Dirac structures and geometry of nonholonomic constraints

Dirac structures and geometry of nonholonomic
constraints
KG (KMMF) Seminarium Geometryczne 13 X 2010 1 / 26

Introduction
References
Hiroaki Yoshimura, Jerrold E. Marsden : ”Dirac structures in
Lagrangian Mechanics”, J. Geom. Phys., 57, (2006) Part I 133–156,
Part II 209-250;
W̷lodzimierz M. Tulczyjew: ”A note on holonomic constraints” in
Revisiting the Foundations of Relativistic Physics, A. Ashtekar et. al.
(eds.), 403-419, 2003 Kluver Academic Press;
Katarzyna Grabowska, Janusz Grabowski: ”Variational calculus with
constraints on general algebroids”, J. Phys. A: Math. Theor. 41,
(2008).
KG (KMMF) Seminarium Geometryczne 13 X 2010 2 / 26

Introduction
References
Hiroaki Yoshimura, Jerrold E. Marsden : ”Dirac structures in
Lagrangian Mechanics”, J. Geom. Phys., 57, (2006) Part I 133–156,
Part II 209-250;
W̷lodzimierz M. Tulczyjew: ”A note on holonomic constraints” in
Revisiting the Foundations of Relativistic Physics, A. Ashtekar et. al.
(eds.), 403-419, 2003 Kluver Academic Press;
Katarzyna Grabowska, Janusz Grabowski: ”Variational calculus with
constraints on general algebroids”, J. Phys. A: Math. Theor. 41,
(2008).
KG (KMMF) Seminarium Geometryczne 13 X 2010 2 / 26

Introduction
References
Hiroaki Yoshimura, Jerrold E. Marsden : ”Dirac structures in
Lagrangian Mechanics”, J. Geom. Phys., 57, (2006) Part I 133–156,
Part II 209-250;
W̷lodzimierz M. Tulczyjew: ”A note on holonomic constraints” in
Revisiting the Foundations of Relativistic Physics, A. Ashtekar et. al.
(eds.), 403-419, 2003 Kluver Academic Press;
Katarzyna Grabowska, Janusz Grabowski: ”Variational calculus with
constraints on general algebroids”, J. Phys. A: Math. Theor. 41,
(2008).
KG (KMMF) Seminarium Geometryczne 13 X 2010 2 / 26

Contents
Dirac structure
Dirac structure on T ∗ Q induced by a distribution on Q.
Nonholonomic constraints and Dirac structures according to Y & M
My point of view
Comments about the terminology (and example)
KG (KMMF) Seminarium Geometryczne 13 X 2010 3 / 26

Contents
Dirac structure
Dirac structure on T ∗ Q induced by a distribution on Q.
Nonholonomic constraints and Dirac structures according to Y & M
My point of view
Comments about the terminology (and example)
KG (KMMF) Seminarium Geometryczne 13 X 2010 3 / 26

Contents
Dirac structure
Dirac structure on T ∗ Q induced by a distribution on Q.
Nonholonomic constraints and Dirac structures according to Y & M
My point of view
Comments about the terminology (and example)
KG (KMMF) Seminarium Geometryczne 13 X 2010 3 / 26

Contents
Dirac structure
Dirac structure on T ∗ Q induced by a distribution on Q.
Nonholonomic constraints and Dirac structures according to Y & M
My point of view
Comments about the terminology (and example)
KG (KMMF) Seminarium Geometryczne 13 X 2010 3 / 26

Contents
Dirac structure
Dirac structure on T ∗ Q induced by a distribution on Q.
Nonholonomic constraints and Dirac structures according to Y & M
My point of view
Comments about the terminology (and example)
KG (KMMF) Seminarium Geometryczne 13 X 2010 3 / 26

Dirac structure
Definition
A Dirac structure L on a manifold M is a vector subbundle of
TM ⊕M T ∗ M which is maximal isotropic with respect to the metric
〈Xp + ϕp | Yp + ψp〉 = 1
2 (〈ψp, Xp〉 + 〈ϕp, Yp〉)
and involutve with respect to Dorfman bracket
[X + ϕ, Y + ψ]D = [X , Y ] + (LX ψ − ıY dϕ).
We call L an almost Dirac structure when it is maximal isotropic and not
involutive.
KG (KMMF) Seminarium Geometryczne 13 X 2010 4 / 26

Dirac structure
Definition
A Dirac structure L on a manifold M is a vector subbundle of
TM ⊕M T ∗ M which is maximal isotropic with respect to the metric
〈Xp + ϕp | Yp + ψp〉 = 1
2 (〈ψp, Xp〉 + 〈ϕp, Yp〉)
and involutve with respect to Dorfman bracket
[X + ϕ, Y + ψ]D = [X , Y ] + (LX ψ − ıY dϕ).
We call L an almost Dirac structure when it is maximal isotropic and not
involutive.
KG (KMMF) Seminarium Geometryczne 13 X 2010 4 / 26

Almost Dirac structure in mechanics
M = T ∗ Q, symplectic form ωQ, associated map βQ:
βQ : TT ∗ Q −→ T ∗ T ∗ Q, β(w) = ωQ(·, w)
∆Q ⊂ TQ is a regular distribution not necessarily involutive,
representing (nonholonomic) constraints,
∆ ◦ Q ⊂ T∗ Q is an anihilator of ∆Q,
πQ : T ∗ Q −→ Q
TπQ : TT ∗ Q −→ TQ
∆T ∗ Q = (TπQ) −1 (∆Q) is a regular distribution on T ∗ Q
∆ ◦ T ∗ Q is an anihilator of ∆T ∗ Q,
KG (KMMF) Seminarium Geometryczne 13 X 2010 5 / 26

Almost Dirac structure in mechanics
M = T ∗ Q, symplectic form ωQ, associated map βQ:
βQ : TT ∗ Q −→ T ∗ T ∗ Q, β(w) = ωQ(·, w)
∆Q ⊂ TQ is a regular distribution not necessarily involutive,
representing (nonholonomic) constraints,
∆ ◦ Q ⊂ T∗ Q is an anihilator of ∆Q,
πQ : T ∗ Q −→ Q
TπQ : TT ∗ Q −→ TQ
∆T ∗ Q = (TπQ) −1 (∆Q) is a regular distribution on T ∗ Q
∆ ◦ T ∗ Q is an anihilator of ∆T ∗ Q,
KG (KMMF) Seminarium Geometryczne 13 X 2010 5 / 26

Almost Dirac structure in mechanics
M = T ∗ Q, symplectic form ωQ, associated map βQ:
βQ : TT ∗ Q −→ T ∗ T ∗ Q, β(w) = ωQ(·, w)
∆Q ⊂ TQ is a regular distribution not necessarily involutive,
representing (nonholonomic) constraints,
∆ ◦ Q ⊂ T∗ Q is an anihilator of ∆Q,
πQ : T ∗ Q −→ Q
TπQ : TT ∗ Q −→ TQ
∆T ∗ Q = (TπQ) −1 (∆Q) is a regular distribution on T ∗ Q
∆ ◦ T ∗ Q is an anihilator of ∆T ∗ Q,
KG (KMMF) Seminarium Geometryczne 13 X 2010 5 / 26

Almost Dirac structure in mechanics
M = T ∗ Q, symplectic form ωQ, associated map βQ:
βQ : TT ∗ Q −→ T ∗ T ∗ Q, β(w) = ωQ(·, w)
∆Q ⊂ TQ is a regular distribution not necessarily involutive,
representing (nonholonomic) constraints,
∆ ◦ Q ⊂ T∗ Q is an anihilator of ∆Q,
πQ : T ∗ Q −→ Q
TπQ : TT ∗ Q −→ TQ
∆T ∗ Q = (TπQ) −1 (∆Q) is a regular distribution on T ∗ Q
∆ ◦ T ∗ Q is an anihilator of ∆T ∗ Q,
KG (KMMF) Seminarium Geometryczne 13 X 2010 5 / 26

Almost Dirac structure in mechanics
M = T ∗ Q, symplectic form ωQ, associated map βQ:
βQ : TT ∗ Q −→ T ∗ T ∗ Q, β(w) = ωQ(·, w)
∆Q ⊂ TQ is a regular distribution not necessarily involutive,
representing (nonholonomic) constraints,
∆ ◦ Q ⊂ T∗ Q is an anihilator of ∆Q,
πQ : T ∗ Q −→ Q
TπQ : TT ∗ Q −→ TQ
∆T ∗ Q = (TπQ) −1 (∆Q) is a regular distribution on T ∗ Q
∆ ◦ T ∗ Q is an anihilator of ∆T ∗ Q,
KG (KMMF) Seminarium Geometryczne 13 X 2010 5 / 26

