Lagrangian Lagrange multiplier, Legendre transform, Hamiltonian
Some questions:
我們從問題出發
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The relationship between Lagrangian, Hamiltonian, and Legendre transformation.
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The relationship between Lagrangian and Lagrange multiplier.
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The physical interpretation of Lagrangian and Lagrange multiplier.
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The geometric interpretation of Lagrange multiplier.
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Add optimization Lagrangian dual problem!
Duality (optimization) - Wikipedia
Usually the term “dual problem” refers to the Lagrangian dual problem but other dual problems are used – for example, the Wolfe dual problem and the Fenchel dual problem. The Lagrangian dual problem is obtained by forming the Lagrangian of a minimization problem by using nonnegative Lagrange multipliers to add the constraints to the objective function, and then solving for the primal variable values that minimize the original objective function. This solution gives the primal variables as functions of the Lagrange multipliers, which are called dual variables, so that the new problem is to maximize the objective function with respect to the dual variables under the derived constraints on the dual variables (including at least the nonnegativity constraints).
Relation (and physical interpretation) between Lagrangian, Hamiltonian, and Legendre Transformation
1, Lagrangian = T - V (KineTic energy - Potential energy)
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generalized Lagrangian
Hamiltonian: why it equal to least action? but action is the integration of Lagrangian, not Hamiltonian?
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Why the Lagrangian and Hamiltonian is linked by Legendre transformation?
Relation of Lagrangian and Lagrange multiplier, and the physical interpretation of Lagrange multiplier.
Answer 0: 只是巧合。除非 Lagrange 的研究都是獨立的題目。不然應該不是如此。
Answer 1: Lagrangian 一般是在 non-constraint 下的定義。大多數物理問題都有 constraints, 例如 surface, 剛體的限制。boundary. 這些限制就成為 Lagrangian multiplier 的 constraints. Lagrange 可能看到這些 constraint 應而發展除 Lagrange multiplier. 不過這只是很表面的關聯。
Answer 2: 這些 constraints, physically 可以視為 generalized force!!
Lagrange multiplier Physical interpretation
wiki Lagrange multiplier: generalized force. (Wiki Lagrangian multiplier)!!
Lagrangian coefficients $\lambda_k$ 就是 generalized force!
Answer 3: Lagrangian 本身的定義就應該從 (Physically) Lagrange Multiplier 出發！
https://physics.stackexchange.com/questions/590960/application-of-lagrange-multipliers-in-action-principle
https://physics.stackexchange.com/questions/622727/connection-between-different-kinds-of-lagrangian
In Goldstein’s Classical Mechanics, he suggests the use of Lagrange Multipliers to introduce certain types of non-holonomic and holonomic constraints into our action. The method he suggests is to define a modified Lagrangian𝐿′(𝑞˙,𝑞;𝑡)=𝐿(𝑞˙,𝑞;𝑡)+∑𝑖=1𝑚𝜆𝑖𝑓𝑖(𝑞˙,𝑞;𝑡),
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It should be stressed that the constraints
𝑓ℓ(𝑞,𝑞˙,𝑡),ℓ ∈ {1,…,𝑚}fℓ(q,q˙,t),ℓ ∈ {1,…,m}
depends implicitly (and possible explicitly) of time
其實 1/2/3 都很類似。只是說法不同。
Answer 4: Lagrangian 本身的定義就應該從 (Based on optimization theory) Lagrange Multiplier 出發！
這是從更廣泛的 optimization 出發。 Physical Lagrangian mechanics 只是大自然的 optimization theory.
General optimization 一定會有 constrains, Lagrange multiplier 是 optimization theory 的基本。另外有 Lagrangian dual problem (primal and dual optimization). 參見 convex optimization from Steven Boyd.
Usually the term “dual problem” refers to the Lagrangian dual problem but other dual problems are used – for example, the Wolfe dual problem and the Fenchel dual problem. The Lagrangian dual problem is obtained by forming the Lagrangian of a minimization problem by using nonnegative Lagrange multipliers to add the constraints to the objective function, and then solving for the primal variable values that minimize the original objective function. This solution gives the primal variables as functions of the Lagrange multipliers, which are called dual variables, so that the new problem is to maximize the objective function with respect to the dual variables under the derived constraints on the dual variables (including at least the nonnegativity constraints).
Lagrange multiplier Geometry interpretation
這個反而簡單。
constraint equation 就是等高線。 Lagrangian multiplier 就是 f(x,y) 和 g(x,y)=k 的 gradient 要平行。
https://www.youtube.com/watch?v=5A39Ht9Wcu0&ab_channel=SerpentineIntegral
不錯的 geometry interpretation, 不過沒有物理的意義。No relation to Lagrangian.
$\lambda$ 只是一個 scaling constant!
