noun

Levi-Civita notation: Parity permutation of 1,-1,0. See also permutation groups.

Kroneeker notation: represented by delta. Equals 0 when the two metrics are equal.

e(i) × e(j) = e(ijk) e(k) . k is a dummy index.

Natural coordinate system: Use the parameters of the motion track itself to define the coordinate system. For example, the tangential direction and the normal direction of a two-dimensional curve can be used as a set of natural coordinate systems.

Tangent Vector: A unit vector representing the tangent direction at a point. formula

Principal Normal: Indicates the direction in which the curve bends. formula

Secondary normal vector : The third set of normal vectors defined by cross product. formula

Rotation frequency: \(f=\sqrt{I}{m}\)

Curvature: The degree to which the curve bends.

Functional: Y is a set of functions, if for each y(x) there is a number J corresponding to it. Then the variable J is a functional of y(x).

Variation: The difference of the function caused by the change of the independent variable and the change of y(x).

scattering cross section

Seven Conserved Quantities of Inverse Square Psychic Force Field

Center of mass: a physical quantity used to simplify a system of particles, the formula

Center of Mass Velocity: The speed at which the center of mass moves (?)

König’s theorem: The total kinetic energy of a system of particles is equal to the sum of the kinetic energy of the center of mass and the kinetic energy of the motion of all particles relative to the center of mass.

Normal Coordinates: The motion pattern of a many-body system is a superposition of a series of frequency oscillations. The coordinates of the independent harmonic oscillators of this array are normal coordinates

Normal Frequency: The motion pattern of a many-body system is a superposition of a series of frequency oscillations. The frequencies of this array of independent simple harmonic oscillators are normal frequencies

Rigid body: a system of mass points with a constant distance between any two points

Radius of gyration: Moment of inertia divided by the square root of mass. is an inherent property of an object.

Inertia ellipsoid: an ellipse composed of three eigenvalues ​​after the orthogonalization of the moment of inertia tensor.

Main moment of inertia: the moment of inertia in three directions obtained by orthogonalizing the moment of inertia.

Main axis of inertia: three sets of coordinate axes obtained by orthogonalizing the moment of inertia.

Euler Angles: A set of rotational angle transformations can be fully expressed. In terms of groups, it is the rotation group SO(3): the Hermitian matrix of unimodular (3-dimensional).

Classification of Rigid Body Motion

**XX constraints (there are many kinds of constraints, figure out how to classify) **

• Constraints: Constraints are the conditions that limit the motion of particles.

• Stability constraints (steady constraints): constraints without time

• Unstable constraints: time-dependent constraints

• Solvable Constraints: Constraints that the particle can escape from. If the particle rolls above the ground, f(x,y,z)>=0

• Unsolvable Constraint: Constraint that the particle can never escape from. f(x,y,z)=0

• Geometric Constraints: Constraints that only restrict the space-time coordinates of the particle. f(x,y,z,t)=0

• Motion Constraints: Added speed-limiting constraints to geometric constraints

• Ideal constraint: sum of all constraint reactions = 0

Here is another classification:

• Holistic Constraints: Constraints are related to the coordinates and time of the system.

• Incomplete constraints: Solvable constraints that cannot be expressed in terms of equations; constraints that are not geometric constraints after integration of differential constraints.

  This defines the complete system and the incomplete system.

The principle of virtual work: under ideal constraints, the work done by a set of virtual displacements is 0

Virtual Work: The work done by the virtual displacement. An imaginary displacement is an infinitely small displacement imagined when the constraints allow it.

D’Alembert’s principle: the elemental work of the sum of the active and inertial forces at any virtual displacement = 0

Lagrange equation:

• Basic form: Lagrange equation with generalized force Q.

• Conservative system: Because of the introduction of potential energy, it can be written in the form of Euler equation with Lagrange quantity.

Action: the integral of the Lagrangian over time

Cyclic coordinates: Lagrangian function L does not contain a generalized coordinate qβ, that is, ∂L/∂qβ=0, then this generalized coordinate is called ignorable coordinates, also known as cycle cyclic coordinates

Hamilton’s principle (the principle of least action) (and it’s important!!): In the two times of t1 and t2, that is, the two ends of the motion path are fixed, and the isochronous variation delta t=0, the trajectory of the real motion is a functional S(c) takes the path of the extreme value.

Active force: A force that has independent magnitude and direction and is not affected by other forces.

Binding reaction force: The force affected by the active force.

Regular transformation: A set of functional transformations that make regular equations have the same form.

Definition of symplectic group: all matrices A of AJA^T=J

virial theorem: gives the relationship between the average potential energy and kinetic energy =1/2*n

Phase Space: A multidimensional space used to describe the state of a system. Each point in the space corresponds to a state of the system. Physical quantities such as generalized coordinates and generalized momentum will be used as coordinate axes.

Configuration space: We need s generalized coordinates to describe the motion state of a system, and these s generalized coordinates form a configuration space.

Variational method to solve the problem

• Euler’s equation: Variational method to solve problems, pay attention to ** the use of the first integral constant is conditional**

• Exercise 1.12

• The subject will be relatively simple

Normal coordinates and normal frequencies of many-body interactions

Carefully read the 2.3 Many-body Interaction section in the book, and then combine it with Exercise 2.5 (this question was taken last year).

Orthogonal decomposition is used to obtain normal coordinates and normal frequencies.

Prove the conservation of the R-L vector in the inverse square proportional force field

read books. The proof is complicated.

From this, Galileo’s theorem and Binet’s equation can be derived:

Action calculation

• relatively difficult

• Time integral over Lagrange quantities.

Regular equation

Required.

(1) Prove the canonical equation by the relationship between Hamiltonian and Laplace

(2) Prove the regular equation by Hamilton’s principle

(3) Give the Hamiltonian of the free particle in the spherical coordinate system

Note: The representation of Hamiltonian cannot show speed

Poisson brackets

Need to know how to count. Note that the regular equation form under Poisson brackets is uniform.

Solve the problem using the H-J method

Usually in the form of additional questions.

H-J Method: Hamilton-Jacobi Method

Tensor

Required.

• “Tensor Analysis” written by Huang Kezhi and published by Tsinghua University Press under self-study. Chapter One

• It’s easy to get the score of this question, remember how the teacher and the teaching assistant brother proved it, just put it on it, it doesn’t matter if you don’t understand it, you can just memorize it, anyway, the process is very small.

##PS

I found out that Master Xiaoguang’s teacher is Yang Zhenning. Then I am the grandson of the old gentleman, lol.