Question
8-7. [S] Consider a particle of mass m moving in a central potential V(r) in non-relativistic Newtonian mechanics. Write down the Lagrangian for this system in polar coordinates. Show that invariance under rotations about the z-axis implies the conservation of the Z -component of the angular momentum.
Solution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references.
Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.

This is only a preview of the solution.
Please use the purchase button to see the entire solution.
By purchasing this solution you'll be able to access the following files:
Solution.pdf
Purchase Solution
$13.00
Google Pay
Amazon
Paypal
Mastercard
Visacard
Discover
Amex
View Available Physics Tutors 529 tutors matched
ionut
Ionut
(ionut)
Hi! MSc Applied Informatics & Computer Science Engineer. Practical experience in many CS & IT branches.Research work & homework
5/5 (5,654+ sessions)
2 hours avg response
Leo
Leo
(Leo)
Hi! I have been a professor in New York and taught in a math department and in an applied math department.
4.9/5 (5,652+ sessions)
2 hours avg response
Pranay
(math1983)
Ph.D. in mathematics and working as an Assistant Professor in University. I can provide help in mathematics, statistics and allied areas.
4.6/5 (5,512+ sessions)
1 hour avg response

Similar Homework Solutions

8.1.0PHP Version806msRequest Duration45MBMemory UsageGET college-homework-library/{category}/{subject}/{id}Route
    • Booting (483ms)time
    • Application (323ms)time
    • 1 x Booting (59.96%)
      483ms
      1 x Application (40.04%)
      323ms
      • Illuminate\Routing\Events\Routing (1.77ms)
      • Illuminate\Routing\Events\RouteMatched (15.46ms)
      • Illuminate\Foundation\Events\LocaleUpdated (6.37ms)
      • eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (258μs)
      • eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (212μs)
      • Illuminate\Database\Events\ConnectionEstablished (1.42ms)
      • Illuminate\Database\Events\StatementPrepared (29.13ms)
      • Illuminate\Database\Events\QueryExecuted (1.37ms)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (166μs)
      • eloquent.booting: App\Models\Subject (144μs)
      • eloquent.booted: App\Models\Subject (61μs)
      • Illuminate\Database\Events\StatementPrepared (10.77ms)
      • Illuminate\Database\Events\QueryExecuted (1.84ms)
      • eloquent.retrieved: App\Models\Subject (160μs)
      • eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (185μs)
      • eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (60μs)
      • Illuminate\Database\Events\StatementPrepared (1.39ms)
      • Illuminate\Database\Events\QueryExecuted (956μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (158μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (23μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (150μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (16μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (11μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (10μs)
      • eloquent.booting: App\Models\SubjectCat (403μs)
      • eloquent.booted: App\Models\SubjectCat (83μs)
      • Illuminate\Database\Events\StatementPrepared (2.72ms)
      • Illuminate\Database\Events\QueryExecuted (1.17ms)
      • eloquent.retrieved: App\Models\SubjectCat (135μs)
      • Illuminate\Cache\Events\CacheHit (25.26ms)
      • Illuminate\Cache\Events\CacheMissed (324μs)
      • Illuminate\Database\Events\StatementPrepared (1.08ms)
      • Illuminate\Database\Events\QueryExecuted (4.26ms)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (94μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (22μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (14μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (11μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (11μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (10μs)
      • Illuminate\Database\Events\StatementPrepared (1.93ms)
      • Illuminate\Database\Events\QueryExecuted (974μs)
      • eloquent.retrieved: App\Models\Subject (102μs)
      • Illuminate\Cache\Events\KeyWritten (1.04ms)
      • Illuminate\Database\Events\StatementPrepared (3.51ms)
      • Illuminate\Database\Events\QueryExecuted (1.04ms)
      • Illuminate\Database\Events\StatementPrepared (1.11ms)
      • Illuminate\Database\Events\QueryExecuted (1.26ms)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (55μs)
