Conservation Theorems: Sect. 2.5
• Discussion of conservation of
Single Particle Only!
– Linear Momentum
– Angular Momentum
– Total (Mechanical) Energy
• Not new Laws! Direct consequences of
Newton’s Laws!
“Conserved”  “A constant”
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.
• 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.
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)
• 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!
• 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!
N = dL/dt
• If no torques act on the particle,
N=0


L = constant
dL/dt = 0
(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!
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}
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.
• 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
If F is conservative,
W12 is the same for
paths a,b,c, & all others!
Conservative Force.
• For conservative forces, one can define a
Potential Energy U.
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.
• 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!
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.
• 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)
• 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
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.
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!
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!)