Finding Hamilton function which equations do I have to solve to find v(θ,ϕ,θ˙,ϕ)?L(θ,ϕ,θ˙,ϕ˙)=mb22(θ˙2+sin2(θ)ϕ˙2)+mbgcos⁡(θ)

Step 1Your equation for Hamiltonian is slightly off. For a Lagrangian L(q,q˙), the Hamiltonian isH(q,q˙)=∑ipiqi˙−L(q,q˙)In your case q1=θ and q2=ϕ. So you need to calculatepθ=∂L(q,q˙)∂θ˙pϕ=∂L(q,q˙)∂ϕ˙Therefore you getH(θ,ϕ,θ˙,ϕ˙)=[θ˙∂∂θ˙+ϕ˙∂∂ϕ˙−1]L(θ,ϕ,θ˙,ϕ˙)

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High School
Calculus and Analysis
Calculus 2
Differential equations

fernandoval3sbr

Answered question

2022-04-07

Finding Hamilton function which equations do I have to solve to find

v (θ, ϕ, \dot{θ}, ϕ)

L (θ, ϕ, \dot{θ}, \dot{ϕ}) = \frac{m b^{2}}{2} ({\dot{θ}}^{2} + \sin^{2} (θ) {\dot{ϕ}}^{2}) + m b g \cos (θ)

Answer & Explanation

posciad48o

Beginner2022-04-08Added 15 answers

Step 1
Your equation for Hamiltonian is slightly off. For a Lagrangian

L (q, \dot{q})

, the Hamiltonian is

H (q, \dot{q}) = \sum_{i} p_{i} \dot{q_{i}} - L (q, \dot{q})

In your case

q_{1} = θ

and

q_{2} = ϕ

. So you need to calculate

p_{θ} = \frac{\partial L (q, \dot{q})}{\partial \dot{θ}}

p_{ϕ} = \frac{\partial L (q, \dot{q})}{\partial \dot{ϕ}}

Therefore you get

H (θ, ϕ, \dot{θ}, \dot{ϕ})

= [\dot{θ} \frac{\partial}{\partial \dot{θ}} + \dot{ϕ} \frac{\partial}{\partial \dot{ϕ}} - 1] L (θ, ϕ, \dot{θ}, \dot{ϕ})

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