      • Illuminate\Database\Events\StatementPrepared (927μs)
      • Illuminate\Database\Events\QueryExecuted (4.57ms)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (73μs)
      • Illuminate\Cache\Events\CacheHit (772μs)
      • creating: homework.show (316μs)
      • composing: homework.show (123μs)
      • creating: components.breadcrumbs (482μs)
      • composing: components.breadcrumbs (145μs)
      • Illuminate\Database\Events\StatementPrepared (10.51ms)
      • Illuminate\Database\Events\QueryExecuted (1.32ms)
      • eloquent.retrieved: App\Models\SubjectCat (90μs)
      • Illuminate\Cache\Events\CacheHit (3.99ms)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • Illuminate\Cache\Events\CacheHit (113μs)
      • Illuminate\Cache\Events\CacheHit (97μs)
      • Illuminate\Cache\Events\CacheMissed (253μs)
      • Illuminate\Database\Events\StatementPrepared (1.85ms)
      • Illuminate\Database\Events\QueryExecuted (2.65ms)
      • eloquent.retrieved: App\Models\SubjectCat (167μs)
      • Illuminate\Cache\Events\KeyWritten (458μs)
      • Illuminate\Cache\Events\CacheHit (254μs)
      • Illuminate\Cache\Events\CacheHit (238μs)
      • Illuminate\Cache\Events\CacheHit (5.93ms)
      • Illuminate\Cache\Events\CacheHit (422μs)
      • Illuminate\Cache\Events\CacheHit (243μs)
      • Illuminate\Cache\Events\CacheHit (4.73ms)
      • Illuminate\Cache\Events\CacheHit (622μs)
      • Illuminate\Cache\Events\CacheMissed (1.29ms)
      • Illuminate\Database\Events\StatementPrepared (4.4ms)
      • Illuminate\Database\Events\QueryExecuted (1.14ms)
      • eloquent.retrieved: App\Models\SubjectCat (114μs)
      • Illuminate\Cache\Events\KeyWritten (3.77ms)
      • Illuminate\Cache\Events\CacheHit (5.83ms)
      • Illuminate\Cache\Events\CacheHit (1.19ms)
      • Illuminate\Cache\Events\CacheHit (549μs)
      • Illuminate\Cache\Events\CacheHit (263μs)
      • Illuminate\Cache\Events\CacheHit (199μs)
      • Illuminate\Cache\Events\CacheHit (1.59ms)
      • Illuminate\Cache\Events\CacheHit (226μs)
      • Illuminate\Cache\Events\CacheHit (316μs)
      • Illuminate\Cache\Events\CacheHit (196μs)
      • Illuminate\Cache\Events\CacheHit (207μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (203μs)
      • Illuminate\Cache\Events\CacheHit (343μs)
      • Illuminate\Cache\Events\CacheHit (281μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (2.06ms)
      • Illuminate\Cache\Events\CacheHit (289μs)
      • Illuminate\Cache\Events\CacheHit (220μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (301μs)
      • Illuminate\Cache\Events\CacheHit (239μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (362μs)
      • Illuminate\Cache\Events\CacheHit (352μs)
      • Illuminate\Cache\Events\CacheHit (420μs)
      • Illuminate\Cache\Events\CacheHit (311μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (128μs)
      • Illuminate\Cache\Events\CacheHit (247μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheMissed (206μs)
      • Illuminate\Database\Events\StatementPrepared (2.41ms)
      • Illuminate\Database\Events\QueryExecuted (1.09ms)
      • eloquent.retrieved: App\Models\SubjectCat (92μs)
      • Illuminate\Cache\Events\KeyWritten (467μs)
      • Illuminate\Cache\Events\CacheHit (307μs)
      • Illuminate\Cache\Events\CacheHit (220μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (226μs)
      • Illuminate\Cache\Events\CacheHit (267μs)
      • Illuminate\Cache\Events\CacheHit (334μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (3.78ms)
      • Illuminate\Cache\Events\CacheHit (209μs)
      • Illuminate\Cache\Events\CacheHit (1.88ms)
      • Illuminate\Cache\Events\CacheHit (273μs)
      • Illuminate\Cache\Events\CacheHit (358μs)
      • Illuminate\Cache\Events\CacheHit (316μs)
      • Illuminate\Cache\Events\CacheHit (356μs)
      • Illuminate\Cache\Events\CacheHit (770μs)
      • Illuminate\Cache\Events\CacheHit (305μs)
      • Illuminate\Cache\Events\CacheHit (430μs)
      • Illuminate\Cache\Events\CacheHit (225μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (199μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (355μs)
      • Illuminate\Cache\Events\CacheHit (371μs)
      • Illuminate\Cache\Events\CacheHit (358μs)
      • Illuminate\Cache\Events\CacheHit (336μs)
      • Illuminate\Cache\Events\CacheHit (509μs)
      • Illuminate\Cache\Events\CacheHit (1.01ms)
