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Lagrangian and Hamiltonian Dynamics. Chapter 7 Claude Pruneau Physics and Astronomy. Minimal Principles in Physics. Hero of Alexandria 2nd century BC. Law governing light reflection minimizes the path length. Fermat’s Principle

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lagrangian and hamiltonian dynamics

Lagrangian and Hamiltonian Dynamics

Chapter 7

Claude Pruneau

Physics and Astronomy

minimal principles in physics
Minimal Principles in Physics
  • Hero of Alexandria 2nd century BC.
    • Law governing light reflection minimizes the path length.
  • Fermat’s Principle
    • Refraction can be understood as the path that minimizes the time - and Snell’s law.
  • Maupertui’s (1747)
    • Principle of least action.
  • Hamilton (1834, 1835)
hamilton s principle
Hamilton’s Principle

Of all possible paths along which a dynamical system may move from one point to another within a specified time interval (consistent with any constraints), the actual path followed is that which minimizes the time integral of the difference between the kinetic and potential energy.

hamilton s principle1
Hamilton’s Principle
  • In terms of calculus of variations:
  • The d is a shorthand notation which represents a variation as discussed in Chap 6.
  • The kinetic energy of a particle in fixed, rectangular coordinates is of function of 1st order time derivatives of the position
  • The potential energy may in general be a function of both positions and velocities. However if the particle moves in a conservative force field, it is a function of the xi only.
hamilton s principle cont d
Hamilton’s Principle (cont’d)
  • Define the difference of T and U as the Lagrange function or Lagrangian of the particle.
  • The minimization principle (Hamilton’s) may thus be written:
derivation of euler lagrange equations
Derivation of Euler-Lagrange Equations
  • Establish by transformation…
lagrange equations of motion
Lagrange Equations of Motion
  • L is called Lagrange function or Lagrangian for the particle.
  • L is a function of xi and dxi/dt but not t explicitly (at this point…)
example 1 harmonic oscillator
Example 1: Harmonic Oscillator

Problem: Obtain the Lagrange Equation of motion for the one-dimensional harmonic oscillator.


  • Write the usual expression for T and U to determine L.
  • Calculate derivatives.
example 2 plane pendulum
Example 2: Plane Pendulum

Problem: Obtain the Lagrange Equation of motion for the plane pendulum of mass “m”.


  • Solution:
  • Write the expressions for T and U to determine L.
example 2 plane pendulum cont d
Example 2: Plane Pendulum (cont’d)
  • Calculate derivatives of L by treating as if it were a rectangular coordinate.
  • Combine...
  • Example 2 was solved by assuming that  could be treated as a rectangular coordinate and we obtain the same result as one obtains through Newton’s equations.
  • The problem was solved by involving kinetic energy, and potential energy. We did not use the concept of force explicitly.
generalized coordinates
Generalized Coordinates
  • Seek generalization of coordinates.
  • Consider mechanical systems consisting of a collection of n discrete point particles.
  • Rigid bodies will be discussed later…
  • We need n position vectors, I.e. 3n quantities.
  • If there are m constraint equations that limit the motion of particle by for instance relating some of coordinates, then the number of independent coordinates is limited to 3n-m.
  • One then describes the system as having 3n-m degrees of freedom.
generalized coordinates cont d
Generalized Coordinates (cont’d)
  • Important note: if s=3n-m coordinates are required to describe a system, it is NOT necessary these s coordinates be rectangular or curvilinear coordinates.
  • One can choose any combination of independent parameters as long as they completely specify the system.
  • Note further that these coordinates (parameters) need not even have the dimension of length (e.g. q in our previous example).
  • We use the term generalized coordinates to describe any set of coordinates that completely specify the state of a system.
  • Generalized coordinates will be noted: q1, q2, …, qn.
generalized coordinates cont d1
Generalized Coordinates (cont’d)
  • A set of generalized coordinates whose number equals the number s of degrees of freedom of the system, and not restricted by the constraints is called a proper set of generalized coordinates.
  • In some cases, it may be useful/convenient to use generalized coordinates whose number exceeds the number of degrees of freedom, and to explicitly use constraints through Lagrange multipliers.
    • Useful e.g. if one wishes to calculate forces due to constraints.
  • The choice of a set of generalized coordinates is obviously not unique.
    • They are in general (infinitely) many possibilities.
  • In addition to generalized coordinates, we shall also consider time derivatives of the generalized coordinates called generalized velocities.
  • Transformation: The “normal” coordinates can be expressed as functions of the generalized coordinates - and vice-versa.
transformation cont d
Transformation (cont’d)
  • Rectangular components of the velocties depend on the generalized coordinates, the generalized velocities, and the time.
  • Inverse transformations are noted:
  • There are m=3n-s equations of constraint…
example generalized coordinates
Example: Generalized coordinates
  • Question: Find a suitable set of generalized coordinates for a point particle moving on the surface of a hemisphere of radius R whose center is at the origin.
  • Solution: Motion on a spherical surface implies:
  • Choose cosines as generalized coordinates.
example generalized coordinates cont d
Example: Generalized coordinates (cont’d)
  • q1,q2,q3 do not constitute a proper set of generalized of coordinates because they are not independent.
  • One may however choose e.g. q1, q2, and the constraint equation
lagrange eqs in gen d coordinates
Lagrange Eqs in Gen’d Coordinates
  • Of all possible paths along which a dynamical system may move from one point to another in configuration space within a specified time interval, the actual path followed is that which minimizes the time integral of the Lagrangian for the system.
  • Lagrangian defined as the difference between kinetic and potential energies.
  • Energy is a scalar quantity (at least in Galilean relativity).
  • Lagrangian is a scalar function.
  • Implies the lagrangian must be invariant with respect to coordinate transformations.
  • Certain transformations that change the Lagrangian but leave the Eqs of motion unchanged are allowed.
  • E.G. if L is replaced by L+d/dt f(qi,t), for a function with continuous 2nd partial derivatives. (Fixed end points)
  • The choice of reference for U is also irrelevant, one can add a constant to L.
lagrange s eqs
Lagrange’s Eqs
  • The choice of specific coordinates is therefore immaterial
  • Hamilton’s principle becomes
Lagrange’s Eqs

