Goldstein Classical Mechanics Homework Solutions

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Lecture Times: Wednesdays and Fridays, 3:20 PM - 4:55 PMLecture Location: ISB 235 The following classes are canceled: W Sept 28, F Sep 30, F Oct 14, F Nov 18, F Dec 2. Lectures are rescheduled for M Oct 3, M Oct 10, M Oct 17, M Oct 24, M Nov 21, ISB 235, 9:45-11:20 AM.Course descriptionThis course is a graduate-level introduction to the theoretical techniques of classical mechanics. Emphasis will be given in particular to those principles and mathematical constructions relevant to modern physics (including quantum mechanics and general relativity), as well as to more classical physical applications. A list of topics that will be covered in this course includes:Variational PrinciplesLagrangian FormulationApplications: the Central Force Problem, the Motion of Rigid Bodies, Small OscillationsHamiltonian FormulationCanonical TransformationsHamilton-Jacobi Theory and Action-Angle VariablesClassical Field Theory

Course OutlineLect. Topic Reading (Goldstein+P+S; *: Soper)1Preliminary remarks; Lagrangian formalism1.1-1.42Lagrangian methods: examples; Hamilton's principle1.5-1.6; 2.13Calculus of variations; Hamilton's principle with constraints2.2-2.54Symmetries and conservation laws2.6-2.75Central force problem; Closed orbits; Virial theorem3.1-3.76Scattering in a central force field3.10-3.117Lenz vector; Three-body problem; Numerical methods3.9; 3.128Coordinate transformations; Euler angles4.1-4.59Infinitesimal and finite rotations; Coriolis effect4.6-4.1010Inertia tensor; Rigid-body motion5.1-5.611Small oscillations and related examples6.1-6.412Legendre transformation and Hamilton equations of motion8.1-8.2; 8.413Principle of least action; Canonical transformations8.5-8.6; 9.1-9.214Canonical transformations: examples; Poisson brackets9.3-9.715Symmetry groups; Liouville's theorem; Hamilton-Jacobi theory9.8-9.9; 10.1-10.216Hamilton Jacobi theory and applications10.3-10.617Fields and transformation laws; stationary action and fields 1*, 2*18Classical field theory; the electromagnetic field3*, 8*19Further general properties of Field Theories9*19Course ReviewCourse Grading and RequirementsThere will be four homework sets. Each one of the homework sets will count 10% towards the final evaluation. There will be one midterm exam, scheduled for Wednesday November 9, under ``quals'' conditions, which will count 20%. The remaining 40% will be based on your performance in the final exam, also to be held under \"quals conditions\", in December.Homework exercisesHomework Set number(PDF)Due DateSolutionsHW Set #1phys210_HW01.pdfFriday 10/14HW1_solutions.pdf HW Set #2 phys210_HW02.pdf Friday 10/28 HW2 Solutions (by John Tamanas) Midterm phys210_midterm.pdf Wednesday 11/9 Midterm Solutions HW Set #3 phys210_HW03.pdf Monday November 21 phys210_HW3_sol.pdf and Hanwen's solutions HW Set #4 phys210_HW04.pdf Thursday December 8, 11AM, ISB 235 (final exam) Hanwen's solutions Final phys210_final.pdf Thursday December 8 Final Solutions

Course objectives: Classical mechanics is a broad subject, and we will select topics that develop techniques for analyzing nontrivial classical systems, and topics that are essential for understanding other branches of physics. You are expected to master the major topics covered in the course and be able to solve relevant problems. The course syllabus is available here. Topics will include:Basic principles: The subtleties in Newton's lawsD'Alembert's principle, Lagrangians, and Lagrange's equations of motionVariational methods and Hamilton's principleSystems with constraintsSymmetries and conservation laws:Noether's theoremHamiltonians and Hamilton's equations of motionCentral forcesSmall oscillations and normal modesRigid body motion and non-inertial reference framesCanonical transformations; Hamilton-Jacobi theory; action-angle variablesSpecial relativity

Other references:  A.L. Fetter and J.D. Walecka, Theoretical Mechanics of Particles and Continua - an excellent textbook, the one that was used when I was taught the subject. Does not cover special relativity.D. Tong, Lectures on Classical Dynamics - David Tong is the new Landau. He has lecture notes on everything, freely available online. The classical mechanics notes are great, but organized a little differently than this course.L. Landau & E. Lifshitz, Mechanics - a classic, like all of the Landau & Lifshitz textsV.I. Arnold, Mathematical Methods of Classical Mechanics - dives deep into the mathematical aspects of the subjectChegg, and other homework solutions offered online, are NOT acceptable references for the homeworks, but you may use them for studying. You are encouraged to work and study with your classmates, but your homework must reflect your own work.