      • Illuminate\Cache\Events\CacheHit (3.85ms)
      • Illuminate\Cache\Events\CacheHit (444μs)
      • Illuminate\Cache\Events\CacheHit (227μs)
      • Illuminate\Cache\Events\CacheHit (9.08ms)
      • Illuminate\Cache\Events\CacheHit (327μs)
      • Illuminate\Cache\Events\CacheHit (181μs)
      • Illuminate\Cache\Events\CacheHit (341μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (198μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (209μs)
      • Illuminate\Cache\Events\CacheHit (190μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (154μs)
      • Illuminate\Cache\Events\CacheHit (218μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (263μs)
      • Illuminate\Cache\Events\CacheHit (361μs)
      • Illuminate\Cache\Events\CacheHit (818μs)
      • Illuminate\Cache\Events\CacheHit (209μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (240μs)
      • Illuminate\Cache\Events\CacheHit (220μs)
      • Illuminate\Cache\Events\CacheHit (326μs)
      • Illuminate\Cache\Events\CacheHit (971μs)
      • Illuminate\Cache\Events\CacheHit (280μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (151μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (148μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (310μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (703μs)
      • Illuminate\Cache\Events\CacheHit (1.27ms)
      • Illuminate\Cache\Events\CacheHit (208μs)
      • Illuminate\Cache\Events\CacheHit (169μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (181μs)
      • Illuminate\Cache\Events\CacheHit (153μs)
      • Illuminate\Cache\Events\CacheHit (190μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (272μs)
      • Illuminate\Cache\Events\CacheHit (224μs)
      • Illuminate\Cache\Events\CacheHit (266μs)
      • Illuminate\Cache\Events\CacheHit (218μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (239μs)
      • Illuminate\Cache\Events\CacheHit (246μs)
      • Illuminate\Cache\Events\CacheHit (329μs)
      • Illuminate\Cache\Events\CacheHit (6.49ms)
      • Illuminate\Cache\Events\CacheHit (285μs)
      • Illuminate\Cache\Events\CacheHit (269μs)
      • Illuminate\Cache\Events\CacheHit (172μs)
      • Illuminate\Cache\Events\CacheHit (196μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (262μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (195μs)
      • Illuminate\Cache\Events\CacheHit (297μs)
      • Illuminate\Cache\Events\CacheHit (480μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (219μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (171μs)
      • Illuminate\Cache\Events\CacheHit (207μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (220μs)
      • Illuminate\Cache\Events\CacheHit (1.53ms)
      • Illuminate\Cache\Events\CacheHit (505μs)
      • Illuminate\Cache\Events\CacheHit (638μs)
      • Illuminate\Cache\Events\CacheHit (222μs)
      • Illuminate\Cache\Events\CacheHit (372μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (227μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (207μs)
      • Illuminate\Cache\Events\CacheHit (221μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (221μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (215μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (222μs)
      • Illuminate\Cache\Events\CacheHit (176μs)
      • Illuminate\Cache\Events\CacheHit (196μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (206μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (228μs)
      • Illuminate\Cache\Events\CacheHit (201μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (252μs)
      • Illuminate\Cache\Events\CacheHit (224μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (1.02ms)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (426μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (181μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (181μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (176μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (155μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (320μs)
      • Illuminate\Cache\Events\CacheHit (199μs)
      • Illuminate\Cache\Events\CacheHit (291μs)
      • Illuminate\Cache\Events\CacheHit (464μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (251μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (234μs)
      • Illuminate\Cache\Events\CacheHit (297μs)
      • Illuminate\Cache\Events\CacheHit (297μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (271μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (283μs)