“s” equations

“m” constraint equations


Force derivable from one/many potential

Constraint Eqs connect coordinates, may be fct(t)

lagrange eqs cont d
Lagrange Eqs (cont’d)
  • Holonomic constraints
  • Scleronomic constraints
    • Independent of time
  • Rheonomic
    • Dependent on time
example projectile in 2d
Example: Projectile in 2D
  • Question: Consider the motion of a projectile under gravity in two dimensions. Find equations of motion in Cartesian and polar coordinates.
  • Solution in Cartesian coordinates:
example motion in a cone
Example: Motion in a cone
  • Question: A particle of mass “m” is constrained to move on the inside surface of a smooth cone of hal-angle a. The particle is subject to a gravitational force. Determine a set of generalized coordinates and determine the constraints. Find Lagrange’s Eqs of motion.




2 degrees of freedom only!

2 generalized coordinates.



example motion in a cone cont d
Example: Motion in a cone (cont’d)
  • Choose to eliminate “z”.

L is independent of q.

is the angular momentum relative to the axis of the cone.

lagrange s eqs with underdetermined multipliers
Lagrange’s Eqs with underdetermined multipliers
  • Constraints that can be expressed as algebraic equations among the coordinates are holonomic constraints.
  • If a system is subject to such equations, one can always find a set of generalized coordinates in terms of which Eqs of motion are independent of these constraints.
  • Constraints which depend on the velocities have the form

Non holonomic constraints unless eqs can be integrated to yield constrains among the coordinates.

  • Generally non-integrable, unless
  • One thus has:
  • Or…
  • Which yields…
  • So the constraints are actually holonomic…
  • We therefore conclude that if constraints can be expressed
  • Constraints Eqs given in differential form can be integrated in Lagrange Eqs using undetermined multipliers.
  • For:
  • One gets:
forces of constraint
Forces of Constraint
  • The underdetermined multipliers are the forces of constraint:
7 6 equivalence of lagrange s and newton s equations
7.6 Equivalence of Lagrange’s and Newton’s Equations
  • Lagrange and Newton formulations of mechanic are equivalent
  • Different view point, same eqs of motion.
  • Explicit demonstration…
Generalized momentum

Generalized force defined through virtual work dW

7.10 Canonical Equations of Motion – Hamilton Dynamics

Whenever the potential energy is velocity independent:

Result extended to define the Generalized Momenta:

Given Euler-Lagrange Eqs:

One also finds:

The Hamiltonian may then be considered a function of the generalized coordinates, qj, and momenta pj:

… whereas the Lagrangian is considered a function of the generalized coordinates, qj, and their time derivative.

To “convert” from the Lagrange formulation to the Hamiltonian formulation, we consider:

But given:

One can also write:

That must also equal:

We then conclude:

Hamilton Equations

Let’s now rewrite:

And calculate:

Finally conclude:

If :

H is a constant of motion

If, additionally, H=U+T=E, then E is a conserved quantity.:

some remarks
Some remarks
  • The Hamiltonian formulation requires, in general, more work than the Lagrange formulation to derive the equations of motion.
  • The Hamiltonian formulation simplifies the solution of problems whenever cyclic variables are encountered.
    • Cyclic variables are generalized coordinates that do not appear explicitly in the Hamiltonian.
  • The Hamiltonian formulation forms the basis to powerful extensions of classical mechanics to other fields e.g. Beam physics, statistical mechanics, etc.
  • The generalized coordinates and momenta are said to be canonically conjugates – because of the symmetric nature of Hamilton’s equations.
more remarks
More remarks
  • If qk is cyclic, I.e. does not appear in the Hamiltonian, then
  • And pk is then a constant of motion.
  • A coordinate cyclic in H is also cyclic in L.
  • Note: if qk is cyclic, its time derivative “q-dot” appears explicitly in L.
    • No reduction of the number of degrees of freedom in the Lagrange formulation: still “s” 2nd order equations of motion.
    • Reduction by 2 of the number of equations to be solved in the Hamiltonian formulation – since 2 become trivial…
where k is possibly a function of t.

One thus get the simple (trivial) solution:

The solution for a cyclic variable is thus reduced to a simple integral as above.

The simplest solution to a system would occur if one could choose the generalized coordinates in a way they are ALL cyclic. One would then have “s” equations of the form :

Such a choice is possible by applying appropriate transformations – this is known as Hamilton-Jacobi Theory.

Where dqk and are not independent!

Some remarks on the calculus of variation

Hamilton’s Principle:


The above integral becomes after integration by parts:

Which gives rise to Euler-Lagrange equations:

Alternatively, Hamilton’s Principle can be written:

Which evaluates to:


Integrate by parts:

The variation may then be written: