Classical Mechanics Pusan National University 201883202 Park seung-chan

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

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

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

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

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

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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