (2) (time independent) • (2) is a vector relation. It applies component by component.

4. An alternative formulation: Let s a constant vector such that the component of the total force F along the s direction vanishes.  Fs = 0 Newton’s 2nd Law  (dp/dt)s = 0  ps = constant The component of linear momentum in a direction in which the force vanishes is a constant in time.

5. Angular Momentum • Consider an arbitrary coordinate system. Origin O. Mass m, position r, velocity v (momentum p = mv). Define: Angular Momentum L(about O): L  r  p = r  (mv)

6. Consider an arbitrary coordinate system. Origin O. Mass m, position r, velocity v (momentum p = mv). Define: Torque(moment of force) N(about O): N  r  F = r  (dp/dt) = r  [d(mv)/dt]  Newton’s 2nd Law!

7. So: L = r  p, N = r  F • From Newton’s 2nd Law: N = r  (dp/dt) = r  m(dv/dt) • Consider the time derivative of L: (dL/dt) = d(r  p)/dt = (dr/dt)  p + r  (dp/dt) = v  mv + r  (dp/dt) = 0 + N or:(dL/dt) = N This is Newton’s 2nd Law - Rotational motion version!

8. N = dL/dt • If no torques act on the particle, N = 0  dL/dt = 0  L = constant (time independent) • The total angular momentum L = r  mv of a particle is conserved when the total torque acting on it is zero. • Reminder:Choice of origin is arbitrary! A careful choice can save effort in solving a problem!

9. Work & Energy • Definition:A particle is acted on by a total force F. The Work done onthe particle in moving it from (arbitrary) position 1 to (arbitrary) position 2 in space isdefined as line integral (limits from 1 to 2): W12  ∫ Fdr • By Newton’s 2nd Law (using chain rule of differentiation): Fdr = (dp/dt)(dr/dt) dt = m(dv/dt)v dt = (½)m [d(vv)/dt] dt = (½)m (dv2/dt) dt = [d{(½)mv2}/dt] dt = d[(½)mv2]  W12 = ∫[d{(½)mv2}/dt] dt = ∫d{(½)mv2}

10. Kinetic Energy  W12 = ∫d{(½)mv2} = (½)m(v2)2 – (½) m(v1)2 • Defining the Kinetic Energy of the particle: T  (½)mv2, This becomes: W12 = T2 -T1 = T  The work done by the total force on a particle is equal to the change in the particle’s kinetic energy.  The Work-Energy Theorem.

11. Look at the work integral: W12 = ∫Fdr • Often, the work done by F in going from 1 to 2 is independent of the path taken from 1 to 2: • In such cases, F is said to be a Conservative Force. • For conservative forces, one can define a Potential EnergyU. If F is conservative, W12 is the same for paths a,b,c, & all others!

12. Potential Energy • For conservative forces,(& only for conservative forces!) we define a Potential Energy function U(r). By definition: W12  ∫Fdr  U1 - U2  - U • The Potential Energyof a particle = its capacity to do work(conservative forces only!).The work done in moving the particle from 1 to 2 = - change in potential energy.

13. For conservative forces ∫Fdr = U1 - U2  - U (1) • Math: This can be satisfied if & only if the force has the form:F  - U (2) • (1) & (2) will hold if U = U(r,t) only (& not for U = U(r,v,t)) • Note: potential energy is defined only to within an additive constant because force = derivative of potential energy! Absolute potential energy has no meaning! • Similarly, because the velocity of a particle changes from one inertial frame to another, absolute kinetic energy has no meaning!

14. Mechanical Energy Conservation • In general: W12 = ∫Fdr • KE Definition: T  (½)mv2 • Work-Energy Theorem: W12 = T2 -T1 = T • Conservative ForcesF: W12= U1 - U2 = - U • Combining gives: T = - U or T2 -T1 = U1 - U2 or T + U = 0 or T1 + U1 = T2 + U2  For conservative forces, the sum of the kinetic and potential energies of a particle is a constant.T + U = constant Conservation of kinetic plus potential energy.

15. Define:Total (Mechanical) Energyof a particle (in the presence of conservative forces): E  T + U • We just showed that (for conservative forces) the total mechanical energy is conserved (const., time independent). • Can show this another way. Consider the total time derivative: (dE/dt) = (dT/dt) + (dU/dt) • From previous results: dT = d[(½)mv2] = Fdr  (dT/dt) = F(dr/dt) = Fr = Fv • Also (assuming U = U(r,t) & using chain rule) (dU/dt) = ∑i (∂U/∂xi)(dxi/dt) + (∂U/∂t)

16. Rewriting (using the definition of ): (dU/dt) = Uv + (∂U/∂t) • So: (dE/dt) = (dT/dt) + (dU/dt) or (dE/dt) = Fv + Uv + (∂U/∂t) • But, F = - U  The first 2 terms cancel & we have: (dE/dt) = (∂U/∂t) • If U is not an explicit function of time, (∂U/∂t) = 0 = (dE/dt)  E = T + U = constant

17. SUMMARY The total mechanical energy E of a particle in a conservative force field is a constant in time. • Note:We have not proven conservation laws! We have derived them using Newton’s Laws under certain conditions. They aren’t new Laws, but just Newton’s Laws in a different language.

18. Conservation Laws • Brief Philosophical Discussion(p. 81): Conservation “postulates” rather than “Laws”? Physicists now insist that any physical theory satisfy conservation laws in order for it to be valid. • For example, introduce different kinds of energy into a theory to ensure that conservation of energy holds.(e.g., Energy in EM field). • Consistent with experimental facts!

19. Conservation Laws & Symmetry Principles(not in text!) In all of physics (not just mechanics) it can be shown: • Each Conservation Law implies an underlying symmetry of the system. • Conversely, each system symmetry implies a Conservation Law:Can show: Translational Symmetry  Linear Momentum Conservation Rotational Symmetry  Angular Momentum Conservation Time Reversal Symmetry  Energy Conservation Inversion Symmetry  Parity Conservation (Parity is a concept in QM!)