      • Illuminate\Cache\Events\CacheHit (223μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (171μs)
      • Illuminate\Cache\Events\CacheHit (203μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (227μs)
      • Illuminate\Cache\Events\CacheHit (231μs)
      • Illuminate\Cache\Events\CacheHit (254μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (150μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (154μs)
      • Illuminate\Cache\Events\CacheHit (193μs)
      • Illuminate\Cache\Events\CacheHit (330μs)
      • Illuminate\Cache\Events\CacheHit (203μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheMissed (2.93ms)
      • Illuminate\Database\Events\StatementPrepared (7.08ms)
      • Illuminate\Database\Events\QueryExecuted (1.38ms)
      • eloquent.retrieved: App\Models\SubjectCat (109μs)
      • Illuminate\Cache\Events\KeyWritten (381μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (350μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (289μs)
      • Illuminate\Cache\Events\CacheHit (247μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (190μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • creating: site.layouts.app (503μs)
      • composing: site.layouts.app (25μs)
      • creating: components.canonical (547μs)
      • composing: components.canonical (114μs)
      • creating: components.open-graph (313μs)
      • composing: components.open-graph (96μs)
      • creating: site.headers.header (363μs)
      • composing: site.headers.header (85μs)
      • Illuminate\Cache\Events\CacheHit (7.33ms)
      • creating: components.footer (113μs)
      • composing: components.footer (193μs)
      • Illuminate\Cache\Events\CacheHit (1.22ms)
      • Illuminate\Cache\Events\CacheMissed (470μs)
      • Illuminate\Database\Events\StatementPrepared (7.31ms)
      • Illuminate\Database\Events\QueryExecuted (1.67ms)
      • eloquent.retrieved: App\Models\SubjectCat (84μs)
      • Illuminate\Cache\Events\KeyWritten (382μs)
      • Illuminate\Cache\Events\CacheHit (483μs)
      • Illuminate\Cache\Events\CacheHit (247μs)
      • Illuminate\Cache\Events\CacheHit (215μs)
      • Illuminate\Cache\Events\CacheHit (280μs)
      • Illuminate\Cache\Events\CacheHit (220μs)
      • Illuminate\Cache\Events\CacheHit (438μs)
      • Illuminate\Cache\Events\CacheHit (357μs)
      • Illuminate\Cache\Events\CacheHit (242μs)
      • Illuminate\Cache\Events\CacheHit (321μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (196μs)
      • Illuminate\Cache\Events\CacheHit (284μs)
      • creating: components.forms.contact-us (313μs)
      • composing: components.forms.contact-us (198μs)
      • creating: components.forms.get-started (214μs)
      • composing: components.forms.get-started (91μs)
      • creating: components.forms.free-tool-download (145μs)
      • composing: components.forms.free-tool-download (76μs)
      • creating: components.forms.claim-free-worksheet (126μs)
      • composing: components.forms.claim-free-worksheet (60μs)
      • creating: components.forms.tutor-subscription-waitlist (122μs)
      • composing: components.forms.tutor-subscription-waitlist (67μs)
      • creating: components.forms.tutor-subscription-join (120μs)
      • composing: components.forms.tutor-subscription-join (64μs)
      • creating: components.forms.tutor-support (134μs)
      • composing: components.forms.tutor-support (63μs)
      • 316 x Illuminate\Cache\Events\CacheHit (19.6%)
        158ms
        15 x Illuminate\Database\Events\StatementPrepared (10.69%)
        86.12ms
        15 x Illuminate\Database\Events\QueryExecuted (3.31%)
        26.68ms
        1 x Illuminate\Routing\Events\RouteMatched (1.92%)
        15.46ms
        6 x Illuminate\Cache\Events\KeyWritten (0.81%)
        6.50ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.79%)
        6.37ms
        6 x Illuminate\Cache\Events\CacheMissed (0.68%)
        5.47ms
        1 x Illuminate\Routing\Events\Routing (0.22%)
        1.77ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.18%)
        1.42ms
        7 x eloquent.retrieved: App\Models\SubjectCat (0.1%)
        791μs
        1 x creating: components.canonical (0.07%)
        547μs
        1 x creating: site.layouts.app (0.06%)
        503μs
        8 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.06%)
        496μs
