Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all...

41
Variational principles for dissipative systems W. M. Tulczyjew and P. Urba ´ nski [email protected] Faculty of Physics University of Warsaw Madrid, 6.09.2006 – p. 1/1

Transcript of Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all...

Page 1: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Variational principles for dissipative systems

W. M. Tulczyjew and P. Urbanski

[email protected]

Faculty of Physics

University of Warsaw

Madrid, 6.09.2006 – p. 1/17

Page 2: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Credo

• A master model for all variational principles of classicalphysics is provided by the principle of virtual work wellknown in statics of mechanical systems.

Madrid, 6.09.2006 – p. 2/17

Page 3: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Credo

• A master model for all variational principles of classicalphysics is provided by the principle of virtual work wellknown in statics of mechanical systems.

• Ingredients:

◦ configuration manifold Q,◦ constrained set C1 ⊂ TQ,◦ virtual work function σ : C1 → R,

Madrid, 6.09.2006 – p. 2/17

Page 4: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Credo

• A master model for all variational principles of classicalphysics is provided by the principle of virtual work wellknown in statics of mechanical systems.

• Ingredients:

◦ configuration manifold Q,◦ constrained set C1 ⊂ TQ,◦ virtual work function σ : C1 → R,

• with the properties:◦ for each q ∈ C0 = τQ(C1) the set C1

q = TqQ ∩ C1 is acone, i.e. λv ∈ C1

q for each v ∈ C1q , λ > 0,

◦ virtual work function is positive homogeneous, i.e.σ : (λv) = λσ(v) for λ > 0.

Madrid, 6.09.2006 – p. 2/17

Page 5: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

The principle of virtual work

is incorporated in the definition of the constitutive set

S = {f ∈ T∗Q; q = πQ(f) ∈ C0, ∀v∈C1

qσ(v) − 〈f, v〉 > 0}.

Madrid, 6.09.2006 – p. 3/17

Page 6: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

The main reference

Włodzimierz M. TULCZYJEW

"The Origin of Variational Principles"

Banach Center Publications 59, "Classical and QuantumIntegrability", Warszawa 2003also math-ph/0405041

Madrid, 6.09.2006 – p. 4/17

Page 7: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Legendre-Fenchel transformation

Letσ : C → R

be a positive homogeneous function defined on a cone C in avector space V .

Madrid, 6.09.2006 – p. 5/17

Page 8: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Legendre-Fenchel transformation

Letσ : C → R

be a positive homogeneous function defined on a cone C in avector space V .The set

S = {F ∈ V ∗; ∀v∈C σ(v) − 〈f, v〉 > 0}

is the Legendre-Fenchel transform of σ.

Madrid, 6.09.2006 – p. 5/17

Page 9: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Legendre-Fenchel transformation

Letσ : C → R

be a positive homogeneous function defined on a cone C in avector space V .The set

S = {F ∈ V ∗; ∀v∈C σ(v) − 〈f, v〉 > 0}

is the Legendre-Fenchel transform of σ.The constitutive set S derived from the work function σ isobtained by applying the Legendre-Fenchel transformation tofunctions

σq : C1q → R.

The Legendre-Fenchel transforms Sq are then combined

S =⋃

q∈C0

Sq

Madrid, 6.09.2006 – p. 5/17

Page 10: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

The inverse Legendre-Fenchel transformation

Let S be a subset of V ∗. We introduce the set

C = {v ∈ V ; supf∈S

〈f, v〉 < ∞}.

Madrid, 6.09.2006 – p. 6/17

Page 11: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

The inverse Legendre-Fenchel transformation

Let S be a subset of V ∗. We introduce the set

C = {v ∈ V ; supf∈S

〈f, v〉 < ∞}.

and the functionσ : C → sup

f∈S

〈f, v〉

Madrid, 6.09.2006 – p. 6/17

Page 12: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Properties of the L-F transformation

• The L-F transform S of a positive homogeneous functionσ : C → R is convex and closed.

