Hamiltonian Dynamics - Lecture 1, Study notes of Dynamics

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Hamiltonian Dynamics

Lecture 1

David Kelliher

RAL

November 12, 2019

Bibliography

The Variational Principles of Mechanics - Lanczos Classical Mechanics - Goldstein, Poole and Safko A Student’s Guide to Lagrangians and Hamiltonians - Hamill Classical Mechanics, The Theoretical Minimum - Susskind and Hrabovsky Theory and Design of Charged Particle Beams - Reiser Accelerator Physics - Lee Particle Accelerator Physics II - Wiedemann Mathematical Methods in the Physical Sciences - Boas Beam Dynamics in High Energy Particle Accelerators - Wolski

Configuration space

q 2

q 3

q 1

t 1

t 2

The state of the system at a time t can be given by the value of the n generalised coordinates qi. This can be represented by a point in an n dimensional space which is called “configuration space” (the system is said to have n degrees of free- dom). The motion of the system as a whole is then characterised by the line this system point maps out in configuration space.

Newtonian Mechanics

The equation of motion of a particle of mass m subject to a force F is

d dt (m˙r) = F(r, ˙r, t) (1)

In Newtonian mechanics, the dynamics of the system are defined by the force F, which in general is a function of position r, velocity ˙r and time t. The dynamics are determined by solving N second order differential equations as a function of time. Note: coordinates can be the vector spatial coordinates ri(t) or generalised coordinates qi (t).

Action

y

x

t 1

t 2

The action S is the integral of L along the trajectory

S =

∫ (^) t 2

t 1

L(q, q˙, t)t (4)

Principle of least action

The principle of least action or Hamilton’s principle holds that the system evolves such that the action S is stationary. It can be shown that the Euler-Lagrange equation defines a path for which.

δS = δ

[∫ (^) t 2

t 1

L(q, q˙, t)t

]

Advantages of Lagrangian approach

The Euler-Lagrange is true regardless of the choice of coordinate system (including non-inertial coordinate systems). We can transform to convenient variables that best describe the symmetry of the system. It is easy to incorporate constraints. We formulate the Lagrangian in a configuration space where ignorable coordinates are removed (e.g. a mass constrained to a surface), thereby incorporating the constraint from the outset.

Conservative force

In the case of a convervative force field the Lagrangian is the difference of the kinetic and potential energies

L(q, q˙) = T (q, q˙) − V (q) (10)

where F =

∂V (q) ∂q

Relativistic free particle - Lagrangian

The momentum for a free particle is

pi = γm x˙i , i = 1, 2 , 3; γ =

1 − β^2

where m = m 0 the rest mass and β is the velocity relative to c. To ensure

pi =

∂L

∂ x˙i

m x˙i √ 1 − β^2

the Lagrangian should be of the form

L(x, x˙, t) = −mc^2

1 − β^2 = −mc^2

c^2

x˙ 12 + ˙x 22 + ˙x 32

General electromagnetic fields

Now include general EM fields U(x, x˙, t) = e(φ − v · A)

L(x, x˙, t) = −mc^2

1 − β^2 − eφ + ev · A. (20)

The conjugate momentum is

Pi =

∂L

∂ x˙i

m x˙i √ 1 − β^2

• eAi (21)

i.e. the field contributes to the conjugate momentum.

Apply Legendre’s transformation to the Lagrangian

Start with the Lagrangian

L = L(q 1 ,... , qn, q˙ 1 ,... , q˙n, t), (25)

and introduce some new variables we are going to call the pi s

pi =

∂L

∂ q˙i

We can then introduce a new function H defined as

H =

∑^ n

i=

pi q˙i − L (27)

We now have a function which is dependent on q, p and time.

H = H(q 1 ,... , qn, p 1 ,... , pn, t) (28)

L and H have a dual nature:

H =

pi q˙i − L,

pi =

∂L

∂ q˙i

L =

pi q˙i − H,

q ˙i =

∂H

∂pi

Hamilton’s canonical equations

Starting from Lagrange’s equation

d dt

∂L

∂ q˙i

∂L

∂q

and combining with pi =

∂L

∂ q˙i leads to p˙i =

∂L

∂qi

∂H

∂qi

So we have

q˙i =

∂H

∂pi (32) p˙i = −

∂H

∂qi

which are called Hamilton’s canonical equations. They are the equations of motion of the system expressed as 2n first order differential equations.