        1 x creating: components.breadcrumbs (0.06%)
        482μs
        1 x eloquent.booting: App\Models\SubjectCat (0.05%)
        403μs
        1 x creating: site.headers.header (0.05%)
        363μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        328μs
        1 x creating: homework.show (0.04%)
        316μs
        1 x creating: components.forms.contact-us (0.04%)
        313μs
        1 x creating: components.open-graph (0.04%)
        313μs
        2 x eloquent.retrieved: App\Models\Subject (0.03%)
        262μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.03%)
        258μs
        1 x creating: components.forms.get-started (0.03%)
        214μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.03%)
        212μs
        1 x composing: components.forms.contact-us (0.02%)
        198μs
        1 x composing: components.footer (0.02%)
        193μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        185μs
        1 x composing: components.breadcrumbs (0.02%)
        145μs
        1 x creating: components.forms.free-tool-download (0.02%)
        145μs
        1 x eloquent.booting: App\Models\Subject (0.02%)
        144μs
        1 x creating: components.forms.tutor-support (0.02%)
        134μs
        1 x creating: components.forms.claim-free-worksheet (0.02%)
        126μs
        1 x composing: homework.show (0.02%)
        123μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.02%)
        122μs
        1 x creating: components.forms.tutor-subscription-join (0.01%)
        120μs
        1 x composing: components.canonical (0.01%)
        114μs
        1 x creating: components.footer (0.01%)
        113μs
        1 x composing: components.open-graph (0.01%)
        96μs
        1 x composing: components.forms.get-started (0.01%)
        91μs
        1 x composing: site.headers.header (0.01%)
        85μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        83μs
        1 x composing: components.forms.free-tool-download (0.01%)
        76μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        67μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        64μs
        1 x composing: components.forms.tutor-support (0.01%)
        63μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        61μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        60μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        60μs
        1 x composing: site.layouts.app (0%)
        25μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      15 statements were executed, 4 of which were duplicates, 11 unique. Show only duplicated89.04ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '62349' limit 1
        29.38mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 62349
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (226)
        4.26mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (62349)
        1.6mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        2.98mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 34
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 62349 and `subject` = 226 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        4.31mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 62349
        • 1: 226
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (226)
        2.23mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 62349 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 62349
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 62349 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        1.52mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 62349
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 62349 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        4.41mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 62349
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        10.5mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 34
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        3.87mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        4.52mstwenty4_siteSubject.php#100
        Bindings
        • 0: 4
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        2.81mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        7.56mstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        8.09mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      8HomeworkLibraryFile.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\SubjectCat
