Phillip Stanley-Marbell

Foundations of Embedded Systems

Department of Engineering, University of Cambridge

http://physcomp.eng.cam.ac.uk

Topic 13: Physical Invariants, Principle of Stationary Action, Noether’s Theorem

(~45 minutes)

Version 0.2020

Pre-Recorded

Video

26

Intended Learning Outcomes for This Topic

2

➌ Deﬁne the action and Lagrangian for a system

By the end of this topic, you should be able to:

➋ Deﬁne generalized velocities for a system

➍ Derive equations of motion for a system in terms of its generalized coordinates

➊ Deﬁne generalized coordinates for a system

➎ Apply the concept of generalized coordinates to measurands in embedded systems

26

Coordinate Systems

4

z

x

y

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Generalized Coordinates

5

If we consider red box’s location and temperature, its generalized coordinates

would be latitude, longitude, elevation, and temperature:

Generalized coordinate space for the TI SensorTag:

{✓,a

x

,a

y

,a

z

,g

x

,g

y

,g

z

,m

x

,m

y

,m

z

,h,p,l,M,spl}

z

x

y

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Deﬁnitions

6

Conﬁguration

Set of sensors for the measurands of interest

Conﬁguration space

Set of possible conﬁgurations

Dimension / degrees of freedom

Smallest number of sensors needed to recreate a given conﬁguration

Coordinate space

Set of possible measurand values

Generalized velocity

Rate of change of measurand coordinates

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Example: Ball with Embedded Sensors

7

time

y-velocity height, h(t)

Measurements for a real instance:

The measurements are a path in the generalized

coordinate space with conﬁguration

{x-accel., y-accel. z-accel.}

…

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An Example

8

26

Example: Ball with Embedded Sensors

9

- - -

    

-

-







  (  )

 ()

- - -

    

-

-

-









  (  )

 ()

time

y-velocity height, h(t)

Recall: measurements are a path in the

generalized coordinate space

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Feasible Paths in Generalized Coordinate Space

11

For any physical system, there will be certain paths that do not occur

time

height, h(t)

true h(t)

h

1

(t)

h

2

(t)

t

0

t

1

For the ball, looking at a single coordinate (height) as a function of time:

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Feasible Paths Minimize the “Action”, S

12

S =

Z

r

[K(t)  U (t)] dt

=

Z

t

2

t

1

[K(t)  U (t)] dt.

Let K(t) be the generalized kinetic energy

and let U(t) be the generalized potential energy

For physically feasible paths, S is stationary

K(t) − U(t) is called the Lagrangian

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Extremal Values of the Functional S

13

time

height, h(t)

path, h(t), that

minimizes the

functional S

h

1

(t)

t

0

t

1

h

2

(t)

x

f(x)

x

min

x

min

−∆x

x

min

+∆x

value of x that

minimizes the

function f(x)

S is a functional: A mapping from functions to real values

Extremals of functionals are the analogs of maxima and minima of functions

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Feasible Paths Minimize the “Action”, S

14

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Principle of Stationary Action

15

(Also known as principle of least action, but that’s a misnomer)

Maupertuis 1744, Euler 1744

The action, S, is a functional.

It has dimensions energy × time and is extremal at the Lagrangian 𝓛 = K(t) − U(t)

S =

Z

t

2

t

1

L( t, r,

˙

r)dt

26

16

For mechanical systems, we already know

the form of the Lagrangian to be K(t) - U(t)

In what follows, we will work through the expression for

the action functional and we will see some interesting

physical laws which fall out of the algebra

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How Deviations from True Path Aﬀect S

17

⇣

n

(t)=h

n

(t)  h

¯

(t).

time

height, h(t)

true h(t)

h

1

(t)

h

2

(t)

t

0

t

1

t*

ζ

1

(t*)

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How Deviations from True Path Aﬀect S

18

Kinetic energy

1

2

m

dh

¯

dt

!

2

Potential energy

mgh

¯

(t)

time

height, h(t)

true h(t)

h

1

(t)

h

2

(t)

t

0

t

1

ˆ

S =

Z

t

1

t

0

2

4

1

2

m

dh

¯

dt

+

d⇣

dt

!

2

 mg

⇣

h

¯

(t)+⇣( t)

⌘

3

5

dt

=

Z

t

1

t

0

2

4

1

2

m

dh

¯

dt

!

2

+ m

dh

¯

dt

d⇣

dt

!

+ m

✓

d⇣

dt

◆

2

 mg

⇣

h

¯

(t)+⇣( t)

⌘

3

5

dt.

Action

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Getting S in a Nicer Form

19

mg

⇣

h

¯

(t)+⇣(t)

⌘

Taylor series expansion:

Potential energy

mgh

¯

(t)+mg⇣

dh

¯

(t)

dh

+ mg

⇣

2

d

2

h

¯

(t)

dh

2

+ ...

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Stationary Action Leads to Equations of Motion

20

Simplifying, we get

For to be zero regardless of

⇣(t)

it must be that

S

S =

Z

h

1

h

0

"

m

d

2

h

¯

dt

2



d P.E.

dh

#

⇣(t)dt.

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m

d

2

h

¯

dt

2

=

d P.E.

dh

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26

21

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From Lagrangian to Motion in Generalized Coordinates

22

d

dt

0

@

@L(t, w, ˙w)

@ ˙w

i



w=r(t)

˙w=

˙

r(t)

1

A

=

@L(t, w, ˙w)

@w

i



w=r(t)

˙w=

˙

r(t)

The Euler-Lagrange equations let us get the equations of motion from Lagrangian

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Noether’s Theorem

23

If under the transformation

t

0

= t +

N

X

i=1

"⌧

i

r

0

= r +

N

X

i=1

"⇣

i

the functional

 =

Z

b

a

L(t, r,

˙

r)dt

is both invariant and extremal, then the value

C =

@L

@

˙

r

⇣ 

✓

@L

@

˙

r

dr

dt

 L(t, r,

˙

r)

◆

⌧

remains constant (i.e., the quantity C is conserved)

t

0

= t +

N

X

i=1

"⌧

i

r

0

= r +

N

X

i=1

"⇣

i

and

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