As developed by Leibniz, D’Alembert, Lagrange, Hamilton and others, motion can be analyzed in terms of energy rather than force.

A key concept is Hamilton’s variational principle, which states that J = ò L dt has a minimum (or maximum) value, so that varying J by changing any parameter which affects the history of the particle results in d J = 0 .

L is called the Lagrangian and is T – V , T and V being the kinetic and potential energies.

The integral is carried out over the path taken by the moving object.

Applied to a free-falling particle of mass m, L = (1/2)m(y’)² – mgy (where y’= dy/dt)

d J = d [ò [(1/2)m(y’)² – mgy] dt] = ò [m(y’) d (y’) – mg d y] dt

= ò [m(y’)(d/dt) d y – mg d y] dt = ò [-m(y’’)– mg] d y dt

which could have any value (other than the true value = 0) unless [-m(y’’) – mg] = 0

Therefore (y’’) = - g as expected.

Note that this derivation makes use of differentiation under the integral sign, integration by parts, and d y’ =d/dt(d y).

It can be shown (pp.394-397) that other solutions for y(t), y’(t) or y’’(t) yield larger values of ò L dt .

Generalized coordinates q_i are any set of independent variables which just suffice to specify (uniquely) the state or configuration of a system, leading to an easy interpretation of its motion.

They may include Cartesian coordinates (x,y,z) or velocities (x’,y’,z’)

However, the q’s should independent, not connected by conditions of constraint, as for example when two particles held a fixed distance apart.

The number of generalized coordinates required is the number of degrees of freedom of the system. For a system of N particles this number is n = 3N – m , where m is the number of holonomic constraints (equalities which are integrable).

If the particle is confined to a surface (plane or otherwise), the number of coordinates is reduced from 3 to 2. The condition of rolling, which is non-holonomic, does not further reduce the number, as explained on p.399. When the body rolls, at least two coordinates must change.

But for a rigid body which does not rotate, the coordinates x_i, y_i and z_i of the component particles are each fixed relative to each other, leaving only 3 degrees of freedom (3N-3 constraints), so the body can be treated as if it were a particle.

Typically, the kinetic energy T depends on the square of velocity coordinates, while the potential energy V depends directly on space coordinates.

For a pointlike mass m attached to a sliding mass M by a string of length r (Fig.10.3.1 on p.400), there are 3 ´ 2 = 6 spatial coordinates (x,y,z), (X,Y,Z) but four holonomic constraints:

z = Z = 0 =Y and (x-X)² + y² = r²

and therefore two degrees of freedom.

If we express T and V in terms of Cartesian coordinates:

T = (1/2)M(X’)² + (1/2)m[(x’)² + (y’)²]

V = mgy (negative in this case)

we have four coordinates, which clearly must be related.

Transforming to spherical coordinates (r, q ), using:

x = X – r sinq , y = - r cosq , x’ = X’ + r q ’cosq , y = - r q ’sinq

gives Eqs.(10.3.4) on p.401, involving only X’ and q

(q ’ and q count as one generalized coordinate).

Although T and V are scalar quantities, vector displacements and velocities are often useful, since we can express T as ½mv.v .

Expressing Hamilton’s principle in terms of generalized coordinates and repeating the arguments of Section 10.1 leads to the Lagrange equations of motion:

d L/d q_i – [d/dt][d L/d q_i’] = 0 for each generalized coordinate i.

Hints for applying these equations are given on p.404. They are best understood by taking examples.

Example 10.5.1: 1-D Harmonic Oscillator (body of mass m, force constant k)

Linear motion implies a single generalized coordinate, x (i=1).

L = T – V = ½ m(x’)² – [-ò (-kx) dx] = ½ m(x’)² - ½ k(x)²

d L/d x = - kx,

[d L/d (x)’] = mx’

d L/d x – [d/dt][d L/d x’] = - kx – mx’’

so the Lagrange equation gives: kx + mx’’ = 0 , as obtained from Newton’s law.

Example 10.5.1: Particle held in orbit by a central force: f(r) = -V(r)

The analysis (p.405) shows that angular momentum is conserved

and that mr’’ = mr(q ’)² + f(r) , as derived on p.212, Eq.(6.5.2a)

Example 10.5.3: Attwood machine

x is measured downwards from the center of the pulley.