• The inverse L-F transform σ : C → R of a subset S ⊂ V ∗ isconvex and closed (the overgraph of σ is closed).

• The L-F transformation and the inverse L-F transformationestablish a one to one correspondence between positivehomogeneous closed convex functions defined on cones inV and non empty closed convex subsets of V ∗.

It follows that the constitutive setprovides a complete characterization ofa convex static system

Madrid, 6.09.2006 – p. 7/17

Page 13: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Partially controlled systems

There is the internal configuration space Q and the controlconfiguration space Q

Q

η ��Q

, (1)

�� //

Madrid, 6.09.2006 – p. 8/17

Page 14: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Partially controlled systems

There is the internal configuration space Q and the controlconfiguration space Q

Q

η �� U //R

Q

, (3)

Madrid, 6.09.2006 – p. 8/17

Page 15: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Partially controlled systems

There is the internal configuration space Q and the controlconfiguration space Q

Q

η �� U //R

Q

, (4)

The constitutive set S derived from the potential U :

S = {f ∈ T∗Q;∃q∈Q η(q) = πQ(f),∀v∈TqQ 〈dU, v〉 = 〈f, Tη(v)〉}

Madrid, 6.09.2006 – p. 8/17

Page 16: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Partially controlled systems

There is the internal configuration space Q and the controlconfiguration space Q

Q

η �� U //R

Q

, (5)

The constitutive set S derived from the potential U :

S = {f ∈ T∗Q;∃q∈Q η(q) = πQ(f),∀v∈TqQ 〈dU, v〉 = 〈f, Tη(v)〉}

A point q ∈ Q ’contributes’ to S if and only if 〈dU, v〉 = 0 for eachvertical v ∈ TqQ.

Madrid, 6.09.2006 – p. 8/17

Page 17: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Generating families

The potential U is interpreted as a family of functions defined onfibres of the fibration η.

Madrid, 6.09.2006 – p. 9/17

Page 18: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Generating families

The potential U is interpreted as a family of functions defined onfibres of the fibration η.

The family (U, η) is called a generating family of the set S.

Madrid, 6.09.2006 – p. 9/17

Page 19: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Generating families

The potential U is interpreted as a family of functions defined onfibres of the fibration η.

The family (U, η) is called a generating family of the set S.

The critical set of the family (U, η):

Cr(U, η) = {q ∈ Q;∀v∈TqQ if Tη(v) = 0 then 〈dU, v〉 = 0}

Madrid, 6.09.2006 – p. 9/17

Page 20: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Convex generating families

Simplified version, without constraints.

�� oo //

Madrid, 6.09.2006 – p. 10/17

Page 21: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Convex generating families

Simplified version, without constraints.

Q

η �� TQτQ

oo σ //R

Q

, (7)

For each q the function σq : TqQ → R is convex.

Madrid, 6.09.2006 – p. 10/17

Page 22: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Convex generating families

Simplified version, without constraints.

Q

η �� TQτQ

oo σ //R

Q

, (8)

For each q the function σq : TqQ → R is convex.

S = {f ∈ T∗Q;∃q∈Q η(q) = πQ(f),∀v∈TqQ 〈σ, v〉 > 〈f, Tη(v)〉}

Madrid, 6.09.2006 – p. 10/17

Page 23: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Convex generating families

Simplified version, without constraints.

Q

η �� TQτQ

oo σ //R

Q

, (9)

For each q the function σq : TqQ → R is convex.