Almost Dirac structure in mechanics
M = T ∗ Q, symplectic form ωQ, associated map βQ:
βQ : TT ∗ Q −→ T ∗ T ∗ Q, β(w) = ωQ(·, w)
∆Q ⊂ TQ is a regular distribution not necessarily involutive,
representing (nonholonomic) constraints,
∆ ◦ Q ⊂ T∗ Q is an anihilator of ∆Q,
πQ : T ∗ Q −→ Q
TπQ : TT ∗ Q −→ TQ
∆T ∗ Q = (TπQ) −1 (∆Q) is a regular distribution on T ∗ Q
∆ ◦ T ∗ Q is an anihilator of ∆T ∗ Q,
KG (KMMF) Seminarium Geometryczne 13 X 2010 5 / 26

Almost Dirac structures in mechanics
Definition
The almost Dirac structure on T ∗ Q induced by a distribution ∆Q is the
subbundle
∆ ⊂ TT ∗ Q ⊕T ∗ Q T ∗ T ∗ Q
∆ = {w + ϕ : w ∈ ∆T ∗ Q, ϕ − βQ(w) ∈ ∆ ◦ T ∗ Q }
If ∆Q = TQ then ∆T ∗ Q = TT ∗ Q and ∆ is the graph of βQ.
If ∆Q is involutive then ∆ is a true Dirac structure.
KG (KMMF) Seminarium Geometryczne 13 X 2010 6 / 26

Almost Dirac structures in mechanics
Definition
The almost Dirac structure on T ∗ Q induced by a distribution ∆Q is the
subbundle
∆ ⊂ TT ∗ Q ⊕T ∗ Q T ∗ T ∗ Q
∆ = {w + ϕ : w ∈ ∆T ∗ Q, ϕ − βQ(w) ∈ ∆ ◦ T ∗ Q }
If ∆Q = TQ then ∆T ∗ Q = TT ∗ Q and ∆ is the graph of βQ.
If ∆Q is involutive then ∆ is a true Dirac structure.
KG (KMMF) Seminarium Geometryczne 13 X 2010 6 / 26

Almost Dirac structures in mechanics
Definition
The almost Dirac structure on T ∗ Q induced by a distribution ∆Q is the
subbundle
∆ ⊂ TT ∗ Q ⊕T ∗ Q T ∗ T ∗ Q
∆ = {w + ϕ : w ∈ ∆T ∗ Q, ϕ − βQ(w) ∈ ∆ ◦ T ∗ Q }
If ∆Q = TQ then ∆T ∗ Q = TT ∗ Q and ∆ is the graph of βQ.
If ∆Q is involutive then ∆ is a true Dirac structure.
KG (KMMF) Seminarium Geometryczne 13 X 2010 6 / 26

Almost Dirac structures in mechanics
Local expressions: For O being an open subset of Q we identify
TO O × V ∋ (q, ˙q),
T ∗ O O × V ∗ ∋ (q, p),
TT ∗ O O × V ∗ × V × V ∗ ∋ (q, p, ˙q, ˙p) = w,
T ∗ T ∗ O O × V ∗ × V ∗ × V ∋ (q, p, a, b) = ϕ.
∆Q = {(q, ˙q) : ˙q ∈ ∆Q(q)}, ∆Q(q) ⊂ V ,
∆ ◦ Q = {(q, ˙q) : ˙q ∈ ∆◦ Q (q)}, ∆◦ Q (q) ⊂ V ∗ ,
∆T ∗ Q = {(q, p, ˙q, ˙p) : ˙q ∈ ∆Q(q)},
∆ ◦ T∗Q = {(q, p, a, b) : a ∈ ∆◦Q (q), b = 0}.
∆ = {(q, p, ˙q, ˙p, a, b) : ˙q ∈ ∆Q(q), a + ˙p ∈ ∆ ◦ Q (q), b = ˙q}.
KG (KMMF) Seminarium Geometryczne 13 X 2010 7 / 26

Almost Dirac structures in mechanics
Local expressions: For O being an open subset of Q we identify
TO O × V ∋ (q, ˙q),
T ∗ O O × V ∗ ∋ (q, p),
TT ∗ O O × V ∗ × V × V ∗ ∋ (q, p, ˙q, ˙p) = w,
T ∗ T ∗ O O × V ∗ × V ∗ × V ∋ (q, p, a, b) = ϕ.
∆Q = {(q, ˙q) : ˙q ∈ ∆Q(q)}, ∆Q(q) ⊂ V ,
∆ ◦ Q = {(q, ˙q) : ˙q ∈ ∆◦ Q (q)}, ∆◦ Q (q) ⊂ V ∗ ,
∆T ∗ Q = {(q, p, ˙q, ˙p) : ˙q ∈ ∆Q(q)},
∆ ◦ T∗Q = {(q, p, a, b) : a ∈ ∆◦Q (q), b = 0}.
∆ = {(q, p, ˙q, ˙p, a, b) : ˙q ∈ ∆Q(q), a + ˙p ∈ ∆ ◦ Q (q), b = ˙q}.
KG (KMMF) Seminarium Geometryczne 13 X 2010 7 / 26

