Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease...
Transcript of Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease...
![Page 1: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/1.jpg)
Is swimming a limit cycle?
Henry O. Jacobs
Geometry, Mechanics, and Dynamics:The Legacy of Jerry Marsden
at the Fields Institute, July 23, 2012THIS IS A BOARD TALK (GET CLOSE)
![Page 2: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/2.jpg)
names
I Erica J. Kim *
I Yu Zheng *
I Dennis Evangelista *
I Sam Burden **
I Alan Weinstein ***
I Jerrold E. Marsden
* Integrative Biology, U.C. Berkeley** EECS, U.C. Berkeley*** Mathematics, U.C. Berkeley
![Page 3: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/3.jpg)
Outline
Motivation
Math
Conclusion
![Page 4: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/4.jpg)
Outline
Motivation
Math
Conclusion
![Page 5: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/5.jpg)
Does this make sense?
Figure : figure courtesy Kakani Katija
Courtesy Nikita Nester
Figure : Jellyfish in Palau (video by Naoki Inoue posted onYouTube Feb 2007)
![Page 6: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/6.jpg)
The Averaging Theorem
TheoremLet
x = f (x)
be a dynamical system with an asymptotically stable fixedpoint at x0. Then for any T -periodic vector field, g(x , t), thedynamical systems
x = f (x) + εg(x , t)
admits a limit cycle near x0 with period T for sufficiently smallε.1
1Guckenheimer & Holmes, Nonlinear Oscillations and Chaos, 2nd Ed,Springer (1983).
![Page 7: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/7.jpg)
The passive system has a stable point
eb
f
b(f)b
Figure : embedding of a dead fish in R3
a motionless corpse in stagnant water is a stable state.
![Page 8: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/8.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?
![Page 9: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/9.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?
![Page 10: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/10.jpg)
An analogous systemConsider the system on R3
x = y
y = −x − νy + ε sin(t)
z = x + xy
The first two equations are that of a forced/damped oscillator.Note that this ODE has z symmetry so we can “ignore” z.(draw diagram on board)
−1 −0.5 0 0.5 1 1.5 2−1
−0.5
0
0.5
1
1.5
Figure : reduced trajectory
−2
−1
0
1
2
−2
−1
0
1
2
−60
−50
−40
−30
−20
−10
0
Figure : unreducedtrajectory
![Page 11: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/11.jpg)
An analogous systemConsider the system on R3
x = y
y = −x − νy + ε sin(t)
z = x + xy
The first two equations are that of a forced/damped oscillator.Note that this ODE has z symmetry so we can “ignore” z.(draw diagram on board)
−1 −0.5 0 0.5 1 1.5 2−1
−0.5
0
0.5
1
1.5
Figure : reduced trajectory
−2
−1
0
1
2
−2
−1
0
1
2
−60
−50
−40
−30
−20
−10
0
Figure : unreducedtrajectory
![Page 12: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/12.jpg)
An analogous systemConsider the system on R3
x = y
y = −x − νy + ε sin(t)
z = x + xy
The first two equations are that of a forced/damped oscillator.Note that this ODE has z symmetry so we can “ignore” z.(draw diagram on board)
−1 −0.5 0 0.5 1 1.5 2−1
−0.5
0
0.5
1
1.5
Figure : reduced trajectory
−2
−1
0
1
2
−2
−1
0
1
2
−60
−50
−40
−30
−20
−10
0
Figure : unreducedtrajectory
![Page 13: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/13.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 14: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/14.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 15: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/15.jpg)
Previous Work1. Liao et. al. Fish exploiting vortices decrease muscle
activity, Science 302 (2003).2. S. Alben, M. J. Shelley, Coherent locomotion as an
attracting state for a free flapping body, PNAS 102(2005).
3. A. Shapere, F. Wilczek, Geometry of self-propulsion atlow Reynolds number, JFM 198 (1989).
4. S. D. Kelly,The mechanics and control of roboticlocomotion with applications to aquatic vehicles, PhDthesis, Caltech, (1998).
5. Kanso et. al., Locomotion of articulated bodies in aperfect fluid, J. Nonlinear Sci 15 (2005).
6. H. Cendra, J. Marsden, T. Ratiu, Lagrangian Reductionby Stages, Memoirs of the AMS, (2001).
7. A. Weinstein, Lagrangian Mechanics on Groupoids,Mechanics Day, Fields Inst, (1995).
![Page 16: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/16.jpg)
Outline
Motivation
Math
Conclusion
![Page 17: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/17.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 18: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/18.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 19: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/19.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 20: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/20.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 21: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/21.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 22: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/22.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 23: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/23.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 24: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/24.jpg)
What just happened?
![Page 25: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/25.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 26: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/26.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 27: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/27.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 28: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/28.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 29: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/29.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 30: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/30.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 31: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/31.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 32: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/32.jpg)
The Big Picture
dead fish
dead fishSE(3)
periodic dead fishSE(3)
reduction
periodic force
locomotion?periodic force
reduction
![Page 33: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/33.jpg)
Some sad news
The Averaging theorem requires that we be in a Banach space.Here are some musings
1. We can use the completion of Q? This involvesnon-differentiable mappings.
2. We can search for a set of feasible perturbations?
3. We may construct a sequence of finite dimensionalmodels.
![Page 34: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/34.jpg)
Outline
Motivation
Math
Conclusion
![Page 35: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/35.jpg)
Conclusion
We found:
1. a dead fish immersed in an ideal fluid is a Lagrangiansystem on TQ.
