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Hamiltonian Formalism

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Legendre transformations Legendre transformation: Adrien-Marie Legendre (1752 –1833)

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What is H? Conjugate momentum Then So

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What is H?

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If Then Kinetic energy In generalized coordinates

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What is H? For scleronomous generalized coordinates Then If

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What is H? For scleronomous generalized coordinates, H is a total mechanical energy of the system (even if H depends explicitly on time) If H does not depend explicitly on time, it is a constant of motion (even if is not a total mechanical energy) In all other cases, H is neither a total mechanical energy, nor a constant of motion

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Hamilton’s equations Hamiltonian: Hamilton’s equations of motion: Sir William Rowan Hamilton (1805 – 1865)

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Hamiltonian formalism For a system with M degrees of freedom, we have 2M independent variables q and p : 2M -dimensional phase space (vs. configuration space in Lagrangian formalism) Instead of M second-order differential equations in the Lagrangian formalism we work with 2M first-order differential equations in the Hamiltonian formalism Hamiltonian approach works best for closed holonomic systems Hamiltonian approach is particularly useful in quantum mechanics, statistical physics, nonlinear physics, perturbation theory

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Hamiltonian formalism for open systems

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Hamilton’s equations in symplectic notation Construct a column matrix (vector) with 2M elements Then Construct a 2M x 2M square matrix as follows:

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Hamilton’s equations in symplectic notation Then the equations of motion will look compact in the symplectic (matrix) notation: Example ( M = 2):

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Lagrangian to Hamiltonian Obtain conjugate momenta from a Lagrangian Write a Hamiltonian Obtain from Plug into the Hamiltonian to make it a function of coordinates, momenta, and time

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Lagrangian to Hamiltonian For a Lagrangian quadratic in generalized velocities Write a symplectic notation: Then a Hamiltonian Conjugate momenta

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Lagrangian to Hamiltonian Inverting this equation Then a Hamiltonian

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Example: electromagnetism

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Hamilton’s equations from the variational principle Action functional : Variations in the phase space :

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Hamilton’s equations from the variational principle Integrating by parts

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Hamilton’s equations from the variational principle For arbitrary independent variations

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Conservation laws If a Hamiltonian does not depend on a certain coordinate explicitly (cyclic), the corresponding conjugate momentum is a constant of motion If a Hamiltonian does not depend on a certain conjugate momentum explicitly (cyclic), the corresponding coordinate is a constant of motion If a Hamiltonian does not depend on time explicitly, this Hamiltonian is a constant of motion

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Higher-derivative Lagrangians Let us recall: Lagrangians with i > 1 occur in many systems and theories: 1.Non-relativistic classical radiating charged particle (see Jackson) 2.Dirac’s relativistic generalization of that 3.Nonlinear dynamics 4.Cosmology 5.String theory 6.Etc.

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Higher-derivative Lagrangians For simplicity, consider a 1D case: Variation Mikhail Vasilievich Ostrogradsky ( )

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Higher-derivative Lagrangians

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Generalized coordinates/momenta:

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Higher-derivative Lagrangians Euler-Lagrange equations: We have formulated a ‘higher-order’ Lagrangian formalism What kind of behavior does it produce?

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Example

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H is conserved and it generates evolution – it is a Hamiltonian! Hamiltonian linear in momentum?!?!?! No low boundary on the total energy – lack of ground state!!! Produces ‘runaway’ solutions: the system becomes highly unstable - collapse and explosion at the same time

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‘Runaway’ solutions Unrestricted low boundary of the total energy produces instabilities Additionally, we generate new degrees of freedom, which require introduction of additional (originally unknown) initial conditions for them These problems are solved by means of introduction of constraints Constraints restrict unstable behavior and eliminate unnecessary new degrees of freedom

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Canonical transformations Recall gauge invariance (leaves the evolution of the system unchanged): Let’s combine gauge invariance with Legendre transformation: K – is the new Hamiltonian (‘Kamiltonian’ ) K may be functionally different from H 9.1

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Canonical transformations Multiplying by the time differential: So 9.1

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Generating functions Such functions are called generating functions of canonical transformations They are functions of both the old and the new canonical variables, so establish a link between the two sets Legendre transformations may yield a variety of other generating functions 9.1

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Generating functions We have three additional choices: Canonical transformations may also be produced by a mixture of the four generating functions 9.1

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An example of a canonical transformation Generalized coordinates are indistinguishable from their conjugate momenta, and the nomenclature for them is arbitrary Bottom-line: generalized coordinates and their conjugate momenta should be treated equally in the phase space 9.2

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Criterion for canonical transformations How to make sure this transformation is canonical? On the other hand If Then 9.4

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Criterion for canonical transformations Similarly, If Then 9.4

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Criterion for canonical transformations So, If 9.4

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Canonical transformations in a symplectic form After transformation On the other hand 9.4

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Canonical transformations in a symplectic form For the transformations to be canonical: Hence, the canonicity criterion is: For the case M = 1, it is reduced to (check yourself) 9.4