Almost Dirac structures in mechanics
Local expressions: For O being an open subset of Q we identify
TO O × V ∋ (q, ˙q),
T ∗ O O × V ∗ ∋ (q, p),
TT ∗ O O × V ∗ × V × V ∗ ∋ (q, p, ˙q, ˙p) = w,
T ∗ T ∗ O O × V ∗ × V ∗ × V ∋ (q, p, a, b) = ϕ.
∆Q = {(q, ˙q) : ˙q ∈ ∆Q(q)}, ∆Q(q) ⊂ V ,
∆ ◦ Q = {(q, ˙q) : ˙q ∈ ∆◦ Q (q)}, ∆◦ Q (q) ⊂ V ∗ ,
∆T ∗ Q = {(q, p, ˙q, ˙p) : ˙q ∈ ∆Q(q)},
∆ ◦ T∗Q = {(q, p, a, b) : a ∈ ∆◦Q (q), b = 0}.
∆ = {(q, p, ˙q, ˙p, a, b) : ˙q ∈ ∆Q(q), a + ˙p ∈ ∆ ◦ Q (q), b = ˙q}.
KG (KMMF) Seminarium Geometryczne 13 X 2010 7 / 26

Constrained Lagrangian system according to Y & M
Let L : TQ → R be a Lagrangian (possibly not hyperregular).
T∗TQ ζ
⑤⑤⑤⑤⑤⑤⑤⑤⑤
❆
π ❆ TQ ❆
T∗Q❇ TQ
❇
πQ τQ ❇
⑥⑥⑥⑥⑥⑥⑥⑥⑥
❲
❆dL
✮
★
❴
❴ ❴ ❴ FL❴
❴ ❴ ❴
Q
Definition
The Legendre map FL : TQ → T ∗ Q is defined by the formula
FL = ζ ◦ dL.
KG (KMMF) Seminarium Geometryczne 13 X 2010 8 / 26

Constrained Lagrangian system according to Y & M
Let L : TQ → R be a Lagrangian (possibly not hyperregular).
T∗TQ ζ
⑤⑤⑤⑤⑤⑤⑤⑤⑤
❆
π ❆ TQ ❆
T∗Q❇ TQ
❇
πQ τQ ❇
⑥⑥⑥⑥⑥⑥⑥⑥⑥
❲
❆dL
✮
★
❴
❴ ❴ ❴ FL❴
❴ ❴ ❴
Q
Definition
The Legendre map FL : TQ → T ∗ Q is defined by the formula
FL = ζ ◦ dL.
KG (KMMF) Seminarium Geometryczne 13 X 2010 8 / 26

Constrained Lagrangian system according to Y & M
There exists the canonical isomorphism
Locally for O ⊂ Q we have
and
γQ : T ∗ TQ −→ T ∗ T ∗ Q
T ∗ TO O × V × V ∗ × V ∗ , T ∗ T ∗ O O × V ∗ × V ∗ × V
γQ(q, ˙q, c, d) = (q, d, −c, ˙q)
It exists for any vector bundle (not only tangent bundle), it is a double
vector bundle isomorphism and antisymplectomorphism.
KG (KMMF) Seminarium Geometryczne 13 X 2010 9 / 26

Constrained Lagrangian system according to Y & M
Definition
For L : TQ → R the composition
γQ ◦ dL : TQ −→ T ∗ T ∗ Q
will be called the Dirac differential of L and denoted by DL
Locally:
DL(q, ˙q) = (q, ∂L
, −∂L , ˙q)
∂ ˙q ∂q
KG (KMMF) Seminarium Geometryczne 13 X 2010 10 / 26

Constrained Lagrangian system according to Y & M
Definition
A partial vector field on T ∗ Q is a map
such that
X : TQ ⊕Q T ∗ Q −→ TT ∗ Q
X (v, p) ∈ TpT ∗ Q.
KG (KMMF) Seminarium Geometryczne 13 X 2010 11 / 26

Constrained Lagrangian system according to Y & M
A Lagrangian system with nonholonomic constraints ∆Q can be described
by a triple
(L, ∆, X )
where
L is a Lagrangian function, possibly not hyperregular,
∆ is the Dirac structure associated to the constraint distribution ∆Q,
X is a partial vector field defined in points
and such that
(v, p) ∈ TQ ⊕ T ∗ Q, such that v ∈ ∆Q, p = FL(v)
X (v, p) + DL(v) ∈ ∆
KG (KMMF) Seminarium Geometryczne 13 X 2010 12 / 26

