Advanced Mechanics 2015
An introduction to Lagrangian mechanics for engineers
Professor Clive Neal-Sturgess
Email: c.e.n.sturgess@bham.ac.uk
Emeritus Professor of Mechanical Engineering
Former HofD (Jaguar Chair for 23 years)
Director Automotive Safety Centre
Research areas:
FE methods, Biomechanics of Trauma, Fracture and Fatigue.
Why does mechanics work?
What are you doing when you model a system? Predicting the future!
How/why can you predict the future?
Principle of least action (PLA) (more accurately Sationary Action)
“When a change occurs in nature, the quantity of action necessary for the change is the least possible” (M. de Maupertuis Lyon 1756)
“People surmised that there must be a Principle of Least Action before they knew how it went” (Wittgenstein: Tractus Philosophicus Logicus # 6.3211)
Underpins: Quantum Mechanics, Electromagnetics, Thermodynamics, Newton’s Laws (1687), FE methods, Relativity, Chemistry, Life - Schrodinger, Evolution
- Law of everything (physical)!!
HAMILTON’S PRINCIPLE (1833) See Wikipedia
Where Uf = Gibbs Free Energy = E - Ts
Hamilton’s Principle: Amongst all the possible trajectories qi(t) which take the system from initial configuration qi(t1) at time t1 to final configuration in Phase Space qi(t2) at time t2, the actual physical trajectory is the one which makes the action an extremum (usually a minimum but could be a saddle point).
Calculus of Variations is a way of determining the extremal (extreme value, maximum, minimum or saddle point) of a Functional (a function described by an integral), an example of which is the Derivation of the Euler-Lagrange equations (1788 Principia 1687)
Define a
Lagrangian such that L= (T-V) (genius step!)
- Where T = Kinetic Energy, V = Potential Energy
Then in terms of Generalised Co-ordinates [explained later]
To derive the Euler-Lagrange equation, examine (see http://mathworld.wolfram.com/Euler-Lagrange)
This is the Euler-Lagrange (E-L) equation.
Spooky! How does the particle (system) know it is going to the final state when it starts out??
Feynman’s explanation: (Sum over all histories QED) at each instant in time the particle follows Least Action. The path is simply the integral (sum) of all these instantaneous actions; the particle does not know where it is going!!
Back to Euler-Lagrange:
Therefore it is Newton's Second Law!
So why bother? Newton was right all along!
Problems with Newtonian mechanics:
▣ Newtonian mechanics, strictly, only apply to point masses
- Rigid body mechanics was down to Euler et.al
▣ Newtonian mechanics are vector mechanics
- Need to specify magnitude and direction of the forces
- Gets very difficult for multi-body problems
- Cannot predetermine the energy split between the multiple bodies (examples)
▣ Newtonian mechanics use Cartesian coordinates
- Can’t be generalised
- Newton (1687) vs Leibnitz
- Fermat 1647
- Momentum vs Energy
- Charles Babbage (1791)
▣ Newton was a tortured genius deriving his equations from observation – but he did not really understand what he was doing. He can be forgiven it took a hundred years to sort it out!
▣ Newtonian mechanics methodology
The Principle of Least Action (PLA) is the Fundamental Principle underlying all of physical science – minimises Free Energy
Conservation Laws:
Reasons why you can predict the future
The Principle of Least Action (PLA) can also be used to derive the Conservation Laws:
▣ Conservation of Energy
▣ Conservation of Momentum
▣ By using Noether’s first theorem (1918):
If one of the generalised coordinates, say qk, does not appear in the Lagrangian, the right-hand side of the E-L equation is zero, and the left-hand side requires that where the momentum is conserved throughout the motion.
Classical mechanics can be generalised to many forces. Newtonian mechanics needs each force and direction to be prescribed. Engineers in UK taught Newtonian Mechanics – simple systems (a mistake)!
BUT whichever way we do it every physical process follows the Principle of Least Action, therefore we can predict the future!!!
