Hamilton’s Principle
Lagrangian & Hamiltonian Dynamics
• Newton’s 2nd Law: F = (dp/dt)
– This is a 100% correct description of particle motion
in an Inertial Frame.  It will give the correct
differential Equations of Motion. When solved (for
given initial conditions) we will get r(t) & v(t).
– For relatively simple motion & in rectangular coordinates,
these equations of motion are relatively simple.
– For more complicated motion & if we use non-rectangular
coordinates, the equations of motion can become
complicated & difficult to deal with.
• e.g., Particle motion on a spherical surface: Recall the Ch.
1 homework on acceleration in spherical coordinates!
Introduction
• Newton’s 2nd Law: F = (dp/dt)
– Consider, for example, particle motion on a surface. 
Forces exist (Forces of Constraint) which keep the particle
in contact with the surface (“Normal Forces”).
– For a smooth, horizontal surface these constraint forces are
simple: Fc = N = - mg
– However, for something more complicated, say, a bead
sliding down a wire: Fc = something very complicated!
Perhaps even impossible to calculate!
2nd Law: F = (dp/dt) F  Ftotal
 To use Newton’s 2nd Law directly, we must know all
forces acting on the particle exactly.
 Sometimes Newton’s 2nd Law is impractical!
Philosophy & History
• To overcome such practical difficulties, several alternate
procedures were developed
Hamilton’s Principle 
Lagrangian Dynamics & Hamiltonian Dynamics
These are the basis of much of the modern theory of matter!
• All such procedures obtain equations of motion which are
100% equivalent to those from Newton’s 2nd Law: F = (dp/dt)
 These alternate procedures are NOT new theories!
They are reformulations of Newtonian Mechanics in a
different mathematical language that sometimes might be more
convenient than a direct application of F = (dp/dt).
• Hamilton’s Principle: Is also applicable outside of particle
mechanics. For example, to fields in E&M.
Hamilton’s Principle (HP):
– Based on experiment!
Philosophical Discussion (again!)
– HP:
–
–
–
–
–
Leads to no new physical theories, only new
formulations of old theories!
HP
can be used to unify several theories: Mechanics,
E&M, Optics, …
HP:
Very elegant & far reaching
HP:
“More fundamental” than Newton’s Laws??!!
HP:
Is given as a (single, simple) postulate.
In the following, we consider conservative systems only.
• HP & Lagrange’s Equations can be extended to nonconservative systems (see Graduate Texts!)
Hamilton’s Principle
Sect. 7.2 Philosophy & History
• There are several “Minimal” Principles in Physics.
– These assume that Nature always minimizes certain
quantities when physical processes take place
– These are very common in the history of physics
• Brief list of others:
– Hero, 200 BC: Optics: Hero’s Principle of Least
Distance: A light ray traveling from one point to another
by reflection from a plane mirror always takes the shortest
path. By geometric construction:
 Law of Reflection: θi = θr
Says nothing about the Law of Refraction!
“Minimal” Principles Continued
– Fermat, 1657: Optics: Fermat’s Principle of Least
Time: A light ray travels in a medium from one point
to another by a path that takes the least time.
Law of Reflection: θi = θr
 Law of Refraction: “Snell’s Law” (Prob. 6.7) n1sinθ1 = n2sinθ2
– Maupertuis, 1747: Mechanics: Maupertuis’s
Principle of Least Action: Dynamical motion takes
place with minimum action
• Action  (Distance)  (Momentum) = (Energy)  (Time)
• Based on Theological Grounds! (???!!!!)
• Lagrange: Put this on a firm mathematical foundation.
• Principle of Least Action  Hamilton’s Principle
Hamilton’s Principle (1834-35)
– Of all of the possible paths of a mechanical
system, the path actually followed is the one
which minimizes the time integral of the
difference in the kinetic & potential energies.
That is, the actual path is the one which makes
the variation of the following integral vanish:
δ∫[T - U] dt = 0
(limits t1 < t < t2)
δ  the arbitrary variation, just discussed in Ch. 6!
The δ of an integral is similar to the derivative of a function.
See sections 6.3 & 6.7!
Hamilton’s Principle
δ∫[T - U] dt = 0
(limits t1 < t < t2)
(1)
Here: T = T(xi) & U = U(xi), (functions of all xi & (dxi/dt)!)
– N particles in 3d,
i = 1,2, ..(3N)
• Define: The Lagrangian L
L  T(xi) - U(xi) = L(xi,xi)
(1)  δ∫Ldt = 0
(2)
• This is identical to the abstract calculus of variations problem
of Ch. 6 with the replacements:
δJ  δ∫Ldt , x  t , yi(x)  xi(t)
yi(x)  (dxi(t)/dt) = xi(t), f[yi(x),yi(x);x]  L(xi,xi;t)
 The Lagrangian satisfies Euler’s eqtns with these replacements!