S = {f ∈ T∗Q;∃q∈Q η(q) = πQ(f),∀v∈TqQ 〈σ, v〉 > 〈f, Tη(v)〉}

The critical set

Cr(σ, η) = {q ∈ Q;∀v∈TqQ if Tη(v) = 0 then σ(v) > 0}

Madrid, 6.09.2006 – p. 10/17

Page 24: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Convex generating families

’Contribution’ of q ∈ Cr(σ, η) to S

Sq = {f ∈ T∗Q;πQ(f) = η(q), ∀v∈TqQ 〈σ, v〉 > 〈f, Tη(v)〉}

Madrid, 6.09.2006 – p. 11/17

Page 25: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Convex generating families

’Contribution’ of q ∈ Cr(σ, η) to S

Sq = {f ∈ T∗Q;πQ(f) = η(q), ∀v∈TqQ 〈σ, v〉 > 〈f, Tη(v)〉}

Proposition.

Sq = {f ∈ T∗Q;πQ(f) = η(q) = q, ∀v∈TqQ 〈σq, v〉 > 〈f, v〉}

where

σq : TqQ → R : v 7→ infv

σ(v), v ∈ TqQ, Tη(v) = v.

Madrid, 6.09.2006 – p. 11/17

Page 26: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Convex generating families

’Contribution’ of q ∈ Cr(σ, η) to S

Sq = {f ∈ T∗Q;πQ(f) = η(q), ∀v∈TqQ 〈σ, v〉 > 〈f, Tη(v)〉}

Proposition.

Sq = {f ∈ T∗Q;πQ(f) = η(q) = q, ∀v∈TqQ 〈σq, v〉 > 〈f, v〉}

where

σq : TqQ → R : v 7→ infv

σ(v), v ∈ TqQ, Tη(v) = v.

σq is well defined and convex.

Madrid, 6.09.2006 – p. 11/17

Page 27: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Convex generating families

’Contribution’ of q ∈ Cr(σ, η) to S

Sq = {f ∈ T∗Q;πQ(f) = η(q), ∀v∈TqQ 〈σ, v〉 > 〈f, Tη(v)〉}

Proposition.

Sq = {f ∈ T∗Q;πQ(f) = η(q) = q, ∀v∈TqQ 〈σq, v〉 > 〈f, v〉}

where

σq : TqQ → R : v 7→ infv

σ(v), v ∈ TqQ, Tη(v) = v.

σq is well defined and convex.

If Cr(σ, η) is a section of η, the family (σ, η) can be reduced to afunction σ : TQ → R

Madrid, 6.09.2006 – p. 11/17

Page 28: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Example 1

A point with configuration q1 is tied to a fixed point q0 with aspring of spring constant k1. Points q1 and q2 are tied with aspring of spring constant k2. The point q1 is subject to frictionand left free.

η : (q1, q2) 7→ (q2)

Madrid, 6.09.2006 – p. 12/17

Page 29: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Example 1

A point with configuration q1 is tied to a fixed point q0 with aspring of spring constant k1. Points q1 and q2 are tied with aspring of spring constant k2. The point q1 is subject to frictionand left free.

η : (q1, q2) 7→ (q2)

The work form of the system is

ϑ(q1, q2, v1, v2) = k1(q1 − q0|v1) + µ‖v1‖ + k2(q2 − q1|v2 − v1)

Madrid, 6.09.2006 – p. 12/17

Page 30: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Example 1

A point with configuration q1 is tied to a fixed point q0 with aspring of spring constant k1. Points q1 and q2 are tied with aspring of spring constant k2. The point q1 is subject to frictionand left free.

η : (q1, q2) 7→ (q2)

The work form of the system is

ϑ(q1, q2, v1, v2) = k1(q1 − q0|v1) + µ‖v1‖ + k2(q2 − q1|v2 − v1)

A point (q1, q2) is critical if ‖k1(q1 − q0) + k2(q1 − q2)‖ 6 µ

Madrid, 6.09.2006 – p. 12/17

Page 31: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Example 1

For a critical point (q1, q2)

infv1

ϑ(q1, q2, v1, v2) = k2(q2 − q1|v2)

Madrid, 6.09.2006 – p. 13/17

Page 32: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Example 1

For a critical point (q1, q2)

infv1

ϑ(q1, q2, v1, v2) = k2(q2 − q1|v2)

S(q1,q2) = {f = k2(q2 − q1)}.