This is a problem I encountered, but not a homework assignment, I'm not only just looking for solutions here... This is my first time encountering questions like this, I'm sure you remember the first time you see mathematical proofs, you need some help with it...That's my situation, please don't report this post...I spent the entire Sunday reading relevant material but something just doesn't click, jumping from introductory physics straight into this is just too hard.....Appreciate any help and hint.

The structure of this course is aligned fairly closely to the textbook by Golstein et al. However, the textbook by Fetter and Walecka has a better treatment of continuum mechanics. The lecture will be based on several sources, including but not limited to these two books. I do not require that you buy any one particular book, but you should seek out additional material beyond what is covered during class time.Learning goalsAfter completing this course, students are expected to be able to apply the Lagrange and Hamilton formalisms to solve mechanics problems withdifferent degrees of freedom and with possible constraints;understand and apply conservation laws in different contexts;calculate the properties of rotating objects and of objects in rotatingcoordinate systems;analyze a large variety of physical systems by approximating them asmulti-dimensional coupled harmonic oscillators;develop fundamental insight into continuum mechanics and the foundations offield theory;master the mathematical tools for solving relevant linear algebra anddifferential equation problems. HomeworkHomework is an essential part of learning the material of this course. Homework will be assigned each week on Friday and collected next week on Friday. You are encouraged to discuss the homework problems with each other after you have tried them to the best of your ability, but you cannot copy the solutions from each other. The homework assignments and solutions will be available for download on CourseWeb.If there are extraordinary circumstances that prevent you from completing an assignment on time, please contact me before the due date so that we canfind a suitable solution.Some of the homework problems will be discussed in class (after they have been graded and returned). For this purpose I will occiasionally pick one student at random to present her/his solution on the board. In this way, you can learn from each other's solutions, and also practice your presentation skills.Grading schemeThere will be one mid-term exam and a comprehensive final exam. The dates for the exams will be announced several weeks in advance.The final grade will be determined by the homework submissions (30%), and mid-term (25%) and final exam (45%). Academic integrityAll students are expected to adhere to the standards of academichonesty. Any student engaged in cheating, plagiarism, or other acts of academic dishonestywould be subject to disciplinary action. Any student suspected of violating this obligationfor any reason during the semester will be required to participate in the procedural process,initiated at the instructor level, as outlined in the University Guidelines on Academic Integrity. This may include, but is not limited to the confiscation of the examination of anyindividual suspected of violating the University Policy.Disability resourcesIf you have a disability for which you are or may be requesting anaccommodation, you are encouraged to contact both your instructor andDisability Resources and Services, 216 William Pitt Union, (412)648-7890/(412) 383-7355 (TTY) as early as possible in the term. DRSwill verify your disability and determine reasonable accommodationsfor this course.

Classical mechanics studies the motion of material bodies. It is amongthe fundamental branches of modern physics and is therefore anessential component of all graduate programs in Physics. This courseintroduces the structure of classical mechanics and discusses some ofits most important applications in modern physics.In the first part of this course, we will start from the Lagrangianformulation of classical mechanics and study the solution of theequations of motion of several systems, from simple one-body systems,to more complex systems acted upon by central forces and rigid bodies,up to scattering problems and oscillations. We will also extend ourdiscussion to include the theory of special relativity.The second part of this course will introduce the hamiltonianformulation of classical mechanics and study its formal and physicalconsequences. More advanced topics, such as non-linear dynamics andcontinuous systems, will be discussed in the third part of thiscourse, depending on time availability.