Constrained Lagrangian system according to Y & M
A Lagrangian system with nonholonomic constraints ∆Q can be described
by a triple
(L, ∆, X )
where
L is a Lagrangian function, possibly not hyperregular,
∆ is the Dirac structure associated to the constraint distribution ∆Q,
X is a partial vector field defined in points
and such that
(v, p) ∈ TQ ⊕ T ∗ Q, such that v ∈ ∆Q, p = FL(v)
X (v, p) + DL(v) ∈ ∆
KG (KMMF) Seminarium Geometryczne 13 X 2010 12 / 26

Constrained Lagrangian system according to Y & M
A Lagrangian system with nonholonomic constraints ∆Q can be described
by a triple
(L, ∆, X )
where
L is a Lagrangian function, possibly not hyperregular,
∆ is the Dirac structure associated to the constraint distribution ∆Q,
X is a partial vector field defined in points
and such that
(v, p) ∈ TQ ⊕ T ∗ Q, such that v ∈ ∆Q, p = FL(v)
X (v, p) + DL(v) ∈ ∆
KG (KMMF) Seminarium Geometryczne 13 X 2010 12 / 26

Constrained Lagrangian system according to Y & M
A Lagrangian system with nonholonomic constraints ∆Q can be described
by a triple
(L, ∆, X )
where
L is a Lagrangian function, possibly not hyperregular,
∆ is the Dirac structure associated to the constraint distribution ∆Q,
X is a partial vector field defined in points
and such that
(v, p) ∈ TQ ⊕ T ∗ Q, such that v ∈ ∆Q, p = FL(v)
X (v, p) + DL(v) ∈ ∆
KG (KMMF) Seminarium Geometryczne 13 X 2010 12 / 26

Constrained Lagrangian system according to Y & M
A Lagrangian system with nonholonomic constraints ∆Q can be described
by a triple
(L, ∆, X )
where
L is a Lagrangian function, possibly not hyperregular,
∆ is the Dirac structure associated to the constraint distribution ∆Q,
X is a partial vector field defined in points
and such that
(v, p) ∈ TQ ⊕ T ∗ Q, such that v ∈ ∆Q, p = FL(v)
X (v, p) + DL(v) ∈ ∆
KG (KMMF) Seminarium Geometryczne 13 X 2010 12 / 26

My point of view
Lagrangian dynamics without constraints according to Y & M (when we
forget partial vector fields)
γQ
T∗T∗Q● TT
✟ ●ξ
✟ ●
πT∗Q ✟
✟
✟
✟
∗ αQ
Q ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴
β ✡
❋ TπQ ❋
✡ ❋
τT∗Q ✡
✡
✡
✡
−1
T
Q
∗
TQ
dL
✡
❋
❋
ζ ✡ π ❋
TQ
TQ
TQ
✡
✡ TQ
✡
☛ ✡ ☛
τQ ✡ τQ ☛ ✡
☛
✡ ☛ τQ
T
☛
✡ ☛ ☛
✡ ☛ ☛
✡
☛
☛
∗Q❍ T
πQ ❍
❍
∗Q● T
πQ ●
●
∗Q● πQ ●
●
Q Q Q
DL = β −1
Q (γQ(dL(TQ)))
KG (KMMF) Seminarium Geometryczne 13 X 2010 13 / 26

My point of view
Lagrangian dynamics without constraints:
DL
T∗T∗Q● TT
✟ ●ξ
✟ ●
πT∗Q ✟
✟
✟
✟
∗ ❴
❴ ❴ ❴ ❴ ❴ ❴ ❴ Q ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴
✡
❋
T
βQ
TπQ
αQ
❋
✡ ❋
τT∗Q ✡
✡
✡
✡
∗ α
TQ
✡
❋
❋
ζ ✡ π ❋
TQ
✡
✡
✡
✡
−1
Q
dL
TQ
TQ
TQ
✡
☛
☛
τQ ✡ τQ ☛ ☛
✡ ☛ τQ
T
☛
✡ ☛ ☛
✡ ☛ ☛
✡
☛
☛
∗Q❍ T
πQ ❍
❍
∗Q● T
πQ ●
●
∗Q● πQ ●
●
Q Q Q
DL = α −1
Q (dL(TQ))
KG (KMMF) Seminarium Geometryczne 13 X 2010 14 / 26

