Updated: 5 May 2016

Textbook: Theoretical Mechanics (second edition) Jinchangyan, Mayongli, higher Education Press

Start time: 6:39 PM

Chapter One Newton's equations of Mechanics 1.1 Newton and the principle

The temporal and spatial view of classical mechanics: absolute time, absolute space and absolute motion.

Inertial Reference System: A reference system for uniform motion relative to absolute space.

Newton's Law of mechanics and Galileo transform

Newton's philosophical rules of reasoning: simplicity, causality, unity, truth.

1.2 Newton's second law expression in common coordinate system

To derive the required knowledge:

1. Write a vector expression of the **base** vector of the coordinate system in a known coordinate system (such as a Cartesian coordinate system)

2. The time derivative of the new base vector is represented by the new base vector.

3. Write out the new expression of the bit vector **R** , to find the speed of the first order of time, to find the second rate of speed, you can substitute Newton's law.

1.3 Particle system

The motion of n-particle k-constrained conditions can be solved by 3n+k equations.

The weighted average of the mass of a particle by **the "centroid"** particle position vector. \ (\textbf{r}_c=\dfrac{\sum\limits_im_i\textbf{r}_i}{\sum\limits_im_i}=\dfrac{\sum\limits_im_i\textbf{r}_i}{m_s }\)

**"centroid system"** follows the reference system of centroid translation of the particle system. Not necessarily the inertial system.

1.4 Momentum theorem

The **change rate of** the particle momentum **theorem** is equal to the **force** that the particle is subjected to. \ (\textbf{f}=\dfrac{d\textbf{p}}{dt}\)

**"The momentum theorem of the particle system"** the change rate of the momentum of the particle system equals the **external force** that the system receives.

**"Particle system momentum"** if the force of the particle system is zero, then the momentum of the particle system is the constant vector.

1.5 theorem of angular momentum

The vector diameter of the **"angular momentum"** particle and the cross product of its momentum. \ (\textbf{l}=\textbf{r}\times m\textbf{v}\) (also known as angular momentum)

The vector diameter of the **"moment"** particle and the cross product of the force it is subjected to. \ (\textbf{m}=\textbf{r}\times \textbf{f}\)

**"Particle angular momentum theorem"** the **change rate of the angular momentum of** a particle is equal to the **moment** the particle is subjected. \ (\textbf{m}=\dfrac{d\textbf{l}}{dt}\)

**"Particle system angular momentum theorem"** the change rate of angular momentum of a particle system equals the **sum of** all the forces acting on the particle system.

1.6 Energy theorem

The **"Particle kinetic energy theorem"** acts on the force of the particle to make a total equal to the increase of the pointing kinetic energy. \ (Dt=\textbf{f}\cdot d\textbf{r}\)

**"Particle system kinetic energy theorem"** The increase of kinetic energy is equal to the sum of work done by external forces and internal forces.

The total kinetic energy of the **"Konig theorem"** is equal to the kinetic energy of the mass concentration at centroid and the movement of the centroid velocity, plus the kinetic energy of each particle relative to the centroid system.

The force in **the "potential energy"** conservative field is regarded as the **negative gradient** of the potential function.

The sum of kinetic energy and potential energy of **"mechanical energy"** .

Conservation of mechanical energy in the conservative field of **the conservation of mechanical energy** . Mechanical energy is conserved when the force of a particle is zero.

1.7 Variable mass Motion equation

**"Michilski equation"**

**"Tsiolkovsky number"**

Motion of charged particles in 1.8 plasmas

Specific applications

Second Zhangla Grange equation 2.1 ideal Constraint

**"Real Displacement"** in the dt time the actual displacement occurred, recorded as D**R**.

**"Virtual displacement"** at a certain moment the particle takes place in a **constrained** infinite Small displacement, which is recorded as Δ**R**. Note that the virtual displacement is a false displacement and does not require time.

**the "Time-variant"**δ function is 0 on T and the other times the same as D.

**"Virtual**" \ (δw=\textbf{f}\cdotδ\textbf{r}\)

**the "ideal constraint"** internal and external binding of the virtual force is zero. \ (\sum\limits_i\textbf{f}_{ni}\cdotδ\textbf{r}_i=0\) where the binding is recorded as **F**N

Common Ideal constraints:

1. Particles moving along smooth surfaces

2. Two particles are connected by a rigid light rod

3. Two rigid bodies with smooth table contact

More general criteria: as long as the connection between the objects is rigid, all the contact surfaces are ideal smooth or absolute rough.

The general equation of the dynamics of **the ideal restraint system of "Dallembert equation"**

\ (\sum\limits_i (\textbf{f}_i-m_i\ddot{\textbf{r}}_i) \cdot \delta \textbf{r}_i=0\)

To eliminate the binding force in the formula.

Multiple dimensions of \ (\delta \textbf{r}_i\) are orthogonal to each other, and multiple motion equations can be obtained respectively.

2.2 Full constraints

**"Complete Constraint"** describes a single constraint that is only related to the coordinate *R*I and time *t* of each particle of the system. The constraint equation can be written

\ (f (r_1,r_2,\cdots,r_n,t) =0\)

**Emphasis** is independent of speed or generalized speed.

Each complete constraint can be algebraic to eliminate an independent coordinate.

The number of independent coordinates for **"degrees of freedom"** . \ (s=3n-k\)

**"Incomplete constraint"** cannot eliminate the non-independent coordinates.

1. Non-integrable differential constraint with time derivative

2. Solvable constraints/single-sided constraints with inequalities

To solve a single-sided constraint method:

1. Constraint is not solvable, the constraint equation takes the equation

2. Remove the constraint and add an independent coordinate

**the "generalized coordinates"** still describe the spatial location. The number is the same as the Freedom *s* . Spanned *s* **-dimensional space** (contact phase space). The generalized coordinate selection method is not unique.

