1. Classical Mechanics Pusan National University 201883202
Park seung-chan

2. Hamiltonian Mechanics
Classical mechanics
Classical mechanics
Physics to explain the relationship between force and motion acting on an object
Newtonian Mechanics
Lagrangian Mechanics
Hamiltonian Mechanics
16C
18C
19C

3. Newton (1642 ~ 1727) Newton’s mechanics
The interaction between the objects that cause the movement of objects is defined as ‘Force’, and Newton’s law of motion describes the state of motion
The first Law – The law of inertia : When there is no external force, the object remains in motion 𝑭 =𝟎 ↔ 𝒅𝒗 𝒅𝒕 =𝟎
The second Law – The law of acceleration : When external force is applied, it changes the motion state of the object 𝐅= 𝒅𝒑 𝒅𝒕 = 𝒅 𝒎𝒗 𝒅𝒕 =𝒎𝒂
The third Law – The law of action reaction : When object A exerts a force on another object B, object B applies a force to object A that is the same in magnitude and opposite in direction. 𝑭 𝑨𝑩 =− 𝑭 𝑩𝑨

4. 𝐸𝑢𝑙𝑒𝑟−𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑖𝑎𝑛 𝐸𝑞 : 𝑑 𝑑𝑡 𝜕ℒ 𝜕 𝑥 − 𝜕ℒ 𝜕𝑥 =0
Lagrangian (1736 ~ 1813)
Lagrangian mechanics
Lagrangian can be obtained by solving Lagrangian equations and finding the trajectory of the object.
𝓛 is function of position & speed, scalar value
𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑖𝑎𝑛 :ℒ=ℒ 𝑞 𝑖 , 𝑞 𝑖 =𝑇−𝑉
𝐸𝑢𝑙𝑒𝑟−𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑖𝑎𝑛 𝐸𝑞 : 𝑑 𝑑𝑡 𝜕ℒ 𝜕 𝑥 − 𝜕ℒ 𝜕𝑥 =0
∙𝑇=kinematic energy
∙𝑉=potential energy
∙ 𝑞 𝑖 =𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠
∙ 𝑞 𝑖 =𝑡𝑖𝑚𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑓𝑜𝑟 𝑞 𝑖

5. Lagrangian (1736 ~ 1813) Lagrangian mechanics
𝑵𝒆𝒘𝒕𝒐 𝒏 ′ 𝒔 𝑳𝒂𝒘
𝑬𝒖𝒍𝒆𝒓−𝑳𝒂𝒈𝒓𝒂𝒏𝒈𝒆 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏
𝑭 =𝒎𝒂=−𝒎𝒈 𝑜𝑛𝑙𝑦 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 𝐹 𝑔
𝒅 𝒅𝒕 𝝏𝓛 𝝏 𝒙 − 𝝏𝓛 𝝏𝒙 =𝟎 𝑥
𝑑 𝑑𝑡 𝜕ℒ 𝜕 𝑥 − 𝜕ℒ 𝜕𝑥 =0
ℒ=𝑇−𝑉
∙ 𝑑 𝑑𝑡 𝜕ℒ 𝜕 𝑥 =𝑚 𝑥
∙ 𝜕ℒ 𝜕𝑥 =−𝑚𝑔
𝑇= 1 2 𝑚 𝑥 2
𝑑 𝑑𝑡 𝜕ℒ 𝜕 𝑥 − 𝜕ℒ 𝜕𝑥 =𝑚 𝑥 − −𝑚𝑔 =0
𝑉=𝑚𝑔𝑥
⇒𝑚 𝑥 = −𝑚𝑔 =𝑚𝑎 (∵𝑎= 𝑥 )
𝑎= 𝑥 =−𝑔
⇒∑𝐹=𝑚𝑎
∑𝐹=𝑚𝑎 ≡ 𝑑 𝑑𝑡 𝜕ℒ 𝜕 𝑥 − 𝜕ℒ 𝜕𝑥
𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛

6. Hamilton (1805 ~ 1865) Hamiltonian mechanics
William Rowan Hamilton introduced in 1833 based on the existing Lagrangian mechanics
An analytical mechanics theory that treats a classical mechanical system as a phase space composed of coordinates and corresponding momentum
H is function of position & momentum, phase space
𝐻𝑎𝑚𝑖𝑙𝑡𝑜𝑛𝑖𝑎𝑛 :𝐻=𝐻 𝑞 𝑖 , 𝑝 𝑖 = 𝑖 𝑝 𝑖 𝑞 𝑖 −𝐿( 𝑞 𝑖 , 𝑞 𝑖 )
𝐻𝑎𝑚𝑖𝑙𝑡𝑜 𝑛 ′ 𝑠 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 : 𝑞 𝑖 = 𝜕𝐻 𝜕 𝑝 𝑖 , 𝑝 𝑖 =− 𝜕𝐻 𝜕 𝑞 𝑖
∙ 𝑝 𝑖 =𝑚𝑜𝑚𝑒𝑡𝑢𝑚

7. Classical mechanics Newtonian mechanics Lagrangian mechanics
Focus on external force
Using the Cartesian coordinate system
x, y, z, t(vector equation)
Newtonian mechanics
ℒ is function of position & speed, scalar value
Using the Generalized coordinate system
𝑞, 𝑞 , 𝑡(position, velocity, time)
Lagrangian mechanics
H is fuction of position & momentum, phase space
Legendre transformation of Lagrangian
𝑞, 𝑝 , 𝑡(position, momentum, time)
Hamiltonian mechanics

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