The string provides a holonomic constraint which links x₁ and x₂.

d L/d x’ = (m₁ + m₂ + I/a²) x’, giving d/dt[d L/d x’] = (m₁ + m₂ + I/a²) x’’

The answer shows that the pulley is equivalent to an inertia mass of I/a²

= M/2 for a uniform disk (where M is the pulley mass)

Example 10.5.4: Double Attwood machine

Position of right-hand pulley is x_p = l + p a – x ; p a can be omitted since it contributes to L only a constant term which disappears upon taking the derivative (see Example 10.5.4).

Making similar omission:

x₁ = x

x₂ = l - x + x’

x₃ = (l - x) + (l’ – x’)

There are two Lagrange equations, differential in dx/dt and dx’/dt, since there are two degrees of freedom (x and x’).

Example 10.5.6 Mass sliding down a sliding inclined plane

As an alternative to using vector notation for v and V, can get the expression for T_m by using the cosine rule in the vector triangle in Fig. 10.5.3.

L is independent of x, so d L/d x = 0

Again, two degrees of freedom, therefore two Lagrange equations.

One of these is equivalent to conservation of linear momentum.

Problem 10.3 (p.430). Uniform solid sphere rolling down an inclined fixed plane.

Choose generalized coordinate x = linear displacement down inclined plane.

Second coordinate f = rotation of sphere , but linked to x by:

angular velocity of sphere = df /dt = x’/a [constraint]

L= (7/10) (mx’)² + mgx sinq if V=0=x at initial position of sphere

Lagrange equations give: x’’ = (5/7) g sinq

Problem 10.4 (p.431): connected blocks

Single degree of freedom, generalized coordinate x

= distance of hanging block below edge of table

1. L = m(x’)² + mgx, leading to x’’ = g/2
2. L = (m + m’/2)(x’)² + mgx + ½ (m’g/l)x² , l = string length

Lagrangian approach leads to conserved quantities which we can identify as momentum (or angular momentum).

Another example, free particle: L = T = ½ m (x’)² , independent of x

d/dt[d L/d x’] = d/dt[mx’]

[d L/d x] = 0

Therefore d/dt[mx’] = 0, mx’ = constant

In general, we define p_i º d L/d q_i as the generalized momentum conjugate to q_i .

If L is independent of q_i , then q_i is known as an ignorable coordinate and

the generalized momentum p_i associated with that coordinate is constant (conserved).

The relevant Lagrange equation can be rewritten: d/dt[p_i] = d L/d q_i

= 0 if q_i is ignorable, giving p_i = constant.

Example 10.6.1: Pendulum attached to movable support (continued)

From p.401, L is as given on p.412;

the mass-M coordinate X is ignorable, so d L/d X’ is constant.

Examination of this term shows that it is the x-axis linear momentum.

Coordinate q is not ignorable; instead,

d L/d q = (m/2) 2X’rq ’sinq - mgr sinq (not as given on p.412)

Equating this to (d/dt)[d L/d q ’] gives the general equation of motion.

p’q = d/dt[dL/dq ’] = d/dt[mr^2q ’ + mX’r cosq ]

For error checking, we can imagine making the block immovable, so that X’=0=X’’

Then the general equation of motion reduces to the equation of a simple pendulum.

To check whether the X’’ term is correct, imagine a solution in which q ’=0=q ’’

The general equation would then imply: tanq = -X’’/g

We can recognize this as a steady state condition in which the block is accelerated uniformly to the right, leaving the pendulum lagging behind (deflected to the left) by a fixed angle.

The mass m has small dimensions, otherwise its moment of inertia is involved.

Two generalized coordinates: q and f .

Radial velocity = l dq /dt , azimuthal velocity = (l sinq )(df /dt)

Eq.(10.6.9) is the general equation of motion, S being a constant, related to the angular momentum pf .

Special cases: (1) f = constant gives simple pendulum

Almost-conical motion: can substitute Eq.(10.6.10) into the general equation Eq.(10.6.9)

With x = q - q ₀ , a SHM equation can be written, corresponding to harmonic oscillation with period T₁ relative to pure conical motion. The corresponding azimuthal angle f ₁ (between maxima, for example) is slightly (for small q ₀) greater than p radians.

