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CMPE-633C / PHYS-401: Geometric Mechanics and Control


Dr Abubakr Muhammad, Assistant Professor of Electrical Engineering

Email: abubakr [at]

Office: Room 9-311A, 3rd Floor, SSE Bldg

Course Details

Year: 2011-12

Semester: Spring

Category: Grad

Credits: 3

Elective course for electrical engineering, computer engineering and physics majors

Course Website:

Course Description

The aim of this course is to provide a unified treatment of nonlinear control theory and mechanical systems subject to motion constraints. Both Lagrangian and Hamiltonian formulations will be developed with an emphasis on solving control problems for nonholonomic and under-actuated systems. Important topics covered in this course include the center manifold theorem for stability, Frobenius and Chow’s theorems, Poisson geometry of nonholonomic systems, symplectic geometry of Hamiltonian flows etc. The students will also learn basic concepts of differential forms, Lie algebras, distributions and Riemannian geometry to formulate these results. Applications of this formulation span controller design in robotics, quantum information, aerospace and other systems that do not yield to non-geometric methods in control.


  • Use of geometric frameworks to understand nonholonomic and under-actuated systems
  • Lagrangian and Hamiltonian formulation of analytical mechanics
  • Differential geometry methods in control problems
  • Applications in robotics, systems theory and mathematical physics

Learning Outcomes

  • Link control theory with a geometric view-point of classical mechanics.
  • Identify holonomic & nonholonomic constraints in physical systems.
  • Understand control limitations due to under-actuation and motion constraints.
  • Appreciate the value of abstract differential geometry methods in solving real-world problems.


  • CMPE-633b (Robot Dynamics & Control) OR
  • PHYS-310 (Classical Mechanics) OR
  • MATH-361 (Dynamical systems) OR
  • By Permission of Instructor

Self check. Before start of class you should be familiar with configuration spaces, Euler-Lagrange equations, phase-plane analysis and fluent with use of vector calculus, linear algebra and differential equations.

Who should take it? EE/CMPE majors seriously interested in robotics and control theory; SSE students interested in pursuing mathematical physics.

Text book

Main text book.

  • Nonholonomic Mechanics and Control by Anthony Bloch, P. Crouch, J. Baillieul, J. Marsden. Interdisciplinary Applied Mathematics Springer-Verlag NY, 2003.

Reference texts.

  • An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised, Volume 120, Second Edition (Pure and Applied Mathematics) by William M. Boothby.
  • Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems (Texts in Applied Mathematics) by Jerrold E. Marsden, Tudor S. Ratiu.
  • Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics) by V.I. Arnold [ geometric mechanics]
  • Riemannian Geometry by Manfredo P. do Carmo , Francis Flaherty.
  • Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems by Francesco Bullo, Andrew D. Lewis.

Grading Scheme

  • Home-works (weekly): 25%
  • Class Participation: 10%
  • Paper Project: 40%
  • Midterm Examination: 25%


Week 1. Jan 23 Lec 1. Configuration spaces; topology and geometry of important configuration spaces; visualization methods for S^n, RP^n, SO(n), T^n; Bloch Ch1;
Week 2. Jan 30 No classes
Week 3. Feb 6 Lec 2. Largrangian mechnaics; variational principles; Hamilton's principle; Euler-Lagrange Equations; Legendre transformation; Hamiltonian mechanics;

Lec 3. Holonomic and nonholonomic constraints; Lagrange D'Alembert's principle; external forces; dynamical and variational nonholonomic equations of motion; optimal control problem; vertical rolling disk example;

Bloch Ch1;
Week 4. Feb 13 Problem solving session. Examples on Lagrangian variational dynamics, optimal control problems

Lec 4. Differential manifolds; differential structures; coordinate charts and atlases; directional derivatives; tangent vectors; tangent space to manifolds; maps between manifolds

do Carmo Ch0;
Week 5. Feb 20

Problem solving session. Group assignments on Lagrangian dynamics

Lec 5. Tangent bundle; Tangent bundle as manifold; vector fields; maps between tangent spaces induced by maps between manifolds; differential map;

do Carmo Ch0;
Week 6. Feb 27

Problem solving session. Coordinate charts on Cartesian product of manifolds; charts for RP^n;

Lec 6. Flows related to vector fields; integral curves; solutions to differential equations; Lie derivative of a vector field; Lie brackets as Lie derivatives; properties of Lie brackets;

do Carmo Ch0;
Week 7. Mar 5 Problem solving session. Coordinate charts on S^n; SO(3): Euler angle, quaternions as charts;

