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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

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 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)

- 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

- Establish by transformation…

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

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

Solution:

- Write the usual expression for T and U to determine L.

- Calculate derivatives.

Example 1: Harmonic Oscillator (cont’d)

- Combine in Lagrange Eq.

Example 2: Plane Pendulum

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

l

- Solution:
- Write the expressions for T and U to determine L.

Example 2: Plane Pendulum (cont’d)

- Calculate derivatives of L by treating as if it were a rectangular coordinate.

- Combine...

Remarks

- 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

- 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)

- 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’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.

Generalized Coordinates (cont’d)

Notation:

Transformation

- Transformation: The “normal” coordinates can be expressed as functions of the generalized coordinates - and vice-versa.

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

- 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)

- 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

- 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.

Remarks

- 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

- The choice of specific coordinates is therefore immaterial

- Hamilton’s principle becomes

Lagrange’s Eqs

“s” equations

“m” constraint equations

Applicability:

Force derivable from one/many potential

Constraint Eqs connect coordinates, may be fct(t)

Lagrange Eqs (cont’d)

- Holonomic constraints

- Scleronomic constraints
- Independent of time
- Rheonomic
- Dependent on time

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: Projectile in 2D (cont’d)

- In polar coordinates…

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.

z

Solution:

Constraint:

2 degrees of freedom only!

2 generalized coordinates.

y

x

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

- 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.

Consider

- Generally non-integrable, unless

- One thus has:

- Or…

- Which yields…

- So the constraints are actually holonomic…

Constraints…

- 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

- The underdetermined multipliers are the forces of constraint:

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:

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

- 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

- 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:

Evaluated:

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:

Consider:

Integrate by parts:

The variation may then be written:

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