Hamiltonian Mechanics

Question 2

What is Newton’s first law?

Answer 2

A body stays in constant motion unless acted upon by a force

Question 3

What is Newton’s second law?

Answer 3

Force is rate of change of momentum
F=ma

Question 4

What is Newton’s third law?

Answer 4

For every force there is an equal and opposite reaction

Question 5

What are the conservation laws?

Answer 5

Momentum
Angular momentum

Question 6

What is the equation for angular momentum?

Answer 6

M=m(r×v)
M=Iω

Question 7

What is the equation for angular velocity?

Answer 7

V=ω×r

Question 8

What is the inertia matrix?

Answer 8

I
Where I_xx=ΣM_i(r_ir_i-x_i²)
I_xy=ΣM_i(-x_iy_i)
I_xz=ΣM_i(-x_iz_i)
etc

Question 9

When does constructive interference occur?

Answer 9

When neighbouring path lengths are the same

Question 10

What is the equation for the action?

Answer 10

A=∫Ldt
From t0 to t1
Where L is the Lagrangian

Question 11

What is the equation for the Lagrangian?

Answer 11

L=T-V
Kinetic energy- potential energy

Question 12

What is the condition for the realised path?

Answer 12

It must extremise the action

Question 13

How do we extremise the action?

Answer 13

x –> x+a(t)
A –> A+δA
Set δA=0

Question 14

What is the Euler Lagrange equation?

Answer 14

∂L/∂x- d/dt ∂l/∂ẋ=0
Where x is the coordinate system

Question 15

What is the generalised Euler-Lagrange equation?

Answer 15

∂L/∂x_i- d/dt ∂l/∂ẋ_i=0

Question 16

What is the equation for the Hamiltonian?

Answer 16

H=T+V
The total energy

Question 17

What is the definition of the Hamiltonian?

Answer 17

H=V∂l/∂V - L
Where L is the Lagrangian

Question 18

What is a conserved property?

Answer 18

d/dt of the property is equal to 0

Question 19

What is Noether’s law?

Answer 19

For each symmetry of the system there is a conserved quantity

Question 20

What is the equation for canonical momentum?

Answer 20

p_i=∂l/∂(d/dt(q_i))
(q_i dot)

Question 21

When is L invariant?

Answer 21

When space is isotropic
δL –> 0

Question 22

What is the generalised equation for the Hamiltonian?

Answer 22

H=-L + ΣP_id/dt(q_i)

Question 23

What is the first of Hamilton’s equations?

Answer 23

d/dt(q_i)=∂H/∂p_i

Question 24

What is the second of Hamilton’s equations?

Answer 24

d/dt(p_i)=-∂H/∂q_i

Question 25

What is the equation of SHM?

Answer 25

d²/dt²(x)=-ω²x

Question 26

What is the equation for d/dt(P_i)?

Answer 26

d/dt(p_i)=∂L/∂q_i

Question 27

What is an Ergodic system?

Answer 27

One which explores all states consistent with energy conservation

Question 28

What is the frequency for a mass spring?

Answer 28

ω=sqrt(k/m)

Question 29

What is the general position equation for a mass spring?

Answer 29

x(t)=De^iωt

Question 30

What are the equations for the forces on a spring?

Answer 30

F=ma (a=d²x/dt² ) F=-kx

Question 31

What is the matrix equation for the spring system?

Answer 31

Mẍ=-kx Where M is the inertia matrix K is the stiffness matrix

Question 32

What is the secular equation for mode frequencies?

Answer 32

(K-ω²M)⋅X = 0 Where M is the inertia matrix K is the stiffness matrix X are the eigenvectors

Question 33

How do we find the mode frequencies?

Answer 33

det(K-ω²M) = 0 ω are the egienvalues