No time is included in **the stability constraint** constraint

2.3 The Lagrange equation of the ideal and complete system

The **"Lagrange equation"** is a kinetic equation directly represented by a generalized coordinate.

**"Generalized force"** corresponds to the generalized force of generalized coordinates \ (q_\alpha\) \ (q_\alpha=\sum\limits_{i=1}{n}\textbf{f}_i\cdot\dfrac{\partial \textbf{r} _i}{\partial q_\alpha}\)

If the system is conservative, then \ (q_\alpha=-\dfrac{\partial v}{\partial Q_\alpha}

The displacement of each particle in the system:

\ (\textbf{r}_i=\textbf{r}_i (q_i,q_2,\cdots,t) \)

Its virtual displacement:

\ (\delta\textbf{r}_i=\sum\limits_{a=1}^s\dfrac{\partial \textbf{r}_i}{\partial q_\alpha}\delta q_\alpha\)

**"Lagrangian function"**\ (l=t-v=l (q,\dot{q},t) \)

**"Lagrange equation"**

Normal form (potential energy is only related to the position and time of the particle, independent of velocity)

\ (\dfrac{d}{dt}\dfrac{\partial l}{\partial \dot{q}_\alpha}-\dfrac{\partial l}{\partial q_\alpha}=0,\quad \alpha=1,2 , \cdots, S\)

General form

\ (\dfrac{d}{dt}\dfrac{\partial t}{\partial\dot{q}_\alpha}-\dfrac{\partial t}{\partial Q_\alpha}=Q_a,\quad \alpha = 1,2,\cdots, S\)

Application of 2.4 Lagrange equation to the equilibrium problem 2.5 The Lagrangian function of the generalized potential energy charged particles in the electromagnetic field * 2.6 Lagrange equation of nonholonomic system * * 2.7 symmetry and conservation law

A mechanical system at the moment the state of T is determined by 2\ (s\) (q\) and \ (q_\alpha\).

**"Motion Integral"** a function of \ (q_\alpha\) and \ (\dot{q}_\alpha\), which remains constant during movement.

**"Generalized momentum"**\ (p_\alpha=\dfrac{\partial l}{\partial \dot{q}_\alpha}\) is called generalized momentum with generalized coordinates \ (q_\alpha\) conjugate.

**"Generalized momentum conservation"** Lagrange function \ (l\) If there is no generalized coordinate \ (q_\alpha\) (can appear its time derivative), namely \ (\dfrac{\partial l}{\partial q_\alpha}=0\), The corresponding generalized momentum is known as constant by Lagrange equation.

**"Generalized energy"**\ (H=\sum\limits_{\alpha=1}^{s}p_\alpha \dot{q}_\alpha-l\) (H is a function of p, Q, T, i.e. Hamiltonian function)

**"Generalized energy integrals"**?

In the **"conserved quantity"** motion integral, the additive becomes the conserved quantity. There are only 7 conserved quantities in classical mechanics.

1. Time uniformity leads to conservation of energy

2. Space uniformity leads to conservation of momentum

3. Angular momentum conservation due to spatial isotropic

2.8 Lagrange equation for instantaneous force problem *

**"Generalized Impulse"**

Chapter III Two body problems

3.1 Two-body problem to single particle problem

3.2 The active potential energy of single particles in the central potential field

3.3 The central potential field inversely proportional to the distance

Stability of particle motion orbits in the 3.4 central potential field *

3.5 Elastic collision

3.6 Scattering Cross-section

Transformation of 3.7-Sphere scattering cross-section from centroid System to laboratory system

3.8 Elastic scattering in the Coulomb potential field * *

3.9 Fission of particles * *

Fourth Chapter rigid Body

Fifth non-inertial reference system

Sixth. Micro-vibration of multi-DOF system

Seventh Chapter damping Motion

The eighth chapter of the Hamiltonian theory of classical mechanics

The Nineth chapter on the application of Hamiltonian theory in physics * * 8.1 regular conjugate coordinates

The non-uniqueness of Lagrange function

8.2 Hamiltonian functions and regular equations

If you follow the definition of chapter II

\ (H=\sum\limits_{\alpha=1}^{s}p_\alpha \dot{q}_\alpha-l\)

Then h is shown as \ (P_\alpha, Q_\alpha, \dot{q}_\alpha\) of three variables (3s\). However, the independent variables only \ (2s\), where the \ (\dot{q}_\alpha\) and \ (l\) can be expressed in P, Q, T, so H can be expressed in P, Q, T.

At this point \ (H (p,q,t) \) is the **Hamiltonian function** .

In the conservative system of "Hamiltonian equation"

\ (\dot{q}_\alpha=\dfrac{\partial h}{\partial p_\alpha},\)

\ (\dot{p}_\alpha=-\dfrac{\partial h}{\partial q_\alpha},\)

Also known as **the regular motion equation** or **the regular equation** . \ (P_\alpha, q_\alpha\) is called a regular variable.

\ (\dfrac{dh}{dt}=\dfrac{\partial h}{\partial T}

Therefore, when \ (h\) does not contain time, it is a constant, namely the **conservation of energy** .

8.3 Euler problems of variational problems

8.4 Hamiltonian principle

8.5 Regular transformations

8.6 Poisson Bracket

8.7 Hamiltonian-Jacobian equation

8.8 using Hamiltonian theory to solve the Doppler problem *

Tenth Chapter Fluid * *

Two-hour Challenge Analysis Mechanics (updated)