δ∫Ldt = 0 (limits t1 < t < t2)
(2)
 The Lagrangian L = L(xi,xi) satisfies
Euler’s equations!
Or, with the changes noted:
(L/xi) - (d/dt)[(L/xi)] = 0
(3)
N particles in 3d: i = 1,2, ..(3N)
 Lagrange’s Equations of Motion
• (3) enables us to get differential equations of motion
without (explicitly) using Newton’s 2nd Law & without
(explicitly) needing to calculate forces! It is a recipe
for getting equations of motion from ENERGY!
Lagrange’s Equations
• Lagrange Equations are most useful in situations where
direct application of Newton’s 2nd Law is difficult or
impossible!
• Recipe for solution using the Lagrangian formalism:
– Step 1: Compute the PE in terms of the xi; U = U(xi). Compute
the KE in terms of the xi = (dxi/dt). T=T(xi). Form the
Lagrangian: L  T(xi) - U(xi) = L(xi,xi)
– Step 2: For each xi, xi; Obtain the Equation of Motion using:
(L/xi) - (d/dt)[(L/xi)] = 0
N particles in 3d: i = 1,2, ..(3N)
• First, we’ll do examples using Lagrange’s Equations in problems where
Newton’s 2nd Law is relatively easy to apply & where we (may) already
know the solution. Later, we’ll do many examples where it would be
difficult or impossible to apply Newton’s 2nd Law.
Simple Example #1
• The 1d simple harmonic oscillator:
– We already know the solution! We’re doing it just to
illustrate the Lagrangian formalism!
– Step 1: Compute U & T: L = T - U
U = (½)kx2, T = (½)mx2, L = T - U = (½)mx2 - (½)kx2
– Step 2: Obtain the Lagrange Equation of motion using:
(L/x) - (d/dt)[(L/x)] = 0
(L/x) = -kx; (L/x) = mx, (d/dt)[(L/x)] = mx

mx + kx = 0 Or x + (k/m)x = 0
Or
x + (ω0)2x = 0
Nothing new! Just used to illustrate the method!
Simple Example #2
• The Plane Pendulum:
– We already know the solution! We do it just to illustrate the
Lagrangian formalism.
– Step 1: Compute U, T; L = T-U
U = mg(1-cosθ), T = (½)m2θ2, L = T - U = (½)m2θ2 - mg(1-cosθ)
– Step 2: Obtain the Lagrange Equation of motion. Treat θ
as if it were a rectangular coordinate! Justification: Next
section! In the formalism, make the replacement x  θ
 the Lagrange Equation is
(L/θ) - (d/dt)[(L/θ)] = 0
(L/θ) = - mgsinθ; (L/θ) = m2θ; (d/dt)[(L/θ)] = m2θ
 m2θ + mgsinθ = 0 Or: θ + (g/)sinθ = 0
Nothing new! Just used to illustrate the method!
Remarks on These 2 Simple examples
• Plane Pendulum:
– We treated θ AS IF it were a rectangular coordinate x!
• Justification: Next section!
– We obtained the known equation of motion!
 Lagrange’s Equations are more general than the
derivation in rectangular coordinates indicates!
• Both Examples:
– Obtained known results using the Lagrange formalism.
– No statements were made regarding forces & there was no
(explicit) calculation of forces!
– Instead of forces (properties of the interaction between the particle &
its environment) we used ENERGY (properties of the particle itself).
 Can get eqtns of motion for a particle using only energy!
Generalized Coordinates
Section 7.3
• The plane pendulum example:
– Showed that we can use the Lagrangian
formalism on coordinates which are
not rectangular!
 This suggests that we can choose any
(complete set) of coordinates we
want to do the problem & then use
Lagrange’s Equations.
– This section is the justification of this.
• Consider a general mechanical system with n
discrete point particles (these might be connected to form
rigid bodies, Ch11). Arbitrary origin.
– The state of the system at time t is specified by
the n position vectors rn(t).
3d  We need 3n coordinates to describe the system.
– Possibly, constraints exist which make the number of
independent coordinates < 3n.
Suppose there are m equations of constraint which
connect some of the 3n coordinates to some others.
