▶Japanese
Post to SNS
- Home
- > School of Science
- > Physics
- > Exercises in Classical Mechanics
- School of Science
-
School of Engineering
- Group 2
- Group 3
- Group 4
- Group 5
- Group 6
- Metallurgical Engineering
- Organic and Polymeric Materials
- Inorganic Materials
- Chemical Engineering Course
- Chemical Engineering Course(Chemical Engineering)
- Applied Chemistry Course(Chemical Engineering)
- Polymer Chemistry
- Mechanical Engineering and Science
- Mechanical and Intelligent Systems Engineering
- Mechano-Aerospace Engineering
- International Development Engineering
- Control and Systems Engineering
- Industrial and Systems Engineering
- Electrical and Electronic Engineering
- Computer Science
- Civil Engineering
- Architecture and Building Engineering
- Social Engineering
- Common Cource of Engineering
- School of Bioscience and Biotechnology
- Common Course of Undergraduate School
-
Graduate School of Science and Engineering
- Mathematics
- Physics(Particle- Nuclear- and Astro-Physics)
- Physics(Condensed Matter Physics)
- Chemistry
- Earth and Planetary Sciences
- Chemistry and Materials Science
- Metallurgy and Ceramics Science
- Organic and Polymeric Materials
- Applied Chemistry
- Chemical Engineering
- Common Course of Mechanical Engineering
- Mechanical Sciences and Engineering
- Mechanical and Control Engineering
- Mechanical and Aerospace Engineering
- Common Course of Electronic Engineering
- Electrical and Electronic Engineering
- Physical Electronics
- Communications and Integrated Systems
- Communications and Computer Engineering
- Civil Engineering
- Architecture and Building Engineering
- International Development Engineering
- Nuclear Engineering
- Graduate School of Bioscience and Biotechnology
-
Interdisciplinary Graduate School of Science and Engineering
- Built Environment
- Energy Sciences
- Computational Intelligence and Systems Science
- Electronic Chemistry
- Materials Science and Engineering
- Innovative and Engineered Materials
- Environmental Chemistry and Engineering
- Environmental Science and Technology
- Mechano-Micro Engineering
- Information Processing
- Electronics and Applied Physics
- Graduate School of Information Science and Engineering
- Graduate School of Decision Science and Technology
- Graduate School of Innovation Management
- Common Course of Graduate School
- Program for Leading Graduate Schools
- Academic unit or major
- Physics
- Instructor(s)
- Nishida Yusuke Adachi Satoshi
- Class Format
- Exercise
- Media-enhanced courses
- Day/Period(Room No.)
- Mon5-6(H102) Thr7-8(H103)
- Group
- a
- Course number
- ZUB.Q212
- Credits
- 2
- Academic year
- 2016
- Offered quarter
- 2Q
- Syllabus updated
- 2016/4/27
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
-
Course description and aims
The variational method to rewrite a given equation is explained. In analytical mechanics, the minimum action principle, which is a variational method, derives Euler-Lagrange equation. The Euler-Lagrange equation is invariant under general coordinate transformations. When a physical system has a symmetry, the corresponding conserved quantity exists. By expressing analytical mechanics in Hamilton formalism, the invariance of equation of motion under a larger class of transformations is obtained.
Both of understanding logic of analytical mechanics and developing ability of calculation in concrete examples are aimed.
Student learning outcomes
Newtonian mechanics is reconstructed on basis of the minimum action principle. The equation of motion is derived from Lagrangian expressed in general coordinate system and is analyzed in several examples. The canonical formalism of analytical mechanics is also explained.
Keywords
principle of variation, minimum action principle, Euler-Lagrange formalism, Lagrange function, general coordinate transformation, symmetry and conserved quantity, Noether's theorem, Hamilton formalism, Hamilton function, canonical transformation, Hamilton=Jacobi formalism, action function, separation of variables
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
Class flow
Materials for exercises are distributed in advance. Students are expected to solve exercises at home and present answers on blackboard. Answers are discussed.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | importance and role of analytical mechanics in physics | The method of analytical mechanics is used everywhere in physics, for example, electromagnetism, quantum mechanics, field theory and so on. |
| Class 2 | variational method | By representing a given equation to fit the variational principle, an invariance with certain range of transformation is obtained automatically. |
| Class 3 | minimum action principle (Euler=Lagrange equation) | The variational method for action function is the minimum action principle. It derives Euler=Lagrange equation as equation of motion. |
| Class 4 | system with constraint | A physical system that has a holonomic constraint can be analyzed by Euler=Lagrange formalism. |
| Class 5 | method of Lagrange multipliers | A physical system with holonomic constrained can be treated by method of Lagrange multipliers, which increases variables to erase the constraint. |
| Class 6 | symmetry and conserved quantity | When a physical system has a continuous symmetry, the corresponding conserved quantity exists according to Noether's theorem. |
| Class 7 | arbitrariness of Lagrange function | Lagrange function has an arbitrariness to include any complete differential with respect to time. |
| Class 8 | small oscillation | Motion around a stable equilibrium is called small oscillation. It is expressed by linear superposition of normal mode oscillations. |
| Class 9 | rigid body motion | Motion of direction of undeformable body in three dimensional space is analyzed. |
| Class 10 | canonical formalism | Hamilton mechanics is the another central formalism of analytical mechanics than Euler=Lagrange mechanics. |
| Class 11 | canonical transformation | Hamilton equation is invariant under the set of canonical transformations, which is larger than that of general coordinate transformations. |
| Class 12 | Poisson bracket | Hamilton equation can be expressed in terms of Poisson bracket. Thereby, the symmetry within the equation becomes more clear. |
| Class 13 | Hamilton=Jacobi equation | A physical system of several degree of freedom that has some symmetry may be solved by separation of variables based on Hamilton=Jacobi equation. |
| Class 14 | phase space and Liouville theorem | State space of dynamical system in the canonical formalism is called phase space. Phase space volume is conserved in time evolution according to Liouvelle theorem. |
| Class 15 | adiabatic invariants | In a slow variation of parameters for a physical system, if a quantity is conserved,, it is called an adiabatic invariant. Action integrals supply adiabatic invariants. |
Textbook(s)
not yet decided
Reference books, course materials, etc.
Landau=Lifshitz,``Mechanics'',(Tokyo tosho)
Yoshirou Oonuki,``Analytical Mechanics'',(Iwanami shoten)
Assessment criteria and methods
Based on blackboard presentation, report and examination.
Related courses
- ZUB.E202 ： Electromagnetism I
- ZUB.E211 ： Exercises in Electromagnetism
- ZUB.Q204 ： Quantum Mechanics I
- ZUB.Q215 ： Exercises in Quantum Mechanics I
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are expected to have learned real analysis and linear algebra in the first school year.