2. We can use particle relabling symmetry to reduce to A.
3. Friction forces produce a stable manifold, S ⊂ A.
4. The system on A is frame invariant.
5. Reduction by frame-invariance projects S to anasymptotically stable point [S ] ∈ [A].
6. The averaging theorem suggests that a periodic force onthe shape of the fish leads to a limit cycle in [A].
7. this implies the existence of a rigid motion (i.e. an SE(3)action) with each period.
8. ... almost.
![Page 36: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/36.jpg)
Conclusion
We found:
1. a dead fish immersed in an ideal fluid is a Lagrangiansystem on TQ.
2. We can use particle relabling symmetry to reduce to A.
3. Friction forces produce a stable manifold, S ⊂ A.
4. The system on A is frame invariant.
5. Reduction by frame-invariance projects S to anasymptotically stable point [S ] ∈ [A].
6. The averaging theorem suggests that a periodic force onthe shape of the fish leads to a limit cycle in [A].
7. this implies the existence of a rigid motion (i.e. an SE(3)action) with each period.
8. ... almost.
![Page 37: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/37.jpg)
Conclusion
We found:
1. a dead fish immersed in an ideal fluid is a Lagrangiansystem on TQ.
2. We can use particle relabling symmetry to reduce to A.
3. Friction forces produce a stable manifold, S ⊂ A.
4. The system on A is frame invariant.
5. Reduction by frame-invariance projects S to anasymptotically stable point [S ] ∈ [A].
6. The averaging theorem suggests that a periodic force onthe shape of the fish leads to a limit cycle in [A].
7. this implies the existence of a rigid motion (i.e. an SE(3)action) with each period.
8. ... almost.
![Page 38: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/38.jpg)
Conclusion
We found:
1. a dead fish immersed in an ideal fluid is a Lagrangiansystem on TQ.
2. We can use particle relabling symmetry to reduce to A.
3. Friction forces produce a stable manifold, S ⊂ A.
4. The system on A is frame invariant.
5. Reduction by frame-invariance projects S to anasymptotically stable point [S ] ∈ [A].
6. The averaging theorem suggests that a periodic force onthe shape of the fish leads to a limit cycle in [A].
7. this implies the existence of a rigid motion (i.e. an SE(3)action) with each period.
8. ... almost.
![Page 39: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/39.jpg)
Conclusion
We found:
1. a dead fish immersed in an ideal fluid is a Lagrangiansystem on TQ.
2. We can use particle relabling symmetry to reduce to A.
3. Friction forces produce a stable manifold, S ⊂ A.
4. The system on A is frame invariant.
5. Reduction by frame-invariance projects S to anasymptotically stable point [S ] ∈ [A].
6. The averaging theorem suggests that a periodic force onthe shape of the fish leads to a limit cycle in [A].
7. this implies the existence of a rigid motion (i.e. an SE(3)action) with each period.
8. ... almost.
![Page 40: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/40.jpg)
Conclusion
We found:
1. a dead fish immersed in an ideal fluid is a Lagrangiansystem on TQ.
2. We can use particle relabling symmetry to reduce to A.
3. Friction forces produce a stable manifold, S ⊂ A.
4. The system on A is frame invariant.
5. Reduction by frame-invariance projects S to anasymptotically stable point [S ] ∈ [A].
6. The averaging theorem suggests that a periodic force onthe shape of the fish leads to a limit cycle in [A].
7. this implies the existence of a rigid motion (i.e. an SE(3)action) with each period.
8. ... almost.
![Page 41: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/41.jpg)
Conclusion
We found:
1. a dead fish immersed in an ideal fluid is a Lagrangiansystem on TQ.
2. We can use particle relabling symmetry to reduce to A.
3. Friction forces produce a stable manifold, S ⊂ A.
4. The system on A is frame invariant.
5. Reduction by frame-invariance projects S to anasymptotically stable point [S ] ∈ [A].
6. The averaging theorem suggests that a periodic force onthe shape of the fish leads to a limit cycle in [A].
7. this implies the existence of a rigid motion (i.e. an SE(3)action) with each period.
8. ... almost.
![Page 42: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/42.jpg)
Conclusion
We found:
1. a dead fish immersed in an ideal fluid is a Lagrangiansystem on TQ.
2. We can use particle relabling symmetry to reduce to A.
3. Friction forces produce a stable manifold, S ⊂ A.
4. The system on A is frame invariant.
5. Reduction by frame-invariance projects S to anasymptotically stable point [S ] ∈ [A].
6. The averaging theorem suggests that a periodic force onthe shape of the fish leads to a limit cycle in [A].
7. this implies the existence of a rigid motion (i.e. an SE(3)action) with each period.
8. ... almost.
![Page 43: Is swimming a limit cycle? - Fields Institute...1.Liao et. al. Fish exploiting vortices decrease muscle activity, Science 302 (2003). 2.S. Alben, M. J. Shelley, Coherent locomotion](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f646e26f6d37a26561a2479/html5/thumbnails/43.jpg)
Primary References
video by Naoki Inoue
Figure : video by Naoki Inoue
1. Lagrangian Reduction by Stages [Cendra, Ratiu &Marsden, 1999].
2. A. Weinstein, Lagrangian mechanics and groupoids,Mechanics Day, Fields Inst. Proc., vol. 7, AMS, 1995.
3. H. J. , Geometric Descriptions of Couplings in Fluids andCircuits , Caltech PhD thesis, 2012.