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.
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.
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
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||✔ Practical and/or problem-solving skills|
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|
|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.|
not yet decided
Yoshirou Oonuki,``Analytical Mechanics'',(Iwanami shoten)
Based on blackboard presentation, report and examination.
Students are expected to have learned real analysis and linear algebra in the first school year.