▣ Compare to Economics!!!
Describing systems using generalised co-ordinates:
In general for any system there are:
No of spatial coordinates (usually 3) x No of particles – No of constraints = No of degrees of freedom = No of generalised coordinates. This gets very difficult for Newtonian mechanics when multiple bodies and dimensions are considered.
You are used to describing systems by Cartesian Co-ordinates (x, y, z) however it is often more convenient to describe systems by more general coordinates, which need not be orthogonal (at 900), for instance for the simple pendulum shown
It would be more convenient to describe the system by the length of the pendulum (l) and the angle (q).
Let such a system of generalised spatial coordinates be qi (i = 1,2,3 ….), and corresponding velocities , then the Lagrangian could be described as:
Where for the pendulum
We will use these types of generalised coordinates throughout the rest of the course.
Constrained motion and Lagrange multipliers:
Constrained motion is often encountered, and when the constraints can be expressed as Holonomic Constraints (all spatial/geometric, not velocities) then a constraint equation can be used to reduce the degrees of freedom (as above), and so reduce the number of generalised coordinates necessary to solve the problem, so making it more simple. [examples given later]
However, if the forces exerted by the constraints are required then it is necessary to use Lagrange Multipliers.
The proof of this is quite complicated – Google it.
I will state the theorem and try to give it physical relevance.
The E-L equation is an equation of force. To introduce the external forces of constraint an undetermined Lagrange multiplier (l) is used in a Modified E-L as:
Where are known as generalised forces of constraint
Where the function is a function expressing the constraints equated to zero, say for a disc rolling down an inclined plane (see diagram):
If the disc is rolling then
The
These relationships can then be used to solve the problem (see later)
As an analogy, imagine the constraint is a spring of stiffness k, then if:
If this were done it would be found that l = 1.
So you can imagine l as the apparent stiffness of the constraint in direction q. It is similar to Castigliano’s theorem.
Solution scheme for solving problems using E-L:
1. Choose generalised coordinates.
- Can be quite difficult at first
2. Determine energies both T and V in Cartesian coordinates
- Not absolutely necessary, but very useful as it is often difficult to determine the energies in the generalised coordinates, as they are not necessarily orthogonal and the energies often involve more that one of the generalised coordinates.
3. Formulate the Lagrangian in generalised coordinates
4. Carry out the partial differentiation.
5. Substitute in E-L Þ equations of motion
6. Use Lagrangian multipliers if you need constraint forces
Lagrangian mechanics leads naturally into Hamiltonian mechanics
The Hamiltonian is the Legendre transform of the Lagrangian: H = T +V = total energy, or
The Hamiltonian is used extensively in Quantum Mechanics – Schrodinger’s wave equation.
References:
1. Numerous Google references, Wikipedia is a good place to start, not to stop!
2. When Least is Best, Paul J Nahin,
Princeton University Press 2004
ISBN: 0-691-07078-4
3. Principle of Least Action:
www.scholarpedia.org/article/Principle_of_least_action
4. What is Life? Erwin Schrodinger
Canto 2007
ISBN: 978-0-521-42708-1 (pbk)
5. Life’s Ratchet: Peter M Hoffmann
Perseus Books 2014
ISBN: in cataloging
6. Chapter 10 – Lagrangian Mechanics
http://people.rit.edu/vwlsps/IntermediateMechanics2/Ch10.pdf
7. Classical Mechanics, R. Douglas Gregory
Cambridge University Press 2006
ISBN: 978-0-521-53409-3 paperback
8. Classical Mechanics, Herbert Goldstein
Addison-Wesley 1950 – the classic
9. An Introduction to Lagrangian Mechanics,
Alain J. Brizard
World Scientific 2008
ISBN: 10 981-281-837-5 (pbk)
10. A Student’s Guide to Lagrangians and Hamiltonians, Patrick Hamill
Cambridge University Press 2014
ISBN:978-1-107-61752-0 (pbk)
For full mathematics see: http//:www.statact.squarespace.com