My point of view
Lagrangian dynamics with constraints according to Y & M
γQ
T∗T∗ ∆ ✤ αQ
Q
●
TT∗Q ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴
✟ ●ξ
❋
T
✡ TπQ ❋
✟ ●
✡ ❋
πT∗Q ✟ τT∗Q ✡
✟ ✡
✟ ✡
✟
✡
∗
TQ
dL
✡
❋
❋
ζ ✡ π ❋
TQ
TQ
TQ
✡
✡ TQ
✡
☛ ✡ ☛
τQ ✡ τQ ☛ ✡
☛
✡ ☛ τQ
T
☛
✡ ☛ ☛
✡ ☛ ☛
✡
☛
☛
∗Q❍ T
πQ ❍
❍
∗Q● T
πQ ●
●
∗Q● πQ ●
●
Q Q Q
DL = ∆(γQ(dL(TQ)))
KG (KMMF) Seminarium Geometryczne 13 X 2010 15 / 26

My point of view
Back
We treat ∆ as a relation T ∗ T ∗ Q −− ⊲ TT ∗ Q
We can take a ”short-cut” identifying T ∗ T ∗ Q with T ∗ TQ
Local expressions:
TT∗Q❊ T
✡ TπQ ❊
✡ ❊
τT∗Q ✡
✡
✡
✡
∗ ˜∆
✤
TQ
dL
✡
❊
❊
ζ ✡ π ❊ TQ
✡
∆Q ✡ ∆Q
✡
✡
T∗Q T∗Q T ∗ TO O × V × V ∗ × V ∗ ∋ (q, ˙q, c, d)
TT ∗ O O × V ∗ × V × V ∗ ∋ (q, p, ˙q, ˙p)
˜∆ : ˙q ∈ ∆Q(q), p = d ˙p − c ∈ ∆ ◦ Q (q)
KG (KMMF) Seminarium Geometryczne 13 X 2010 16 / 26

My point of view
Back
We treat ∆ as a relation T ∗ T ∗ Q −− ⊲ TT ∗ Q
We can take a ”short-cut” identifying T ∗ T ∗ Q with T ∗ TQ
Local expressions:
TT∗Q❊ T
✡ TπQ ❊
✡ ❊
τT∗Q ✡
✡
✡
✡
∗ ˜∆
✤
TQ
dL
✡
❊
❊
ζ ✡ π ❊ TQ
✡
∆Q ✡ ∆Q
✡
✡
T∗Q T∗Q T ∗ TO O × V × V ∗ × V ∗ ∋ (q, ˙q, c, d)
TT ∗ O O × V ∗ × V × V ∗ ∋ (q, p, ˙q, ˙p)
˜∆ : ˙q ∈ ∆Q(q), p = d ˙p − c ∈ ∆ ◦ Q (q)
KG (KMMF) Seminarium Geometryczne 13 X 2010 16 / 26

My point of view
Back
We treat ∆ as a relation T ∗ T ∗ Q −− ⊲ TT ∗ Q
We can take a ”short-cut” identifying T ∗ T ∗ Q with T ∗ TQ
Local expressions:
TT∗Q❊ T
✡ TπQ ❊
✡ ❊
τT∗Q ✡
✡
✡
✡
∗ ˜∆
✤
TQ
dL
✡
❊
❊
ζ ✡ π ❊ TQ
✡
∆Q ✡ ∆Q
✡
✡
T∗Q T∗Q T ∗ TO O × V × V ∗ × V ∗ ∋ (q, ˙q, c, d)
TT ∗ O O × V ∗ × V × V ∗ ∋ (q, p, ˙q, ˙p)
˜∆ : ˙q ∈ ∆Q(q), p = d ˙p − c ∈ ∆ ◦ Q (q)
KG (KMMF) Seminarium Geometryczne 13 X 2010 16 / 26

My point of view
Back
We treat ∆ as a relation T ∗ T ∗ Q −− ⊲ TT ∗ Q
We can take a ”short-cut” identifying T ∗ T ∗ Q with T ∗ TQ
Local expressions:
TT∗Q❊ T
✡ TπQ ❊
✡ ❊
τT∗Q ✡
✡
✡
✡
∗ ˜∆
✤
TQ
dL
✡
❊
❊
ζ ✡ π ❊ TQ
✡
∆Q ✡ ∆Q
✡
✡
T∗Q T∗Q T ∗ TO O × V × V ∗ × V ∗ ∋ (q, ˙q, c, d)
TT ∗ O O × V ∗ × V × V ∗ ∋ (q, p, ˙q, ˙p)
˜∆ : ˙q ∈ ∆Q(q), p = d ˙p − c ∈ ∆ ◦ Q (q)
KG (KMMF) Seminarium Geometryczne 13 X 2010 16 / 26