In other words, the motion is regular (not chaotic) but not self-repeating, a result of the non-linear terms in the equation of motion.

Compare with the apsidal angles for a central-force orbit, p.245.

P.416: can integrate the general equation of motion to give an effective potential U(q )

which contains gravitational and centrifugal components.

Compare with the central-field Eq.(6.11.1) on p.235.

Internal forces of constraint do not appear explicitly in the Lagrange equations; they only affect the choice of generalized coordinates and the number of resulting equations.

We can include constraint forces by omitting the restraint conditions, resulting in a larger number of equations and involving coordinates which are not independent. These coordinates can be made independent by the method of Lagrange multipliers.

Imagine a system with two coordinates which are not independent but linked by an equation of the form: f(q₁,q₂,t) = 0

For a given value of t, d f = 0 and the coordinates are linked as in Eq.(10.7.4).

Substituting into Eq.(10.7.2) (Hamilton’s variational principle) gives Eq.(10.7.5), which contains a square-bracket term multiplied by a single coordinate (q₁), which can be varied arbitarily.

Therefore the square-bracket term must be identical to zero, leading to Eq.(10.7.6) where terms in q₁ and q₂ have been collected on either side to the equals sign. This equality can be maintained only if each side is equal to the same function of time l (t), giving two simultaneous equations represented, by Eq.(10.7.7), which can be solved for the three variables q₁,q₂ and l by invoking a third equation Eq.(10.7.1) – the constraint.

The terms Q₁ = l (t)(¶ f/¶ q₁) and Q₂ = l (t)(¶ f/¶ q₁) are the generalized forces of constraint.

The procedure is illustrated by Example 10.7.1 (p.419-421)

and by end-of-chapter problems e.g. 10.19 (p.432).

Lagrange method can be extended to non-conservative systems. For a static system, imagine simultaneous virtual displacements d r from equilibrium which are consistent with any constraints on the system.

Then total virtual work d W = S F.dr = 0, consistent with energy being minimized.

For a dynamical system, D’Alembert’s Principle replaces F by (F - ¶ p/¶ t), giving:

S (F_i – p_i).d r = 0

By way of proof, D’Alembert’s principle can be used to derive the Lagrange equations of motion (as on pp.422-424).

A general definition of the Hamiltonian function is:

H = S q’_i p_i - L where q’_i = dq/dt

Most mechanical systems satisfy the following two conditions:

(1) The potential energy is a function of one or more q_i (and not explicitly of time);

(2) The kinetic energy T depends on one or more (q’_i)²

Because of this quadratic dependence, ¶ T/¶ q’_i = 2c_i (q’_i) = [2/ q’_i] [c_i (q’_i)²]

Therefore T = S c_i (q’_i)² = S [q’_i/2] (¶ T/¶ q’_i ),

a special case of Euler’s Theorem for a homogeneous function (footnote, p.427).

This allows us to write S q’_i p_i = S q’_i [¶ L/¶ q_i’] = S q’_i [¶ T/¶ q_i’] = 2T

since the V component of L is a function of q_i and not q_i’.

Therefore H = S q’_i p_i - L = 2T – T + V = T + V = E

where E is the total energy of the system.

From the general definition of H ,

d H = S [ p_{i d} q_i + q’_{i d} p_i – (¶ L/¶ q’_i)d q’_i – (¶ L/¶ q_i)d q_i ]

Since p_i = (¶ L/¶ q’_i) and p’_i = (¶ L/¶ q_i), d H = S [q’_{i d} p_i – p_i’d q_i ]

But in general, d H = S (¶ H/¶ p_i) d p_i + (¶ H/¶ q_i) d q_i

So therefore (¶ H/¶ p_i) = q’_i and (¶ H/¶ q_i) = p’_i

These are Hamilton’s canonical equations of motion.

Simple example: free particle. T = ½ m (x’)², V = 0, L = ½ m (x’)²

Need p º ¶ L/¶ x’ = mx’ , giving x’ = p/m

H = T + V = ½ m (x’)² = p²/2m

The Hamilton equations are: (¶ H/¶ p) = p/m = x’ (as we already know)

and (¶ H/¶ q_i) = 0 = dp/dt i.e. p = constant.