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

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 Define: Angular Momentum L (about O):
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. Torque (moment of force) N (about O):
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. N = r  (dp/dt) = r  m(dv/dt)
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.  L = constant (time independent)
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 on the particle in moving it from (arbitrary) position 1 to (arbitrary) position 2 in space is defined 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 Energy U.
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 Energy of 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. Absolute potential energy has no meaning!
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 Forces F: 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) Energy of 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!)