      7SubjectCat.php
      App\Models\Subject
      2Subject.php
          _token
          FW1wHSogCYkQFJIdTLn9LF8Di2nqNYiE2BRnLLkH
          utm_source
          direct
          redirectUrl
          /college-homework-library/Physics/Mathematical-Physics/62349
          _previous
          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Physics/Mathema...
          _flash
          array:2 [ "old" => [] "new" => [] ]
          PHPDEBUGBAR_STACK_DATA
          []
          path_info
          /college-homework-library/Physics/Mathematical-Physics/62349
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
          request_headers
          0 of 0
          array:21 [ "priority" => array:1 [ 0 => "u=0, i" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] "x-amzn-trace-id" => array:1 [ 0 => "Root=1-67ffadd0-400d00222694380a3cbd64d1" ] "host" => array:1 [ 0 => "staging.dev.24houranswers.com" ] "x-forwarded-port" => array:1 [ 0 => "443" ] "x-forwarded-proto" => array:1 [ 0 => "https" ] "x-forwarded-for" => array:1 [ 0 => "18.220.171.141" ] "content-length" => array:1 [ 0 => "" ] "content-type" => array:1 [ 0 => "" ] ]
          request_cookies
          []
          
          response_headers
          0 of 0
          array:5 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "no-cache, private" ] "date" => array:1 [ 0 => "Wed, 16 Apr 2025 13:17:05 GMT" ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6InlnTmpxWktHZkFVeVpEbGRwcUoyZ2c9PSIsInZhbHVlIjoiVVNJbFF4REk5Y29LcEhZQnpDMEtqS3VJOENDdXR1VzQycHlwVTZTOFVuckF4RldYN2dkUXVMNi9oOE9PZktUOXVkQzNFMG90Rks4OXhyMTBuZkEvL2tBeWdpNnF5cEJUdVVzZkFiNUtsL2N4SkIzNndUaXErVHhuaVFEakpHYWgiLCJtYWMiOiJhOWVkZmJmZDA1OWRjYTE3OWE0YjNiZmQ3NTEyNjMwYzZjNDJmNDc4ZTRhY2UzZTY0YTAxMzUzMTdmNWY2Mjg0IiwidGFnIjoiIn0%3D; expires=Wed, 16 Apr 2025 15:17:05 GMT; Max-Age=7200; path=/; domain=.24houranswers.com; samesite=laxXSRF-TOKEN=eyJpdiI6InlnTmpxWktHZkFVeVpEbGRwcUoyZ2c9PSIsInZhbHVlIjoiVVNJbFF4REk5Y29LcEhZQnpDMEtqS3VJOENDdXR1VzQycHlwVTZTOFVuckF4RldYN2dkUXVMNi9oOE9PZktUOXVkQzNFM" 1 => "24houranswers_session=eyJpdiI6ImlpRmM0Z2FJUy9UY0dGUlc4UTRwSVE9PSIsInZhbHVlIjoiR1dnMU5JSEJmOHdFZ0k5NWk2dFBKMDYzSUdBYUJHOGNUV3JFUStwK1AyUitaVFdLQzBBVkdoemVnSjh5VnJjU0dFZ3JTNTV2QVowOHdPZ1dWaklYcEYxNWw0blZrM21RRnRiK0ljQTFXamZSVDNNd2pNbmZNMDdzNVg2Z05Jdm8iLCJtYWMiOiI2MTNhOTk3ZWQ0Yzg4MGEzYzdjMTZhNWYzYzU5MGEzMDUxYmI2NzQ5MThmYTFiNGNkYWVlN2ViYjFhODJmZjc3IiwidGFnIjoiIn0%3D; expires=Wed, 16 Apr 2025 15:17:05 GMT; Max-Age=7200; path=/; domain=.24houranswers.com; httponly; samesite=lax24houranswers_session=eyJpdiI6ImlpRmM0Z2FJUy9UY0dGUlc4UTRwSVE9PSIsInZhbHVlIjoiR1dnMU5JSEJmOHdFZ0k5NWk2dFBKMDYzSUdBYUJHOGNUV3JFUStwK1AyUitaVFdLQzBBVkdoemVnSjh5Vn" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6InlnTmpxWktHZkFVeVpEbGRwcUoyZ2c9PSIsInZhbHVlIjoiVVNJbFF4REk5Y29LcEhZQnpDMEtqS3VJOENDdXR1VzQycHlwVTZTOFVuckF4RldYN2dkUXVMNi9oOE9PZktUOXVkQzNFMG90Rks4OXhyMTBuZkEvL2tBeWdpNnF5cEJUdVVzZkFiNUtsL2N4SkIzNndUaXErVHhuaVFEakpHYWgiLCJtYWMiOiJhOWVkZmJmZDA1OWRjYTE3OWE0YjNiZmQ3NTEyNjMwYzZjNDJmNDc4ZTRhY2UzZTY0YTAxMzUzMTdmNWY2Mjg0IiwidGFnIjoiIn0%3D; expires=Wed, 16-Apr-2025 15:17:05 GMT; domain=.24houranswers.com; path=/XSRF-TOKEN=eyJpdiI6InlnTmpxWktHZkFVeVpEbGRwcUoyZ2c9PSIsInZhbHVlIjoiVVNJbFF4REk5Y29LcEhZQnpDMEtqS3VJOENDdXR1VzQycHlwVTZTOFVuckF4RldYN2dkUXVMNi9oOE9PZktUOXVkQzNFM" 1 => "24houranswers_session=eyJpdiI6ImlpRmM0Z2FJUy9UY0dGUlc4UTRwSVE9PSIsInZhbHVlIjoiR1dnMU5JSEJmOHdFZ0k5NWk2dFBKMDYzSUdBYUJHOGNUV3JFUStwK1AyUitaVFdLQzBBVkdoemVnSjh5VnJjU0dFZ3JTNTV2QVowOHdPZ1dWaklYcEYxNWw0blZrM21RRnRiK0ljQTFXamZSVDNNd2pNbmZNMDdzNVg2Z05Jdm8iLCJtYWMiOiI2MTNhOTk3ZWQ0Yzg4MGEzYzdjMTZhNWYzYzU5MGEzMDUxYmI2NzQ5MThmYTFiNGNkYWVlN2ViYjFhODJmZjc3IiwidGFnIjoiIn0%3D; expires=Wed, 16-Apr-2025 15:17:05 GMT; domain=.24houranswers.com; path=/; httponly24houranswers_session=eyJpdiI6ImlpRmM0Z2FJUy9UY0dGUlc4UTRwSVE9PSIsInZhbHVlIjoiR1dnMU5JSEJmOHdFZ0k5NWk2dFBKMDYzSUdBYUJHOGNUV3JFUStwK1AyUitaVFdLQzBBVkdoemVnSjh5Vn" ] ]
          session_attributes
          0 of 0
          array:6 [ "_token" => "FW1wHSogCYkQFJIdTLn9LF8Di2nqNYiE2BRnLLkH" "utm_source" => "direct" "redirectUrl" => "/college-homework-library/Physics/Mathematical-Physics/62349" "_previous" => array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Physics/Mathematical-Physics/62349" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]