My point of view
How to get ˜ ∆ not going to the ”Hamiltonian” side of the diagram?
Many of the ideas od Analytical Mechanics come from statics,
In statics constraints are described by admissible configurations and
admissible virtual displacements (WMT),
We observed in ”Variational ... in general algebroids” that different
set of admissible virtual displacements produce different E-L
equations (vaconomic, nonholonomic) for the same set of admissible
configurations.
How to put virtual displacements into the game?
KG (KMMF) Seminarium Geometryczne 13 X 2010 17 / 26

My point of view
How to get ˜ ∆ not going to the ”Hamiltonian” side of the diagram?
Many of the ideas od Analytical Mechanics come from statics,
In statics constraints are described by admissible configurations and
admissible virtual displacements (WMT),
We observed in ”Variational ... in general algebroids” that different
set of admissible virtual displacements produce different E-L
equations (vaconomic, nonholonomic) for the same set of admissible
configurations.
How to put virtual displacements into the game?
KG (KMMF) Seminarium Geometryczne 13 X 2010 17 / 26

My point of view
How to get ˜ ∆ not going to the ”Hamiltonian” side of the diagram?
Many of the ideas od Analytical Mechanics come from statics,
In statics constraints are described by admissible configurations and
admissible virtual displacements (WMT),
We observed in ”Variational ... in general algebroids” that different
set of admissible virtual displacements produce different E-L
equations (vaconomic, nonholonomic) for the same set of admissible
configurations.
How to put virtual displacements into the game?
KG (KMMF) Seminarium Geometryczne 13 X 2010 17 / 26

My point of view
How to get ˜ ∆ not going to the ”Hamiltonian” side of the diagram?
Many of the ideas od Analytical Mechanics come from statics,
In statics constraints are described by admissible configurations and
admissible virtual displacements (WMT),
We observed in ”Variational ... in general algebroids” that different
set of admissible virtual displacements produce different E-L
equations (vaconomic, nonholonomic) for the same set of admissible
configurations.
How to put virtual displacements into the game?
KG (KMMF) Seminarium Geometryczne 13 X 2010 17 / 26

My point of view
How to get ˜ ∆ not going to the ”Hamiltonian” side of the diagram?
Many of the ideas od Analytical Mechanics come from statics,
In statics constraints are described by admissible configurations and
admissible virtual displacements (WMT),
We observed in ”Variational ... in general algebroids” that different
set of admissible virtual displacements produce different E-L
equations (vaconomic, nonholonomic) for the same set of admissible
configurations.
How to put virtual displacements into the game?
KG (KMMF) Seminarium Geometryczne 13 X 2010 17 / 26

My point of view
The unconstrained case: all virtual displacements are admissible
KG (KMMF) Seminarium Geometryczne 13 X 2010 18 / 26

My point of view
The constrained case: we choose admissible displacements
Locally:
U = {u ∈ TTQ : τTQ(u) ∈ ∆Q, TτQ(u) ∈ ∆Q}
u = (q, ˙q, δq, δ ˙q) : ˙q ∈ ∆Q(q), δq ∈ ∆Q(q)
κQ(U) = U
We treat κQ |U as a relation in TTQ and we look for the dual relation with
respect to the evaluations 〈·, ·〉, 〈〈·, ·〉〉.
KG (KMMF) Seminarium Geometryczne 13 X 2010 19 / 26

My point of view
The constrained case: we choose admissible displacements
Locally:
U = {u ∈ TTQ : τTQ(u) ∈ ∆Q, TτQ(u) ∈ ∆Q}
u = (q, ˙q, δq, δ ˙q) : ˙q ∈ ∆Q(q), δq ∈ ∆Q(q)
κQ(U) = U
We treat κQ |U as a relation in TTQ and we look for the dual relation with
respect to the evaluations 〈·, ·〉, 〈〈·, ·〉〉.
KG (KMMF) Seminarium Geometryczne 13 X 2010 19 / 26

My point of view
The constrained case: we choose admissible displacements
Locally:
U = {u ∈ TTQ : τTQ(u) ∈ ∆Q, TτQ(u) ∈ ∆Q}
u = (q, ˙q, δq, δ ˙q) : ˙q ∈ ∆Q(q), δq ∈ ∆Q(q)
κQ(U) = U
We treat κQ |U as a relation in TTQ and we look for the dual relation with
respect to the evaluations 〈·, ·〉, 〈〈·, ·〉〉.
KG (KMMF) Seminarium Geometryczne 13 X 2010 19 / 26