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1D harmonic oscillator Let us find a conserved canonical momentum Generating function 9.3

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1D harmonic oscillator Nonlinear partial differential equation for F Let’s try to separate variables Let’s try 9.3

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1D harmonic oscillator We found a generating function! 9.3

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1D harmonic oscillator 9.3

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1D harmonic oscillator 9.3

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Canonical invariants What remains invariant after a canonical transformation? Matrix A is a Jacobian of a space transformation From calculus, for elementary volumes: Transformation is canonical if 9.5

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Canonical invariants For a volume in the phase space Magnitude of volume in the phase space is invariant with respect to canonical transformations: 9.5

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Canonical invariants What else remains invariant after canonical transformations? 9.5

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Canonical invariants For M = 1 For many variables 9.5

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Poisson brackets Poisson brackets: Poisson brackets are invariant with respect to any canonical transformation 9.5 Siméon Denis Poisson (1781 – 1840)

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Poisson brackets Properties of Poisson brackets : 9.5

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Poisson brackets In matrix element notation: In quantum mechanics, for the commutators of coordinate and momentum operators: 9.5

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Poisson brackets and equations of motion 9.6

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Poisson brackets and conservation laws If u is a constant of motion If u has no explicit time dependence In quantum mechanics, conserved quantities commute with the Hamiltonian 9.6

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Poisson brackets and conservation laws If u and v are constants of motion with no explicit time dependence For Poisson brackets: If we know at least two constants of motion, we can obtain further constants of motion 9.6

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Infinitesimal canonical transformations Let us consider a canonical transformation with the following generating function ( ε – small parameter): Then 9.4

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Infinitesimal canonical transformations Multiplying by dt Then 9.4

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Infinitesimal canonical transformations Infinitesimal canonical transformations: In symplectic notation: 9.4

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Evolution generation Motion of the system in time interval dt can be described as an infinitesimal transformation generated by the Hamiltonian The system motion in a finite time interval is a succession of infinitesimal transformations, equivalent to a single finite canonical transformation Evolution of the system is a canonical transformation!!! 9.6

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Application to statistical mechanics In statistical mechanics we deal with huge numbers of particles Instead of describing each particle separately, we describe a given state of the system Each state of the system represents a point in the phase space We cannot determine the initial conditions exactly Instead, we study a certain phase volume – ensemble – as it evolves in time 9.9

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Application to statistical mechanics Ensemble can be described by its density – a number of representative points in a given phase volume The number of representative points does not change Ensemble evolution can be thought as a canonical transformation generated by the Hamiltonian Volume of a phase space is a constant for a canonical transformation 9.9

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Application to statistical mechanics Ensemble is evolving so its density is evolving too On the other hand Liouville’s theorem In statistical equilibrium 9.9 Joseph Liouville ( )

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Hamilton–Jacobi theory We can look for the following canonical transformation, relating the constant (e.g. initial) values of the variables with the current ones: The reverse transformations will give us a complete solution 10.1

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Hamilton–Jacobi theory Let us assume that the Kamiltonian is identically zero Then Choosing the following generating function Then, for such canonical transformation: 10.1

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Hamilton–Jacobi theory Hamilton–Jacobi equation Conventionally: Hamilton’s principal function Partial differential equation First order differential equation Number of variables: M Sir William Rowan Hamilton (1805 – 1865) Karl Gustav Jacob Jacobi (1804 – 1851)

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Hamilton–Jacobi theory Suppose the solution exists, so it will produce M + 1 constants of integration: One constant is evident: We chose those M constants to be the new momenta While the old momenta 10.1

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Hamilton–Jacobi theory We relate the constants with the initial values of our old variables: The new coordinates are defined as: Inverting those formulas we solve our problem 10.1

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Have we met before? Remember action? 10.1

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Hamilton’s characteristic function When the Hamiltonian does not depend on time explicitly Generating function (Hamilton’s characteristic function) 10.1

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Hamilton’s characteristic function Now we require: So: Detailed comparison of Hamilton’s characteristic vs. Hamilton’s principal is given in a textbook (10.3) 10.3

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Hamilton’s characteristic function What is the relationship between S and W ? One of possible relationships (the most conventional): 10.3

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Periodic motion For energies small enough we have periodic oscillations (librations) – green curves For energies great enough we msy have periodic rotations – red curves Blue curve – separatrix trajectory – bifurcation transition from librations to rotations 10.6

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Action-angle variables For either type of periodic motion let us introduce a new variable – action variable (don’t confuse with action!): A generalized coordinate conjugate to action variable is the angle variable: The equation of motion for the angle variable: 10.6

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Action-angle variables In a compete cycle This is a frequency of the periodic motion 10.6

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Example: 1D Harmonic oscillator 10.2

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Example: 1D Harmonic oscillator 10.2

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Action-angle variables for 1D harmonic oscillator Therefore, for the frequency: 10.6

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Separation of variables in the Hamilton- Jacobi equation Sometimes, the principal function can be successfully separated in the following way: For the Hamiltonian without an explicit time dependence: Functions H i may or may not be Hamiltonians 10.4

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