1. First of all, Lagrangian is a mathematical quantity which has no physical meaning but Hamiltonian is physical (for example, it is total energy of the system, in some case) and all quantities in Hamiltonian mechanics has physical meanings which makes easier to have physical intuition.

2. In Hamiltonian mechanics you have canonical transformations which allows you change coordinates and find an easier canonical coordinates and momenta in which it is easier to solve problem.

3. The best of all is, Lagrangian is a powerful mathematical method to solve problems in classical mechanic but Hamiltonian is a powerful method to solve problems in classical mechanics, quantum mechanics, statistical mechanics, thermodynamics... etc actually almost all physics...

For example: In thermodynamics: Gibbs free energy, Helmhotz free energy... are all canonical transformations of Hemiltonian..

1. First of all, Lagrangian is a mathematical quantity which has no physical meaning but Hamiltonian is physical (for example, it is total energy of the system, in some case) and all quantities in Hamiltonian mechanics has physical meanings which makes easier to have physical intuition.

2. In Hamiltonian mechanics you have canonical transformations which allows you change coordinates and find an easier canonical coordinates and momenta in which it is easier to solve problem.

3. The best of all is, Lagrangian is a powerful mathematical method to solve problems in classical mechanic but Hamiltonian is a powerful method to solve problems in classical mechanics, quantum mechanics, statistical mechanics, thermodynamics... etc actually almost all physics...

For example: In thermodynamics: Gibbs free energy, Helmholtz free energy... are all canonical transformations of Hamiltonian..

1. First of all, Lagrangian is a mathematical quantity which has no physical meaning but Hamiltonian is physical (It is total energy of the system) and all quantities in Hamiltonian mechanics has physical meanings which makes easier to have physical intuition.

2. In Hamiltonian mechanics you have canonical transformations which allows you change coordinates and find an easier canonical coordinates and momenta in which it is easier to solve problem.

3. The best of all is, Lagrangian is a powerful mathematical method to solve problems in classical mechanic but Hamiltonian is a powerful method to solve problems in classical mechanics, quantum mechanics, statistical mechanics, thermodynamics... etc actually almost all physics...

For example: In thermodynamics: Gibbs free energy, Helmhotz free energy... are all canonical transformations of Hemiltonian..

1. First of all, Lagrangian is a mathematical quantity which has no physical meaning but Hamiltonian is physical (for example, it is total energy of the system, in some case) and all quantities in Hamiltonian mechanics has physical meanings which makes easier to have physical intuition.

2. In Hamiltonian mechanics you have canonical transformations which allows you change coordinates and find an easier canonical coordinates and momenta in which it is easier to solve problem.

3. The best of all is, Lagrangian is a powerful mathematical method to solve problems in classical mechanic but Hamiltonian is a powerful method to solve problems in classical mechanics, quantum mechanics, statistical mechanics, thermodynamics... etc actually almost all physics...

For example: In thermodynamics: Gibbs free energy, Helmhotz free energy... are all canonical transformations of Hemiltonian..