A common pitfall in deriving Hamiltonian from Lagrangian mechanics

Lagrangian mechanics

Lagrange’s equations, and the action functional

Legendre transformation

The mechanism of Legendre transformation
Definition of “H”, note that the first and the second equation are the same but with a different perspective

The “pitfall”

Definition of “Hamiltonian” by Legendre transformation, this equality is built upon an invertible relation between “ω” and “p”
Partial derivatives of Hamiltonian along with its variables, be careful that the second equation is erroneous

The fix

Lagrange equation and partially differentiated Hamiltonian along q_i with all conditions listed
Expanding the last term of the partially differentiated Hamiltonian along q_i and replaced red-colored term by Lagrange equation
Deriving Hamiltonian mechanics from Lagrange mechanics through Legendre transformation

Total derivative, an alternative approach

Further insight

References

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Autonomous System Researcher | M.Sc. | Robotics and Mechatronics enthusiasm| www.linkedin.com/in/sheng-han-hsieh-6197bb150

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Hsieh, Sheng-Han

Hsieh, Sheng-Han

Autonomous System Researcher | M.Sc. | Robotics and Mechatronics enthusiasm| www.linkedin.com/in/sheng-han-hsieh-6197bb150

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