My point of view
The constrained case: we choose admissible displacements
Locally:
U = {u ∈ TTQ : τTQ(u) ∈ ∆Q, TτQ(u) ∈ ∆Q}
u = (q, ˙q, δq, δ ˙q) : ˙q ∈ ∆Q(q), δq ∈ ∆Q(q)
κQ(U) = U
We treat κQ |U as a relation in TTQ and we look for the dual relation with
respect to the evaluations 〈·, ·〉, 〈〈·, ·〉〉.
KG (KMMF) Seminarium Geometryczne 13 X 2010 19 / 26

My point of view
˜∆
KG (KMMF) Seminarium Geometryczne 13 X 2010 20 / 26

My point of view
Conclusion
The relation ˜∆ is dual to the relation κQ restricted to the set of admissible
virtual displacements.
Equations
The equations for a curve t ↦−→ ℘(t) in the phase space T ∗ Q are
Local expressions:
t ↦−→ (q(t), p(t))
t℘(t) ∈ ˜ ∆ ◦ dL(∆Q)
˙q ∈ ∆Q(q(t)), p(t) = ∂L
∂L
(q, ˙q), ˙p(t) ∈
∂ ˙q ∂q (q, ˙q) + ∆◦Q (q(t))
KG (KMMF) Seminarium Geometryczne 13 X 2010 21 / 26

My point of view
Conclusion
The relation ˜∆ is dual to the relation κQ restricted to the set of admissible
virtual displacements.
Equations
The equations for a curve t ↦−→ ℘(t) in the phase space T ∗ Q are
Local expressions:
t ↦−→ (q(t), p(t))
t℘(t) ∈ ˜ ∆ ◦ dL(∆Q)
˙q ∈ ∆Q(q(t)), p(t) = ∂L
∂L
(q, ˙q), ˙p(t) ∈
∂ ˙q ∂q (q, ˙q) + ∆◦Q (q(t))
KG (KMMF) Seminarium Geometryczne 13 X 2010 21 / 26

Example
Lagrangian system with constraints: a sleigh on a horizontal plane
Q = R 2 × S 1 ∋ (x, y, ϕ),
∆Q(x, y, ϕ) = {( ˙x, ˙y, ˙ϕ) : ˙x sin(ϕ) = ˙y cos(ϕ)},
∆Q = 〈 ∂ ∂
, cos(ϕ)
∂ϕ ∂x
+ sin(ϕ) ∂
∂y 〉
L(x, y, ϕ, ˙x, ˙y, ˙ϕ) = m
2 ( ˙x 2 + ˙y 2 ) + I
2 ˙ϕ2
KG (KMMF) Seminarium Geometryczne 13 X 2010 22 / 26

Example
Equations for a curve t ↦−→ (x(t), y(t), ϕ(t), px(t), py (t), π(t))
˙x = px/m, ˙px = µ(t) sin ϕ
˙y = py /m, ˙py = −µ(t) cos ϕ
˙ϕ = π/m, ˙π = 0
˙x sin ϕ = ˙y cos ϕ
Solution (momentum)
π(t) = π0
px(t) = p0 cos( π0
t + ϕ0)
I
py (t) = p0 sin( π0
t + ϕ0)
I
KG (KMMF) Seminarium Geometryczne 13 X 2010 23 / 26

Example
Modeling constraints: a sleigh subject to strong viscous force
perpendicular to the sleigh
Equations for a curve t ↦−→ (x(t), y(t), ϕ(t), px(t), py (t), π(t)) (γ < 0)
˙x = px/m, ˙px = (γ/m)(py cos ϕ − px sin ϕ)(− sin ϕ)
˙y = py /m, ˙py = (γ/m)(py cos ϕ − px sin ϕ)(cos ϕ)
˙ϕ = π/I , ˙π = 0
KG (KMMF) Seminarium Geometryczne 13 X 2010 24 / 26

Example
Initial momentum along the sleigh, π0/I = 1, ϕ0 = 0
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
black: solution with constraints
red: γ/m = −3
blue: γ/m = −10
green: γ/m = −100
KG (KMMF) Seminarium Geometryczne 13 X 2010 25 / 26

Example
Initial momentum perpendicular to the sleigh, π0/I = 1, ϕ0 = 0
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
red: γ/m = −3
blue: γ/m = −10
green: γ/m = −100
KG (KMMF) Seminarium Geometryczne 13 X 2010 26 / 26