Madrid, 6.09.2006 – p. 13/17

Page 33: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Example 2

A point with configuration q1 is tied to a fixed point q0 with aspring of spring constant k1. Points q1 and q2 are tied with aspring of spring constant k2. The point q1 is left free and point q2

is subject to friction.

η : (q1, q2) 7→ (q2)

Madrid, 6.09.2006 – p. 14/17

Page 34: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Example 2

A point with configuration q1 is tied to a fixed point q0 with aspring of spring constant k1. Points q1 and q2 are tied with aspring of spring constant k2. The point q1 is left free and point q2

is subject to friction.

η : (q1, q2) 7→ (q2)

The work form of the system is

ϑ(q1, q2, v1, v2) = k1(q1 − q0|v1) + µ‖v2‖ + k2(q2 − q1|v2 − v1)

Madrid, 6.09.2006 – p. 14/17

Page 35: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Example 2

A point with configuration q1 is tied to a fixed point q0 with aspring of spring constant k1. Points q1 and q2 are tied with aspring of spring constant k2. The point q1 is left free and point q2

is subject to friction.

η : (q1, q2) 7→ (q2)

The work form of the system is

ϑ(q1, q2, v1, v2) = k1(q1 − q0|v1) + µ‖v2‖ + k2(q2 − q1|v2 − v1)

There is one critical point in a fibre

q1 = q0 +k2

k1 + k2(q2 − q0).

Madrid, 6.09.2006 – p. 14/17

Page 36: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Example 2

For a critical point (q1, q2)

infv1

ϑ(q1, q2, v1, v2) = µ‖v2‖ +k1k2

k1 + k2(q2 − q1|v2)

Madrid, 6.09.2006 – p. 15/17

Page 37: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Example 2

For a critical point (q1, q2)

infv1

ϑ(q1, q2, v1, v2) = µ‖v2‖ +k1k2

k1 + k2(q2 − q1|v2)

The family can be reduced to the function

(q2, v2) 7→ µ‖v2‖ +k1k2

k1 + k2(q2 − q1|v2).

Madrid, 6.09.2006 – p. 15/17

Page 38: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Convex relation

Let a relation R : T∗Q1 → T

∗Q2 be generated by a convexfunction G : TK → R, K ⊂ Q1 × Q2, i.e.

b ∈ R(a) if 〈b, v2〉 − 〈a, v1〉 6 G(v1, v2)

Madrid, 6.09.2006 – p. 16/17

Page 39: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Convex relation

Let a relation R : T∗Q1 → T

∗Q2 be generated by a convexfunction G : TK → R, K ⊂ Q1 × Q2, i.e.

b ∈ R(a) if 〈b, v2〉 − 〈a, v1〉 6 G(v1, v2)

Let D ⊂ T∗Q1 be generated by a function σ : TC1 → R.

Madrid, 6.09.2006 – p. 16/17

Page 40: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Convex relation

Let a relation R : T∗Q1 → T

∗Q2 be generated by a convexfunction G : TK → R, K ⊂ Q1 × Q2, i.e.

b ∈ R(a) if 〈b, v2〉 − 〈a, v1〉 6 G(v1, v2)

Let D ⊂ T∗Q1 be generated by a function σ : TC1 → R.

Theorem. Let K and C1 × Q2 have clean intersection and letY = K ∩ (C1 × Q2) then the family

ρ : TY → R : (q1, q2, v1, v2) 7→ G(q1, q2, v1, v2) + σ(v1)

is a generating family of R(D).

Madrid, 6.09.2006 – p. 16/17

Page 41: Variational principles for dissipative systemsurbanski/Madrid.pdf · • A master model for all variational principles of classical physics is provided by the principle of virtual

Application

Legendre transformation for dissipative systems.

Madrid, 6.09.2006 – p. 17/17