Lagrangian and Hamiltonian Dynamics

1. Lagrangian and Hamiltonian Dynamics Chapter 7 Claude Pruneau Physics and Astronomy

2. 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)

3. 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.

4. 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.

5. 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:

6. Derivation of Euler-Lagrange Equations • Establish by transformation…

7. 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…)

8. 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.

9. Example 1: Harmonic Oscillator (cont’d) • Combine in Lagrange Eq.

10.  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.

11. Example 2: Plane Pendulum (cont’d) • Calculate derivatives of L by treating as if it were a rectangular coordinate. • Combine...

12. 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.

13. 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.

14. 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.

15. 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.

16. Generalized Coordinates (cont’d) Notation:

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

18. 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…

19. 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.

20. 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

21. 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.

22. 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.

23. Lagrange’s Eqs • The choice of specific coordinates is therefore immaterial • Hamilton’s principle becomes

24. Lagrange’s Eqs “s” equations “m” constraint equations Applicability: Force derivable from one/many potential Constraint Eqs connect coordinates, may be fct(t)

25. Lagrange Eqs (cont’d) • Holonomic constraints • Scleronomic constraints • Independent of time • Rheonomic • Dependent on time

26. 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:

27. Example: Projectile in 2D (cont’d) • In polar coordinates…

28. 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

29. 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.

30. Example: Motion in a cone (cont’d) • For r:

31. 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.

32. Consider • Generally non-integrable, unless • One thus has: • Or… • Which yields… • So the constraints are actually holonomic…

33. 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:

34. Forces of Constraint • The underdetermined multipliers are the forces of constraint:

35. Example: Disk rolling incline plane

36. Example: Motion on a sphere

37. 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…

39. Generalized momentum Generalized force defined through virtual work dW

40. For a conservative system:

44. 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:

45. … 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:

46. That must also equal: We then conclude: Hamilton Equations

47. 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.:

48. 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.

49. 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…

50. 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.