Here is a summary of the topics covered in class lecture by lecture:Date Topics coveredMain reference 08/29 Syllabus. Review of Newtonian mechanics for a single particle. [Text] (Ch. 1, Sec. 1.1) 08/31 System of particles, linear and angular momentum, equations of motion, conservation laws. [Text] (Ch. 1, Sec. 1.2) 09/02 System of particles, conservative forces, mechanical energy, mechanicalenergy conservation. Constraints, generalized coordinates, introduction toLagrangian approach. [Text] (Ch. 1, Sec. 1.2-1.4) 09/07 Lagrangian equations of motion from d'Alembert principle. [Text] (Ch.1, Secs. 1.4-1.5) 09/09 Applications of Lagrangian approach: simple examples. [Text] (Ch.1, Sec. 1.6) 09/12 Derivation of Lagrange's equations from a variationalprinciple (Hamilton's principle). [Text] (Ch.2, Sec. 2.3) 09/14 Applications of Lagrangian approach: more examples. Noether's theorem. [Text] (Ch.2, Sec. 2.6), your notes 09/16 Energy function and conservation of energy. [Text] (Ch.2, Sec. 2.7) 09/19 Hamilton's principle for non-holonomic systems,introduction of Lagrange multipliers. [Text] (Ch.2, Sec. 2.4) 09/19 Problem Session (12:00-1:00 p.m., Keen 707) 09/21 No class. 09/23 Central force problem: reduced system, Lagrangian, equations of motion. [Text] (Ch.3, Sec. 3.1-3.2) 09/26 Central force problem: first integrals, integration ofmotion. Study of the effective potential and classification of theorbits. [Text] (Ch. 3, Sec. 3.3) 09/26 Problem Session (10:30-11:30 a.m., Keen 707) 09/28 Introduction to Kepler problem. [Text] (Ch. 3, Sec. 3.7) 09/30 Continuing the discussion of Kepler problem. Derivation of Kepler's laws. [Text] (Ch. 3, Sec. 3.7) 10/03 The differential equation for the orbit with examples. [Text] (Ch. 3, Sec. 3.5) 10/05 The motion in time in the Kepler problem. TheLaplace-Runge-Lenz vector. [Text] (Ch. 3, Sec. 3.8-3.9) 10/07 FIRST MIDTERM EXAM 10/10 Scattering in a central force field. Rutherford scattering. [Text] (Ch. 3, Sec. 3.10) 10/10 Problem Session (12:00-1:00 p.m., Keen 707) 10/12 Rigid body motion: degrees of freedom, generalized coordinates;linear and angular momentum, kinetic energy. [Text] (Ch. 4, Sec. 4.1) 10/14 Rigid body motion: brief review oforthogonal transformations. [Text] (Ch. 4, Secs. 4.2-4.3) 10/17 Rigid body motion: Euler angles, Euler's Theorem. [Text] (Ch. 4, Secs. 4.4, 4.6) 10/17 Problem Session (12:00-1:00 p.m., Keen 707) 10/19 Rigid body motion: space frame vs body frame; inertia tensor and moments of inertia. [Text] (Ch. 4, Sec. 4.9; Ch. 5, Secs. 5.1-5.3) 10/20 Rigid body motion: principal moments ofinertia. Calculation of simple examples. [Text] (Ch. 5, Secs. 5.3-5.4) 10/21 Rigid body motion: Steiner's theorem. Inertia Ellipsoid. Euler's equations of motion. [Text] (Ch. 5, Secs. 5.3-5.5) 10/24 Rigid body motion: discussion of homework problems. [Text] (Your Notes) 10/26 Rigid body motion: motion of a torque-free rigid body. [Text] (Ch. 5, Sec. 5.6) 10/28 Rigid body motion: motion of a heavy symmetric top with a fixed point, part 1. [Text] (Ch. 5, Sec. 5.7) 10/31 Rigid body motion: motion of a heavy symmetric top with a fixed point, part 2. [Text] (Ch. 5, Sec. 5.7) 11/02 Rigid body motion: stability of rigid body rotations. (your notes) 11/04 SECOND MIDTERM EXAM 11/07 Oscillations: formalism of small oscillations and theeigenvalue problem. [Text] (Ch. 6,Secs. 6.1-6.2) 11/09 No class. 11/14 Oscillations: example of two coupled harmonicoscillators, normal modes. [Text] (Ch. 6, Sec. 6.2 ) 11/16 Oscillations: example of a linear triatomic molecule, longitudinal and transverse oscillations. [Text] (Ch. 6, Sec. 6.4) 11/18 Hamilton's equations of motion. Hamiltonian function andits properties[Text] (Ch. 8, Sec. 8.1-8.2) 11/28 Hamilton's equations of motion derived from a variational principle.Poisson's brackets and their properties. [Text] (Ch. 8, Sec. 8.5; Ch. 9, Sec. 9.5) 11/30 Canonical transformations: introduction. [Text] (Ch. 9, Sec. 9.1) 12/02 Canonical transformations: more properties and examples. [Text] (Ch. 9, Secs. 9.1-9.3) 12/05 Canonical transformations: simplectic approach. [Text] (Ch. 9, Secs. 9.4-9.5) 12/07 Canonical transformations: equations of motion,infinitesimal transformations, Poisson brackets.[Text] (Ch. 9, Secs. 9.5-9.6) 12/09 Canonical transformations: Liouville's Theorem.Problems. [Text] (Ch. 9, Sec. 9.9) 153554b96e

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