 The number of degrees of freedom is s = 3n - m
• For a general mechanical system with
s = 3n - m degrees of freedom
• We needn’t necessarily choose s rectangular
coordinates to describe the system. We are free to choose
any set of s coordinates which completely describes the state
of motion of the system. Depending on the problem:
– We could choose s curvilinear (spherical or cylindrical)
coordinates. Or: We could choose a mixture of rectangular
coordinates (k = # rectangular coordinates) & curvilinear
coordinates (s - k = # curvilinear coords)
– The s coordinates needn’t have units of length! They
could be dimensionless or they could have (almost) any
units.
• Generalized Coordinates  Any set of s quantities
which completely specifies the state of the motion of
the system (for a system with s degrees of freedom).
• Standard Notation: q1, q2, q3, …Or: qj (j = 1,2,… s)
• Proper Set of Generalized Coordinates  Any set
of coordinates whose number = s (# degrees of
freedom) and are not restricted by constraints.
– Sometimes, it might be useful to use generalized
coordinates whose number is > s & to explicitly take into
account constraints using Lagrange multipliers (as in Ch. 6)
– We’ll see: We need to do this to calculate forces of
constraint.
• Note: The choice of the set of s generalized coordinates to
solve a problem is not unique. It depends on the problem &
on personal taste!
• Example: Consider a disk
rolling down an inclined plane .
We could use z = height of the
disk center of mass above the

reference level and y = linear
z
distance rolled down the plane

or z and the angle θ (radians) =
the angular distance rotated. For a disk of radius R, the
equation of constraint is y = Rθ.  y & θ aren’t independent!
• In addition to the generalized coordinates qj, in the Lagrangian
formalism, we also have
Generalized Velocities  (dqj/dt)  qj
• Suppose we start in rectangular coordinates & transform to a
set generalized coordinates. In general, this involves
equations of the form:
xα,i = xα,i (q1,q2,q3,..t) or xα,i = xα,i (qj,t)
α, = 1,2,3 (rectangular indices) i = 1,2,3,…, n; j = 1,2, 3,…,s
• Also, there are velocity transformations: xα,i = xα,i (qj,qj,t)
– The velocities in rectangular coordinates can depend on both the
generalized coordinates & the generalized velocities
• There also might be equations of constraint:
fk(xα,i ,t) = 0, k = 1,2, .. m
• For example, if we choose spherical
coordinates as the generalized coordinates:
x1 = x = r sinθ cosφ, x2 = y = r sinθ sinφ
x3 = z = r cosθ
q1 = r, q2 = θ, q3 = φ, q1 = r, q2 = θ, q3 = φ
x1 = (dx/dt)
= r sinθ cosφ + rθ cosθ cosφ - rφ sinθ sinφ
x2 = (dy/dt)
= r sinθ sinφ + rθ cosθ sinφ + rφ sinθ cosφ
x3 = (dz/dt) = r cosθ - rθ sinθ
xα,i = xα,i (qj,qj)
Example 7.1
• Find a proper set of generalized coordinates for a point particle
moving on the surface of a hemisphere of radius R whose center is at
the origin. Constraint eqtn (motion on the surface of a hemisphere):
x2 + y2 + z2 - R2 = 0 , z  0 (1)
• Following the text, we choose the cosines of the angles between the x,
y, & z axes & a line connecting the particle with the origin:
q1 = (x/R) , q2 = (y/R) , q3 = (z/R)
 (1) becomes:
(q1)2 +(q2)2 + (q3)2 = 1
(2)
• Because of (1), q1, q2, q3 aren’t independent  They aren’t a proper
set of generalized coordinates! ((2) is a constraint eqtn.) For a proper set,
we need only choose 2 of the 3 qj: All that’s needed on a sphere’s surface!
• In general, we can always use the constraint equations to reduce the
number of generalized coordinates needed in a problem!
Example 7.2
• Use the (x,y) coordinate system in
the figure to find the KE T, the PE
U, & the Lagrangian L for a simple
pendulum (length , bob mass m),
moving in the xy plane. Determine
the transformation equations from
the (x,y) system to the coordinate θ.
Find the equation of motion.
• Worked on the board!
Configuration Space
• The state of a system of n particles & subject to m constraints
connecting some of the 3n rectangular coordinates is
completely specified by s = 3n – m generalized coordinates.
 Sometimes its convenient to represent the state of such a
system by a point in an abstract s-dimensional space called
CONFIGURATION SPACE. Each dimension in this space
corresponds to one of the coordinates qj. This point specifies
the CONFIGURATION of the system at a particular time. As
the qj change in time (governed by the eqtns of motion) this
point traces out a curve in configuration space. The exact
curve depends on the initial conditions. Often we speak of
“the path” of the system as it “moves” in configuration space.
• Obviously, this is NOT the same as the particle path as it
moves in ordinary 3d space!