Lec 7. Dual to vector spaces; 1-forms as dual to vector fields; cotangent space; cotangent bundle; Multi-linearity and skew symmetry on tensor fields; Differential k-forms on manifolds; differential forms in local coordinates; k-form basis, their duals and computations;

Block Ch2;
Week 8. Mar 12

Lec 8. Interior product; exterior derivative on forms; differential forms on R^3; dualization operators in R^3 between forms and vector fields; vector calculus using differential forms;

Lec 9. Pull-backs and push-forwards of maps; Lie derivative of differential forms; Cartan's magic formula; important identities on Lie derivatives; integration over manifolds; Stoke's theorem;

Block Ch2;
Week 9. Mar 19

Mid Semester break

Week 10. Mar 26

Review Session. Differential forms revisited.

Lec 10. Distributions, involutive and integrable distributions, Frobenius's theorem, Fibre bundles, Vector bundles, Riemannian metrics, lengths on Riemannian manifolds

Block Ch2;
Week 11. April 2

Problem solving session. Computations on closed & exact forms, Lie derivatives of forms, flows of vector fields etc.

Lec 11. Affine connections; Covariant derivative; parallel transport; metric compatibility; preview to Levi Cevita connections;

Block Ch2; DoCarmo Ch 1,2;
Week 12. April 9

Problem solving session. Review of fibre bundles; examples; top-down and bottom-up constructions; fibres, structure group and triviality;

Lec 13. From affine connection to Levi Cevita connections; metric compatibility conditions; symmetry condition in affine connections; Existence and uniquesness of Levi Cevita connection; Christoffel symbols; geodesic equations;


DoCarmo Ch2l
Week 13. April 16

Lec 14. Prelude to geometric mechanics; dictionary of terms between differential geometry and mechanics; configuration space manifolds; velocities on tangent bundles; rigid body motion; Riemannian metrics arising from kinetic energy definitions;

Lec 15. Euler Lagrange equations from a geometric perspective; Christoffel symbols and Levi Cevita connections revisited; external forces as elements of the cotangent bundle; distributions and co-distributions corresponding to non-holonomic constraints; affine connection modification by non-holonomic constraints; Lagrange-d'Alembert principle in geometric language;

Lecture notes by Francesco Bullo and Andrew Lewis.
Week 14. April 23 Review Session. Lie algebra of vector fields; Lie brackets and its connection to controllability; Chow's theorem; A quick overview of holonomy; vertical lifts in fibre bundles; vertical and horizontal vector fields;

Lec 16. Geometrization of Hamiltonian mechanics; Poisson and symplectic manifolds; Hamiltonian vector fields in Poisson and Symplectic manifolds; volume preservation in Hamiltonian flows; Lioville's theorem; canonical symplectic 2-forms on R^2n;

Bloch Ch 3; Marsden Ch 2;
Week 15. April 30

Lec 17. Canonical 1-forms on the cotangent bundle; derived canonical symplectic 2-forms; cotangent bundles as local structure of symplectic manifolds (Darboux's theorem); Geometrization of Lagrangian mechanics; fibre derivative and coordinate transformation via Legendre transformation; hyperregular Lagrangians; canonical 1-forms and 2-forms correponding to Lagrangians; Lagrangian vector fields; 2nd order vector fields; comparison of Hamiltonian and Lagrangian dynamics;

Block Ch3; Marsden Ch2;
Week 16. May 7 Presentations.

Student Projects

  • Nauman Ahmad, Motion planning with non-holonomic constraints.
  • Syed Moeez Hassan, Berry phase and holonomy.
  • Talal Jawlana, Kinematics of contacts and rolling mechanisms.
  • Mudassir Moosa, Geometric quantum computation.
  • Imran Sajjad, Motion planning with non-holonomic constraints.
  • Muntazir Abidi, Geometric quantization.
  • Ahmed Umer Ashraf, Euler Lagrange cohomological groups on symplectic manifolds.
  • Talha Manzoor, Optimal control of a homogeneous rolling ball.
  • Usama Bin Sikandar, Geometric Motion Planning Analysis of Two Classes of Underactuated Mechanical Systems.
  • Nouman Tariq, Foucalt Pendulum and Geometric Phase.
  • Azeem-ul-Hassan, Geometric quantization.
  • Usman Naseer, Topological quantum field theory.
  • Anadil Saeed, Geometry of the falling cat problem.
  • Alamdar Hussain, Equations of motion of a magnetic monopole: An Example of Phase-Space reductions on Cotangent Bundles.
  • Muhammad Usama, Lagrangian formulation of linear circuits.
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