Lecture notes
Analytical Mechanics and Modelling Lecture Notes (PHY2073)
- Institution
- University Of Surrey (UNIS)
Full module lecture notes with headings and subheaders for easy navigation
[Show more]
Seller
Follow
Content preview
📓Lecture Notes/ Week 1 Lecture 1 & 2
Contents
Introduction and Review
Constraints
Example: A Simple Pendulum
Example: A Parabolic Bowl
Degrees of Freedom
Example - a rigid body with centre of mass fixed
Coordinates and Notation
Generalised Coordinates
Example A system with 2 particles and 0 constraints ∴ 6 degrees of freedom
Introduction and Review
📌 Newton → Lagrange→Hamilton
Newton: F = ma
Lagrange: Lagrangian L =T −V
T = kinetic energy and V = potential energy
Hamilton: Hamiltonian H =T +V
Lagrange and Hamilton deal with scalars not vectors
Constraints
Exist when a system is not able to move freely in 3D
Example: A Simple Pendulum
∣L∣ = L = length of inextensible c
⨀ = out of page
⨂ = into page
Lecture Notes/ Week 1 Lecture 1 & 2 1
, a = fixed vector r = vector describing the position of M
Holonomic: Equations of Constraint
1. Motion is in the x-y plane, z =0
2. Have L = r − a or ∣L∣ = L = ∣r − a∣
Non-Holonomic: Inequalities of Constraint
3. θ is limited to −90° → +90°⇒ −90° < θ < 90°
Example: A Parabolic Bowl
1. A mass m slides on the surface of the bowl (no rotation or friction)
m described by 3 coordinates (x, y, z ) and 1 holonomic constraint
z = ax2 + ay2
2 degrees of freedom
2. A mass moves in the bowl
m described by 3 coordinates (x, y, z ) and 1 non-holonomic constraint
z ≥ ax2 + ay2
Lecture Notes/ Week 1 Lecture 1 & 2 2
, 3 degrees of freedom
In Case 1, system only requires 2 coordinates to describe motion of m
In Case 2, system requires 3 coordinates to describe motion of m
Degrees of Freedom
a particle in 3D with no constraints has 3 degrees of freedom
If we have N particles, then
m is Holonomic Constraints
n = no.DOF = 3N − m
Example - a rigid body with centre of mass fixed
x′ , y′ and z ′ axes are fixed with respect to the object, origin at the centre of mass position
There are 3 rotational degrees of freedom associated with rotation about (x, y, z). Coordinates could
be 3 angles (αx , αy , αz ) of axes x′ , y′ , z ′ with respect to (x, y, z)
Coordinates and Notation
Lecture Notes/ Week 1 Lecture 1 & 2 3
, N = 1 → (1 particle)
m = 2 → (2 constraints)
Generalised Coordinates
q! , q2 , q3 , ..., qn where n is degrees of freedom
Example A system with 2 particles and 0 constraints ∴ 6 degrees of freedom
⎧x1 q1 ⎫
Particle 1: ⎨ y1 q2 ⎬
⎩ ⎭
z1 q3
⎧x2 q4 ⎫
Particle 1: ⎨ y2 q5 ⎬
⎩ ⎭
z2 q6
d
For rate of change dt use
Velocities
dx dθ
= ẋ, = θ̇
dt dt
Accelerations
dẋ dθ̇
= ẍ, = θ̈
dt dt
∴ also write qi , q˙i , q¨i
Lecture Notes/ Week 1 Lecture 1 & 2 4
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller tiffanybiles. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £7.99. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
79789 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy revision notes and other study material for 14 years now